Designing LC Ladder Filters with Lossy Elements
by
Timothy A Brown
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
May 5th, 2013
Keywords: ladder, filters, lossy, synthesis
Copyright 2013 by Timothy A Brown
Approved by
Bogdan M. Wilamowski, Chair, Professor of Electrical and Computer Engineering
Michael Baginski, Professor of Electrical and Computer Engineering
Thaddeus Roppel, Associate Professor of Electrical and Computer Engineering
ii
Abstract
Many publications and many books have been written on LC Ladder filters, however, most of
the material in circulation only deals with ideal (lossless) LC ladder filters. It is assumed that in the
lossless design, all of the reactance values are ideal ones. However, since reactance elements are
actually lossy, the purpose of this thesis is to show how to design LC ladder filters with lossy
components, and achieve optimal responses. Three different methods are presented to correct the lossy
ladder filter. The first method uses the Sum Squared Error (SSE) to minimize the difference between the
poles of the ideal filer and the lossy filter. The second method uses the SSE to minimize the difference
between the coefficients of the ideal filer?s transfer function and the lossy filter?s transfer function. The
final method uses the SSE to minimize the difference between magnitude plot of the ideal filter and
lossy filter. The first and second methods both use the Nelder-Mead algorithm for the optimization
scheme, and the third method uses the Levenberg?Marquardt algorithm. The first two methods work
well but only for low-pass filters. In the case of more complex designs, the number of roots and the
filters order in the ideal filter and lossy filter do not match, so this method cannot be used. In
conclusion, the third method where the frequency magnitude plots of the ideal and lossy filter are being
matched, provide the best solution. As a result, this method allows one to design a ladder filter with
lossy components, while achieving ideal-like results.
iii
Acknowledgments
I would like to thank Dr. Bogdan Wilamowski, for his support and encouragement during my
time here at Auburn, and for letting me work under him as a graduate student. I would also like to thank
my committee members, Dr. Michael Baginski, and Dr.Thaddeus Roppel for being on my committee. I
also would like to thank all of my colleagues in Room 409 in Broun Hall, as well as Michael Pukish and
Xing Wu for helping me on certain projects in MATLAB. Last but not least, I would like to thank my
parents who installed in me the value of education, without their guidance, I would not be where I am
today.
iv
Table of Contents
Abstract ......................................................................................................................................................... ii
Acknowledgments ........................................................................................................................................ iii
Table of Figures ............................................................................................................................................ vi
List of Tables .................................................................................................................................................ix
1 Introduction .......................................................................................................................................... 1
1.1 Thesis Outline ................................................................................................................................ 1
2 Low-pass Filters ..................................................................................................................................... 2
2.1 Low Pass Filter Response .............................................................................................................. 2
2.2 Simple low-pass filter circuit ......................................................................................................... 4
2.3 2nd Order Low-pass Filter .............................................................................................................. 5
3 Synthesis of lossless LC Ladder Filters .................................................................................................. 7
3.1 Background ................................................................................................................................... 7
3.2 Properties of lossless ladders ........................................................................................................ 7
3.3 A synthesis strategy ...................................................................................................................... 9
3.3.1 Ladder Synthesis Example ................................................................................................... 11
3.4 Calculate the Transfer Function of Ladder Circuit ...................................................................... 12
3.5 Frequency Transformation from low-pass to other types of filters ........................................... 16
4 Synthesis of lossy LC Ladder Filters ..................................................................................................... 18
4.1 Introduction ................................................................................................................................ 18
4.2 Background ................................................................................................................................. 18
4.3 Procedure of designing Lossy Ladder Filters ............................................................................... 20
4.3.1 Calculating the transfer function of a lossy ladder filter .................................................... 21
4.3.2 Calculating the Error Function ............................................................................................ 26
4.3.3 Optimization Routine - Nelder Mead Method .................................................................... 27
5 Nelder-Mead Simulation Results ? Error Calculated From Roots ....................................................... 31
5.1 3rd Order Chebyshev Low-pass Filter .......................................................................................... 31
5.2 5th Order Butterworth Low-pass filter ........................................................................................ 34
v
5.3 8th Order Chebyshev Band-pass Filter ......................................................................................... 37
6 Nelder-Mead Simulation Results ? Error Calculated From Coefficients ............................................. 41
6.1 3rd Order Chebyshev Low-pass Filter .......................................................................................... 41
6.2 5th Order Butterworth Low-pass Filter ........................................................................................ 45
6.3 8th Order Chebyshev Band-pass Filter ......................................................................................... 49
6.4 MATLAB Algorithm Limitations ................................................................................................... 51
7 Levenberg ? Marquardt (LM) Simulation Results ............................................................................... 53
7.1 3rd Order Chebyshev Low-pass Filter .......................................................................................... 57
7.2 5th Order Butterworth Low-Pass Filter ........................................................................................ 59
7.3 4th Order Chebyshev Low-Pass Filter .......................................................................................... 61
7.4 8th Order Chebyshev Band-pass Filter ......................................................................................... 63
7.5 8th Order Butterworth Band-pass Filter ...................................................................................... 65
8 Conclusion ........................................................................................................................................... 67
9 References .......................................................................................................................................... 69
10 Appendices .......................................................................................................................................... 71
A Matlab input function: main.m ................................................................................................... 71
B Matlab Impedance Function: impedence.m ............................................................................... 72
C Matlab Error and Plotting Function: error.m .............................................................................. 73
D Matlab Transfer Function: Ttra_poly.m ...................................................................................... 76
E 2nd Order low-pass filter symbolic transfer function .................................................................. 76
F 3rd Order low-pass filter symbolic transfer function .................................................................. 76
G 4th Order low-pass filter symbolic transfer function ................................................................... 77
H 5th Order low-pass filter symbolic transfer function ................................................................... 77
I 6th Order low-pass filter symbolic transfer function ................................................................... 77
J 7th Order low-pass filter symbolic transfer function ................................................................... 78
K 8th Order low-pass filter symbolic transfer function ................................................................... 78
L 9th Order low-pass filter symbolic transfer function ................................................................... 79
vi
Table of Figures
Figure 2-1: Brickwall Low-Pass Filter Amplitude Response .......................................................................... 2
Figure 2-2: More realistic low-pass filter ...................................................................................................... 3
Figure 2-3: Simple Low-pass Filter ................................................................................................................ 4
Figure 2-4: Frequency Response of simple low-pass filter ........................................................................... 5
Figure 2-5: 2nd Order low-pass filter ............................................................................................................. 5
Figure 2-6: Voltage Divider Circuit ................................................................................................................ 6
Figure 3-1 Doubly Terminated lossless ladder: ............................................................................................. 8
Figure 3-2: Ladder Synthesis Example ........................................................................................................ 12
Figure 3-3: 5th order Ideal Chebyshev low-pass filter ................................................................................. 13
Figure 3-4: 5th order ladder with series impedances and shunt admittances ............................................ 13
Figure 3-5: 5th order Chebyshev low-pass filter .......................................................................................... 16
Figure 3-6: Filter transformation from low-pass to other types of filters .................................................. 17
Figure 4-1: Basic Structure of Ladder Filter ................................................................................................ 18
Figure 4-2: Models of Real Capacitors and Real Inductors ......................................................................... 19
Figure 4-3: Movement of Poles due to lossy elements .............................................................................. 20
Figure 4-4: 3rd order lossy ladder filter ....................................................................................................... 21
Figure 4-5: Stages of 3rd order low-pass lossy filter .................................................................................... 23
Figure 4-6: Pole locations for 3rd order ideal and lossy low-pass filter ....................................................... 26
Figure 4-7: The Triangle ABC, the midpoint M and the Reflected point R ................................................. 29
Figure 4-8: The Triangle BGW and point R with extended point E ............................................................. 29
vii
Figure 4-9: The Contraction point C1 or C2 .................................................................................................. 30
Figure 4-10: Shrinkage toward the Triangle B ............................................................................................ 30
Figure 5-1: 3rd order ideal Chebyshev low-pass filter ................................................................................. 32
Figure 5-2: 3rd order lossy Chebyshev low-pass filter ................................................................................. 32
Figure 5-3: 3rd order Chebyshev Pole plot and Frequency Response ? Before Optimization ................... 33
Figure 5-4: 3rd order Chebyshev Pole plot and Frequency Response ? After Optimization ....................... 33
Figure 5-5: 5th order ideal Butterworth low-pass filter ............................................................................... 35
Figure 5-6: 5th order lossy Butterworth low-pass filter ............................................................................... 35
Figure 5-7: 5th order Butterworth Pole plot and Frequency Response ? Before Optimization .................. 36
Figure 5-8: 5th order Butterworth Pole plot and Frequency Response ? After Optimization ..................... 36
Figure 5-9: 8th order Chebyshev band-pass filter ........................................................................................ 38
Figure 5-10: 8th order Lossy Chebyshev band-pass filter ........................................................................... 38
Figure 5-11: 8th order Band-pass Pole plot and Frequency Response ? Before Optimization ................... 40
Figure 5-12: 8th order Band-pass Pole plot and Frequency Response ? After Optimization ...................... 40
Figure 6-1: 3rd order ideal Chebyshev low-pass filter ................................................................................. 41
Figure 6-2: 3rd order lossy Chebyshev low-pass filter ................................................................................. 41
Figure 6-3: 3rd order Chebyshev Coefficient Value Vs Order Plot ? Before Optimization .......................... 43
Figure 6-4: 3rd order Chebyshev Coefficient Value Vs Order Plot ? After Optimization ............................. 43
Figure 6-5: 3rd order Chebyshev filter - Before Optimization ..................................................................... 44
Figure 6-6: 3rd order Chebyshev filter- After Optimization ......................................................................... 44
Figure 6-7: 5th order ideal Butterworth low-pass filter ............................................................................... 45
Figure 6-8: 3rd order lossy Chebyshev low-pass filter ................................................................................. 45
Figure 6-9: 5th order Butterworth Coefficient Value Vs Order Plot ? Before Optimization ........................ 47
Figure 6-10: 5th order Butterworth Coefficient Value Vs Order Plot ? After Optimization ........................ 47
Figure 6-11: 5th order Butterworth filter- Before Optimization .................................................................. 48
Figure 6-12: 5th order Butterworth filter- After Optimization .................................................................... 48
viii
Figure 6-13: 8th order Chebyshev Band-pass filter ...................................................................................... 49
Figure 6-14: 8th order lossy Chebyshev Band-pass filter ............................................................................. 49
Figure 6-15: 8th order Band-pass Coefficient Value Vs Order Plot ? Before Optimization ......................... 50
Figure 6-16: 8th order Band-pass Frequency Response ? Before Optimization .......................................... 50
Figure 6-17: Example of Ideal Bandpass Filter Impedance ......................................................................... 51
Figure 6-18: Example of Lossy Bandpass Filter Impedance ........................................................................ 52
Figure 7-1: Block Diagram and Pseudocode for LM Algorithm ................................................................... 56
Figure 7-2: 3rd Order Chebyshev Low-pass Filter ........................................................................................ 57
Figure 7-3: 3rd Order Chebyshev Low-Pass Filter Frequency Plot ? Before Optimization ......................... 58
Figure 7-4: 3rd Order Chebyshev Low-Pass Filter Frequency Plot ? After Optimization ............................ 58
Figure 7-5: 5th Order Butterworth Low-pass Filter ..................................................................................... 59
Figure 7-6: 5th Order Butterworth Low-Pass Filter Frequency Plot ? Before Optimization ........................ 60
Figure 7-7: 5th Order Butterworth Low-Pass Filter Frequency Plot ? After Optimization .......................... 60
Figure 7-8: 4th Order Chebyshev Low-Pass Filter ........................................................................................ 61
Figure 7-9: 4th Order Chebyshev Low-Pass Filter Frequency Plot ? Before Optimization .......................... 62
Figure 7-10: 4th Order Chebyshev Low-Pass Filter Frequency Plot ? After Optimization ........................... 62
Figure 7-11: 8th Order Chebyshev Band-pass Filter .................................................................................... 63
Figure 7-12: 8th Order Chebyshev Band-pass Filter Frequency Plot ? Before Optimization ....................... 64
Figure 7-13: 8th Order Chebyshev Band-pass Filter Frequency Plot ? After Optimization ......................... 64
Figure 7-14: 8th Order Butterworth Band-pass Filter .................................................................................. 65
Figure 7-15: 8th Order Butterworth Band-pass Filter Frequency Plot ? Before Optimization .................... 66
Figure 7-16: 8th Order Butterworth Band-pass Filter Frequency Plot ? After Optimization ....................... 66
ix
List of Tables
Table 2-1 Classic Analog Filters ..................................................................................................................... 4
Table 3-1: Polynomials D(s) for Butterworth Filter ..................................................................................... 10
Table 3-2: Ladder Synthesis Division and Element Extraction .................................................................... 12
Table 3-3: Element Values for series and shunt ladder .............................................................................. 13
Table 3-4: 5th order transfer function ......................................................................................................... 14
Table 3-5: Coefficients of 5th Chebyshev Transfer Function ....................................................................... 14
Table 3-6: Element Values for 5th Order Low-pass Filter ........................................................................... 15
Table 4-1: 3rd Order Lossy Element Values ................................................................................................. 21
Table 4-2: 3rd Order lossy transfer function ................................................................................................ 22
Table 4-3: 3rd Order Lossy Transfer Function - Mathematical .................................................................... 22
Table 4-4: 3rd Order ideal low-pass filter with symbolic coefficients ? Calculated in MATLAB .................. 24
Table 4-5: 3rd Order lossy low-pass filter with Symbolic coefficients ? Calculated in MATLAB .................. 25
Table 4-6: Ideal Roots and Lossy roots for 3rd order low-pass filter ........................................................... 27
Table 5-1: 3rd Order Chebyshev low-pass filter specifications .................................................................... 31
Table 5-2: 3rd Order Chebyshev low-pass filter element values ................................................................. 32
Table 5-3: Optimized values for 3rd order Chebyshev................................................................................. 33
Table 5-4: 5th Order Butterworth low-pass filter specifications ................................................................. 34
Table 5-5: 5th Order Butterworth low-pass filter element values .............................................................. 35
Table 5-6: Optimized values for 5th order Butterworth .............................................................................. 36
Table 5-7: 8th Order Chebyshev Band-pass low-pass filter specifications .................................................. 37
x
Table 5-8: 8th Order Chebyshev Band-pass low-pass filter element values with Q=10% ........................... 38
Table 5-9: Optimized values for 8th order Band-pass filter ......................................................................... 39
Table 6-1: 3rd Order Chebyshev Filter with Ideal and Optimized Values .................................................... 42
Table 6-2: 5th Order Butterworth Filter with Ideal and Optimized Values ................................................. 46
Table 6-3: Coefficients for the Transfer Function for 4th order Ideal Band-pass Filter with Q = 10% ........ 53
Table 7-1 LM Algorithm Basic Steps ............................................................................................................ 57
Table 7-2: 3rd Order Chebyshev Low-Pass Filter with Ideal, Lossy and Optimized Values ......................... 57
Table 7-3: 5th Order Butterworth Low-Pass Filter with Ideal, Lossy and Optimized Values ....................... 59
Table 7-4: 4th Order Chebyshev Low-Pass Filter with Ideal, Lossy and Optimized Values ........................ 61
Table 7-5: 8th Order Chebyshev Band-pass Filter with Ideal, Lossy and Optimized Values ....................... 63
Table 7-6: 8th Order Butterworth Band-pass Filter with Ideal, Lossy and Optimized Values .................... 66
1 Introduction
LC (or RLC) ladder filters play a role in many electrical systems. They are used in many types of
oscillator circuits and they are also very important when it comes to tuning, such as radio receivers and
television sets, where they are used to select a narrow range of frequencies from a broad spectrum. An LC
ladder filter can be used as a band-pass filter, low pass-filter, high-pass filter, or a band-stop filter. The
frequency response of each one of these types of filters can greatly impact the significance/performance of
the whole system. This is a bad thing and can definitely decrease performance on the whole system. Since
communication systems demand high-performance filters, filters need to have characteristics as close to
ideal as possible.
Since classical filter designs are done with ideal (lossless) components, ideal-like responses are not
possible due to the fact that real inductors and capacitors have losses (reactance). The greater the losses of
the reactance elements used in the filter, the greater the difference between the frequency response of the
filter, and the frequency response that the filter was intended to have. The lack of ideal reactance elements
makes all of the classical design methods inefficient for practical applications. The purpose of this thesis is to
show how to design LC ladder filters, to achieve ideal-like responses, with lossy components. Three different
ways are presented in this thesis to help minimize the error between the ideal filter and lossy filter, as well
as two different optimization schemes to aid in the calculation of the new lossy component values. Using
the algorithms proposed in this thesis allows one to design a ladder filter with lossy components, while
achieving ideal-like results.
1.1 Thesis Outline
A basic overview of low-pass filters is given in Chapter 2. It includes a 2nd order filter and explains
how this basic filter is the basis for all ladder filters. Chapter 3 explains the background of the LC ladder filter
as well as the properties of lossless ladder filters. This chapter also presents a unique synthesis strategy to
design ladder filters, as well as how to calculate the transfer function for an ideal filter, and how to
transform low-pass filters into other types of filters. Chapter 4 explains how to design a lossy ladder filter, as
well as how to determine the lossy transfer function mathematically, and in MATLAB. How to calculate the
2
error function and how the Nelder-Mead optimization function works is also presented in Chapter 3.
Chapter 4 shows the simulation results with the error calculated from the roots of the filter, and Chapter 5
shows the simulation results with the error calculated from the coefficients of the lossy and ideal transfer
functions. This chapter also explains the MATLAB algorithm limitations for more complex filters. The
Levenberg-Marquardt algorithm and simulation results are given in Chapter 6. Finally, the conclusion is
given in Chapter 7, References are in Chapter 8, and the Appendices follow.
2 Low-pass Filters
Low-pass filters have been around for a long time and are perhaps the most well-known filter when
it comes to LC passive ladder filters. However, what most people do not know is that the low pass filter
design approach is the basic prototype design when it comes to other types of filter responses. Some of
these responses include the high-pass filter, the band-pass filter, and the band-stop filter. Since the low-pass
filter is the basic building block for all other filter designs, I will discuss how the LC low-pass filter operates.
2.1 Low Pass Filter Response
The brickwall low-pass frequency response is shown in Figure 2-1. The vertical axis is attenuation,
and increases downwards. As long as the frequency is below the cut-off frequency, ?c, inputs
Figure 2-1: Brickwall Low-Pass Filter Amplitude Response
?
0 ?c Frequency
0 dB
passband
stopband
Attenuation
(dB)
Increasing
3
are passed in with no attenuation. However, inputs above ?c are attenuated. Frequencies up to ?c are
called the ?passband?, while frequencies after ?c are called the ?stopband?. The passband is the frequency
band of the wanted signal, while the stopband is the frequency band of the un-wanted signal.
As can be seen from Figure 2-1 the transition from passband to stopband is infinitesimal. This type
of narrow response in unachievable in practice, but it represents the ideal response of a low pass filter. The
more realistic response of a low-pass filter is shown in Figure 2-2. The un-shaded areas show
Figure 2-2: More realistic low-pass filter
the degree of freedom that the designer has for the frequency response. As can be seen in Fgure2-2, the
passband does not have to be perfectly flat. There can many variations in the passband, from perfectly flat,
to ripples in the passband. There also can be variations in the stopband, from a perfectly flat response, to
ripples in the stopband. There are four main types of classic analog low pass filters that create these
different kinds of responses. A brief explanation of each type of filter is given in Table 2-1
? 0 ?c
Frequency
Attenuation
Increasing
Passband Transition Band Stopband ?
?s
4
Filter Frequency Response
Butterworth Flattest pass-band - poor roll-off rate
Chebyshev Pass-band ripple - better roll-off rate
Elliptic Pass- and ?stop band ripple ? steepest roll off rate
Inverse Chebyshev Ripple in stop band
Table 2-1 Classic Analog Filters
2.2 Simple low-pass filter circuit
A simple low-pass filter is shown in Figure 2-3. It contains one component which is the capacitance,
whose reactance varies with the frequency, which in turn causes the output voltage to change as the
frequency changes. The reactance is given by Xc = 1/?C, where ? is the angular frequency, and decreases as
the frequency increases, causing a voltage divider action with the resistor (R1). The
Figure 2-3: Simple Low-pass Filter
capacitor has reactance and blocks low frequency signals allowing them to go to the load, but at higher
frequencies, the reactance drops and the capacitor acts like a short circuit. The frequency response of the
circuit is shown in Figure 2-4.
5
Figure 2-4: Frequency Response of simple low-pass filter
The cutoff frequency is defined by ?C = 1/RC, where ?C is in rad/s, R is in ohms, and C in Farads. This
configuration has a single pole and a 3-dB cutoff frequency of 1kHz as shown in the figure above.
2.3 2nd Order Low-pass Filter
`A simple low pass filter like this is mostly used in decoupling, or to prevent AC interference generated by
power supplies. However, a single pole circuit may be inadequate. A second pole can be added by
Figure 2-5: 2nd Order low-pass filter
adding a series inductor to the circuit, as shown in Figure 2 This circuit now has two components to stop AC
from passing from the input to the output. The increasing reactance of the inductor, Xi = ?L, and the
decreasing reactance of the capacitor, Xc = 1/?C, make this a two-pole or 2nd order circuit. Once again at low
frequencies the inductor is a short and the capacitor is an open, this couples the signal at the output. At
resonance (when the voltage across the inductor is equal to the voltage across the capacitor), the 3-dB
6
cutoff frequency is set. Once again at high frequencies, the inductors reactance increases, and the capacitors
reactance decreases, decouple the input from the output, which shorts the output signal to ground.
The transfer function for this circuit is seen as a voltage divider, where Z1 = sL1 + R1, and Y2 = sC2 +G2.
The transfer function is shown in Equation (2.1).
Figure 2-6: Voltage Divider Circuit
g1848g1867g1873g1872
g1848g1861g1866 uni0009=
1
g1851g2870
1
g1851g2870 + g1852g2869
(2.1 )
Then if we plug in and simplify:
g1848g1867g1873g1872
g1848g1861g1866 uni0009=
1
g1871g2870g4666g1838g2869g1829g2870g4667 + g1871g4666g1838g2869g1833g2870 + g1829g2870g1844g2869g4667 + 1 + g1844g2869g1833g2870uni0009 (2.2 )
The simplified transfer function is shown in equation (2.2). Since the variable ?s? is raised to the second
order, this is proof that this is a 2nd order filter, and has two complex poles. Also, since this is a 2nd order
filter, it will roll off at 40 dB/dec and the steepness and bandwidth of the filter is controlled by the qualify
factor (Q). The 2nd order ladder filter is the proto-type design for all higher order ladder filters. To achieve an
even higher order filter, all one must do is keep adding inductors in series, and capacitors in parallel; this is
the beauty of the LC Ladder network circuit. However, the inductors and capacitors much be chosen so that
the filter has the correct cutoff frequency at ?C, and is matched with the load.
Z1
Y2
7
3 Synthesis of lossless LC Ladder Filters
3.1 Background
Many methods exist for synthesizing low-pass filters. One method is software based. The user
selects what kind of filter to implement (low-pass, high-pass, bandpass, etc), and then selects the
appropriate attenuation in the pass-band (?p), attenuation in the stopband (?s), the passband frequency
(?p), and the stopband frequency (?s). After these values are selected, the type of approximation for the
filter (Butterworth, Chebyshev, inverse Chebyshev, etc), and the order ?N,? of the filter are selected. Then
finally, the program would calculate all of the results, including the pole locations and zeros, the transfer
function, as well as the frequency and phase response. One such program that does this is Professor
Wilamowski?s LADDER software, which has been used in this thesis. The other method to synthesize LC
Ladder filters is the mathematical method. For passive ladder filters, multiplication, division, design
equations, and the use of look up tables, is all one needs to design a filter. This chapter explains how to
calculate transfer functions mathematically, and also introduces another unique method to synthesize
ladder filters.
3.2 Properties of lossless ladders
Lossless ladders have the form of the circuit shown in Figure 3-1. Since the signal is supplied to the
ladder by a resistive source, and since the load is resistive as well, the ladder is said to be doubly terminated
(the source and the load resistance are both the same). Doubly terminated ladder filters have a lower
sensitivity to component variations.
8
Figure 3-1 Doubly Terminated lossless ladder:
a) General Representation; b) 5th order low-pass
As can be seen in Figure 3-1b, this is a low-pass ladder filter. At low frequencies the inductors will be shorts
and the capacitors will be open, so the filter simply reduces down to two parallel wires, as shown in Figure
3-1a. In general, if the ladder has n branches, with one inductor in each series branch, and one capacitor in
each shunt branch, the transfer function is an nth-order polynomial D(s), and the transfer function once
again is of order n:
g1846g4666g1871g4667 = g1848g1867g1873g1872
g1848g1861g1866 uni0009=
g1837
g1830g4666g1871g4667 =uni0009
g1837
g1871g3041 + g1853g3041g2879g2869g1871g3041g2879g2869 + uni22EF g1853g2869g1871 +uni0009g1853g2868 (3.1)
At high frequencies, the inductors act like opens, and capacitors are short circuits, which means the
signal is grounded, and this is exactly what was explained in Chapter 1. No matter the order of the filter, all
low-pass ladder circuits reduce to Equations (3.2) at dc and at high frequencies.
g1846g4666uni0030g4667 = g1837
g1853g2868 uni0009=
g1844g2870
g1844g2869 + g1844g2870uni0009 uni0009uni0009uni0009uni0009uni0009g1853g1866g1856uni0009uni0009uni0009uni0009uni0009g1846g4666g1871g4667uni007Cg3046uni2192g2998 =
g1837
g1871g3041 = uni0030 (3.2)
(a)
Lossless Ladder
(b)
Zin
9
3.3 A synthesis strategy
For the circuit shown in Figure 3-1, we have identified the current from the source I1, and its direction,
we have also identified Zin as the input impedance of the RLC circuit, which is made up of the lossless ladder
terminated into R2. Just like the poles of the circuit, the input impedance has a real and imaginary
components (see equation 3.3)
Zin = Rin + jXin (3.3)
and the current at the input is shown below (see equation 3.4)
g1835
g2869 =
g1848g3046
g1844g2869 + g1852g3036g3041 (3.4)
Since this is a lossless LC circuit, we can assume that the power at the input is equal to the power at
the load (P1 = P2). Then,
g1842
g2869 =
|g1848g2869g4666g1862g2033g4667|g2870
4g1844g2869 (3.5)
g1842
g2870 = g1844g3036g3041 ? |g1835g2869g4666g1862g2033g4667|g2870 =
|g1848g2870g4666g1862g2033g4667|g2870
g1844g2870 (3.6)
After substituting Equation 3.4 into Equation 3.6 and after proper manipulation, the desired transfer
function for the ladder circuit is shown below (see Equation 3.7)
|g1846g4666g1871g4667|g2870 = g1844g2870g1844g3036g3041
|g1844g2869 + g1852g3036g3041|g2870 (3.7)
Once again P1 = P2, because there is no power is loss in this lossless LC Circuit, if we take the ratio of both
power expressions, we have the proposed transfer function for LC filters (see Equation 3.8), which is defined
with Equations (3.5 - 3.7).
|g1834g4666g1862g1875g4667|g2870 = 4g1844g2869
g1844g2870
|g1848g2870g4666g1862g2033g4667|g2870
|g1848g2869g4666g1862g2033g4667|g2870 =
4g1844g2869g1844g3036g3041
|g1844g2869 + g1852g3036g3041|g2870 ? 1 (3.8)
If we substitute the impedance (Equation 3.3), into Equation 3.8, H(j?) takes on the form
10
|g1834g4666g1862g1875g4667|g2870 = 1 ? |g1844g2869 ? g1852g3036g3041|g2870
|g1844g2869 + g1852g3036g3041|g2870 = 1 ? |g2025g4666g1862g2033g4667|
g2870 (3.9)
Where ?(s) is called the auxiliary function, which is also known as the reflection coefficient. Like in
electromagnetic wave theory, ?(s) is a measure of how well R1 and Zin are matched. After some algebra it
can be shown that Equation 3.9 can be solved for Zin, in terms of ?(s) and R1.
g1852
g3036g3041 = g1844g2869
uni00091 ? g2025g4666g1871g4667
uni00091 + g2025g4666g1871g4667 (3.10)
Since ?, is determined from the prescribed transfer function H, we have reduced the problem to
determining a lossless ladder circuit terminated into R2, from a specified Zin. It is known that the Butterworth
response is:
|g1834g4666g1862g2033g4667|g2870 = uni00091
uni00091 + g2033g2870g3041 (3.11)
then from equation 3.1 we have
|g1834g4666g1862g2033g4667|g2870 = uni0009g1837g3009
uni0009g1830g4666g1871g4667g1830g4666?g1871g4667 (3.12)
Where KH is equal to K from Equation 3.1, and D(s) is the known denominator from the Butterworth
function. The polynomials D(s) for n = 1, 2 and 3 are shown in Table 3-1
Order |g1830g4666g1862g1875g4667|g2870 Dg4666sg4667
1 1 + g2033g2870 g1871 + 1
2 1 + g2033g2872 g1871g2870 + uni221A2g1871 + 1
3 1 + g2033g2874 g1871g2871 + 2g1871g2870 + 2g1871 + 1
Table 3-1: Polynomials D(s) for Butterworth Filter
Since H(s) is known (Equation 3.8), we can rewrite the auxiliary function using Equation 3.9 as
11
|g2025g4666g1871g4667|g2870 = 1 ? uni00091
uni00091 + g2033g2870g3041 =
g2033g2870g3041
1 + g2033g2870g3041 =
g1871g3041
g1830g4666g1871g4667uni0009 (3.13)
Finally, if we substitute Equation 3.11, into Equation 3.9 we obtain Equation 3.14, which allows us to use this
unique synthesis to calculate the elements for a ladder filter by using polynomial division.
g1852
g3036g3041 = g1844g2869[
uni00091 ? g2025g4666g1871g4667
uni00091 + g2025g4666g1871g4667uni0009]
?g2869 = g1844g2869[uni0009g1830g4666g1871g4667 ? g1871
g3041
uni0009g1830g4666g1871g4667 + g1871g3041uni0009]
?g2869 (3.14)
3.3.1 Ladder Synthesis Example
If we let n=3, and by setting R1 = 1, and by using Equation 3.14 and Table 3-1, we find that the
auxiliary function is
g2025g4666g1871g4667 = g1871g2871
g1871g2871 + 2g1871g2870 + 2g1871 + 1 = (3.15)
and then we fine that Zin
g1852
g3036g3041 =
2g1871g2870 + 2g1871 + 1
2g1871g2871 + 2g1871g2870 + 2g1871 + 1 = (3.16)
In order to determine the ladder circuit, Zin needs to be expanded. To expand the circuit, this is done by
dividing the polynomial of higher degree by the polynomial of the lower degree. This is done for one step,
then you have to invert the polynomial and divide again, then you have to invert the polynomial again, and
once again divide again. This process is repeated over and over until there is no remainder. After each
division, the first term of each quotient is an element value for the ladder circuit. An example of the
polynomial division and the element extraction is shown below.
Division Quotient Element
2g1871g2870 + 2g1871 + 1g3493g1871g2871 + 2g1871g2870 + 2g1871 + 1uni0009 = g1871 + 1 g1871uni0009 g1871g1838g2869
g1871 + 1g34932g1871g2870 + 2g1871 + 1 = 1 2g1871 g1871g1829g2870
1uni0009uni221Ag1871 + 1uni0009 = 1 g1871 g1871g1838g2871
12
1uni221A1 = uni0030 1 g1844g2870
Table 3-2: Ladder Synthesis Division and Element Extraction
or
g1852
g3036g3041 =
uni0031
g1871 g3397 uni0031
uni0032g1871 g3397uni0009 uni0031
g1871 g3397uni0009uni0031uni0031
g3404 uni0031
g1871g1838g2869 g3397 uni0031
g1871g1829g2870 g3397uni0009 uni0031
g1871g1838g2871 g3397uni0009 uni0031g1833
g2870
uni0009
(3.17)
Now that the polynomial division is complete, the circuit realization is extracted from quotient, and
the corresponding circuit is shown in Figure 3-2. The normalized element values can then be found from
ladder tables. Ladder circuits can be developed this way for any nth order low pass filter. In the next section,
the more traditional mathematical approach to designing nth other filters will be explained.
Figure 3-2: Ladder Synthesis Example
3.4 Calculate the Transfer Function of Ladder Circuit
To calculate the transfer function of a ladder filter, consider the 5th order low pass filter shown in
Figure 3-3. In order to analyze this circuit, I will introduce the notation Yj (admittance), for the parallel
Zin
13
Figure 3-3: 5th order Ideal Chebyshev low-pass filter
branches, and Zi (impedance), for the series branches as shown in Figure 3-4 and in Table 3-3. Then using
Kirchoff?s voltage laws, KCL and KVL, and also Ohm?s law, I can calculate the transfer function by starting at
Vout, and working back to Vs.
Z1 = sL1
Y2 = sC2
Z3 = sL3
Y4 = sC4
Z5 = sL5
Table 3-3: Element Values for series and shunt ladder
Figure 3-4: 5th order ladder with series impedances and shunt admittances
Ohm? s Law: I6 = I5 = Vout * R2 (3.18)
KVL V3 = Vout*g46661 + s*L5*R2g4667 (3.19)
KCL:
I3 = I4 + I5 = sC4*V3 + Vout*R2
I3 = Vout[s*C4g46661+sL5*R2g4667 + R2]
(3.20)
KVL
V2 = I3*sL3 + V3
V2 = Vout[s*C4g46661+sL5*R2g4667 + R2]* sL3 + Vout*g46661 + s*L5*R2g4667
(3.21)
Z1 Z3 Z5
Y2
Y4
R2
R1 V2 V3 V1
I1
I3
I2 I4
I5
Vout
Vs
14
V2 = Vout[sL3[s*C4g46661+sL5*R2g4667+ R2] +g46661 + s*L5*R2g4667]
KCL:
I1 = I2 + I3
I1 = sC2*V2 + Vout[s*C4g46661+sL5*R2g4667 + R2]
I1 = Vout*{sC2*[sL3[s*C4g46661+sL5*R2g4667+ R2] +g46661 + s*L5*R2g4667] +
[s*C4g46661+sL5*R2g4667 + R2]}
(3.22)
KVL Vs = I1R1 + sL1I1 +V2 (3.23)
Table 3-4: 5th order transfer function
After expansion and simplification of Equation (3.24), the coefficients of the 5th order transfer function are
shown in Table 3-5.
S5
L1*C2*L3*C4*L5*R2
S4
L1*C2*L3*C4+R1*C2*L3*C4*L5*R2
S3
L3*C4*L5*R2+L1*C4*L5*R2+L1*C2*L5*R2+L1*C2*L3*R2+R1*C2*L3*C4
S2
L3*C4+L1*C4+L1*C2+R1*C4*L5*R2+R1*C2*L5*R2+R1*C2*L3*R2
S1
L5*R2+L3*R2+L1*R2+R1*C4+R1*C2
S0
1+R1*R2
Table 3-5: Coefficients of 5th Chebyshev Transfer Function
15
If we consider a 5th order low pass filter with the following characteristics ?p=1 dB, ?s=30 dB, ?p=1rad/sec,
?s=2rad/sec and designed with the following elements (see Table 3-6 ), and then plugging these values into
the coefficients of the transfer function shown in Table 3-5, the transfer function of the circuit can be
calculated.
Element Value
L1 3.481829
C2 0.761919
L3 4.537546
C4 0.761919
L5 3.481829
R1 1
R2 1
Table 3-6: Element Values for 5th Order Low-pass Filter
Transfer function is shown in Equation 3.25:
g1846g1832 = 1
31.93g1871g2873 + 18.42g1871g2872+45.18g1871g2871 + 17.53g1871g2870 + 13.02g1871 + 2 (3.25)
The frequency response is show in Figure 3-55, this plot was generated in MATLAB by plotting the transfer
function (Equation 3.8), and as you can see from the plot, there are ripples in the passband.
16
Figure 3-5: 5th order Chebyshev low-pass filter
3.5 Frequency Transformation from low-pass to other types of filters
The low-pass filter is a prototype filter for other types of filters. There is no need to calculate the
transfer function of the desired filter, it can all be done from the low-pass filter. One method is to use the
Foster Reactance Function. To use the Foster Function, the user substitutes the appropriate Foster Function
into the transfer function of the low-pass filter, this intern transforms the filter into a high-pass, band-pass
or any other type of filter. The Foster Function is mostly used in active filters. Another approach to filter
transformation is element substitution. In this approach, individual elements of the original low-pass filter
are replaced with sub-circuits (see Figure 3-6).
-150
-100
-50
0
Ma
gn
itu
de
(d
B)
10-3 10-2 10-1 100 101
-450
-360
-270
-180
-90
0
Ph
as
e (
de
g)
Bode Diagram
Frequency (rad/sec)
17
After the sub circuits have been inserted, the new element values are then re-scaled to the
appropriate values by the equations shown in Figure 3-6 where:
L = Inductor Value
C = Capacitance Value
?c = Center Frequency
?0 = Cut-off Frequency
B = Bandwidth of Band-pass filter
Low-pass Highpass Bandpass Bandstop
Z
g3013
g3104g3278
g2869
g3013g3104g3278
g3013
g3003uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009
g3003
g3013g3104g3116g3118
g1828g1838 g2033g2868g2870uni2044
uni0009uni0031 g1828g1838uni2044
Y
g3004
g3104g3278
g2869
g3004g3104g3278
g1828 g1829g2033g2868g2870uni2044
uni0009g1829 g1828uni2044
g2869
g3003g3004uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009
g3003g3004
g3104g3116g3118
Figure 3-6: Filter transformation from low-pass to other types of filters
The two main reasons for scaling are to produce more practical/ feasible values for the elements
e.g. load resistance (magnitude scaling), and/or to move the filter to the appropriate 3-dB cutoff point
(frequency scaling). This form of frequency transformation from low-pass filters is quick and efficient, and is
mostly used in passive designs
18
4 Synthesis of lossy LC Ladder Filters
4.1 Introduction
For almost all prototype low-pass filters, the values of the elements have already been derived and
put into tables. While this type of design from tables works fine, it represents an ideal response, which is
insufficient for the real world or for practical applications. In practical applications, the inductors and
capacitors will each introduce losses into the circuit. These losses will affect the frequency response of the
filter significantly, and if the frequency response changes, this can greatly impact the performance of the
whole system. This next section describes how to design filters with lossy components, and how to optimize
the filter to achieve ideal like results.
4.2 Background
The basic structure of the lossless ladder is shown in Figure 4-1,where the parallel branches are
Figure 4-1: Basic Structure of Ladder Filter
the admittances (Y), and the series branches are the impedances (Z), which are purely imaginary. However,
this time the filter includes the losses. In order to include the losses in the filter, all of the inductors and
capacitors in the circuit must be replaced with their lossy counterpart, see Figure 4-2.
Z2-1 Zn-1
Y1
Yn-2
Yn
R1
Vs
R2
19
Figure 4-2: Models of Real Capacitors and Real Inductors
As can be seen from Fig. 4-2, each lossy inductor can be represented by an ideal inductor L, in series
with R L (which is measured in ohms), and each lossy capacitor can be represented by an ideal capacitor C, in
parallel with a conductance Gc (which is measured in siemens). The resistance R L, and the conductance Gc,
are assumed to be constants, i.e. they do now depend on the frequency ?. This is equivalent to saying that
the quality factor (Q) of each inductor and capacitor varies linearly with frequency. The complex impedance
Z(s) of the lossy inductor is shown in Equation 4.1, and
g1852g4666g1871g4667 = g1871g1838 + g1870 (4.1)
the complex admittance Y(s) of the lossy capacitors is shown in Equation 3.2
g1851g4666g1871g4667 = g1871g1829 + g1859 (4.2)
In ideal filters, the lower the Q, the less sharper the response, and the higher the Q, the more
idealistic the response. The same thing goes for the lossy filter, the higher the losses of the reactance
elements used in the design, the greater the difference between the lossy frequency response of the filter,
and the ideal frequency response of the filter.
The difference in the ideal response and lossy frequency response of the circuit is due to the
reactance of the elements, which intern change the location of the poles on the complex plane (see Figure
4-3). In order to gain ideal like characteristics, one must minimize the distance between the poles of the
20
ideal filter and the lossy filter, this is done optimizing the reactance elements of the lossy filter, so that they
match the ideal filter. The different methods of optimization will be explained in the following chapters.
Figure 4-3: Movement of Poles due to lossy elements
4.3 Procedure of designing Lossy Ladder Filters
This chapter describes how to design passive low-pass filter ladder filter, with lossy elements. Results
are included for the Butterworth, Chebyshev, and Bandpass ladder filters. Just like in the ideal case, one can
use software, or tables to design the filter, however, for each ideal element, it must be replaced with its
lossy component type (see Figure 4-2). The three main steps in designing a lossy ladder filter are listed
below.
1. Determine the transfer function for the ideal and lossy ladder filter
2. Calculate the error function
3. Minimize the error function using optimization techniques
Determining the transfer function mathematically for the filter can be done as explain in Chapter 2.
However, for higher order N filters, and for band-pass filters, determining the transfer function
mathematically can be rather complex. In this Chapter, the method to calculate the transfer function by
hand, and in MATLAB, both in the symbolic form and in the coefficient form will be explained. After the
transfer function is calculated, the next step is to minimize the error between the ideal filter response and
-0.5 -0.4 -0.3-1
-0.5
0
0.5
1
Ideal Roots
Lossy Roots
21
the lossy filter response. To minimize the error between both responses, the least squares method is used.
Finally the error function is minimized using different optimization techniques.
4.3.1 Calculating the transfer function of a lossy ladder filter
Consider a 3rd order lossy low-pass filter show in Figure 4-4, where ?p=1 dB, ?s=30 dB, ?p=1rad/sec,
?s=2rad/sec, designed using ideal reactance elements, L1=2.023593, C2=0.994102, L3=2.023593, Rin=
Rout=1 and with lossy elements R1 = G2 = R3 = 0.1.
Figure 4-4: 3rd order lossy ladder filter
The impedance and admittance equations are shown in Table 4-1
Z1 = sL1 +R1
Y2 = sC2 +G2
Z3 = sL3 +R3
Rin = Rout = 1
Table 4-1: 3rd Order Lossy Element Values
The mathematically calculated lossy transfer function is shown below:
Ohm?s Law: I4 = I3 + 0 = Vout * Rout (4.3)
KVL:
V2 =Vout + I3Z3
V2 = Vout + Vout*Rout*Z3
V2 = Vout g46661 +Z3* Routg4667
(4.4)
22
KCL
I1 = I2 + I3
I1 = Y2*[Vout*g46661 +Z3*Routg4667] +Vout*Rout
I1 = Vout[Y2g46661+Z3*Routg4667 +Rout]
(4.5)
KVL
Vin = I1R1 +Z1I1 +V2
Vin = g4666Vout[Y2g46661+Z3*Routg4667 +Rout]g4667R1 +
Z1* Vout[Y2g46661+Z3*Routg4667 +Rout]+ Vout g46661 +Z3* Routg4667
(4.6)
Table 4-2: 3rd Order lossy transfer function
After simplification, and substituting the equations from Table 4-1 into Equation 4.6, the final
transfer function is shown in Table 4-3. Equation 4.7 shows transfer function, with the coefficients a,b,c & d,
representing the elements for each power ?s?, in Table 4-1.
S3
a = L1*C2*L3
S2
b =R1*C2*L3+L1*G2*L3+L1*C2+L1*C2*R3+C2*L3
S1
c =L3+R1*G2*L3+R1*C2+R1*C2*R3+L1+L1*G2+L1*G2*R3+G2*L3+C2+C2*R3
S0
d = 2+R3+R1+R1*G2+R1*G2*R3+G2+G2*R3
Table 4-3: 3rd Order Lossy Transfer Function - Mathematical
g1846g1832 = 1
g1853g1871g2871 + g1854g1871g2870 + g1855g1871 + g1856 (4.7)
23
The final transfer function with all values substituted into a, b c, and d is show in in Equation 4.8
g1846g1832 = 1
4.062g1871g2871 + 4.019g1871g2870 + 5.037g1871 + 2 (4.8)
Calculating the lossy transfer function mathematically like this is straight forward for small order N
filters, but for higher order filters, calculating the transfer function mathematically can be extremely
tedious, especially when it comes to band-pass filters. Band-pass filters introduce a complex numerator into
the transfer function, which make calculating the transfer function a lot more intense. Another way to
calculate the transfer function is to use MATLAB, which is explained below.
Every ladder filter has several reoccurring stages as shown in Figure 4-5. If we assume the output of
Stage 1 is 1V (Vo =1V), and then if we work backwards through each stage, we can develop the transfer
function for any type of filter, whether it be lossy or ideal. However upon completion, we must remember
that the final transfer function will be the inverse of Vin.
Figure 4-5: Stages of 3rd order low-pass lossy filter
Consider the 3rd order low-pass lossy lady filter shown in Figure 4-5. For the lossy ladder filter shown
above, we regard the voltage or current of each stage, as the generic variable State (U), as well as the
resistance of each stage as a generic resistance, Coefficients (Co). Where Coefficients (Co) is going to be the
impedance of a series branch, as well as the admittance of a shunt branch. Now by definition, each stage has
Stage 4 Stage 2 Stage 3 Stage 1
24
one State (U) variable, and one Coefficients (Co) variable, and every stage of the transfer function can be
calculated by adding up all of the previous stages, as shown in Equation 4.9
Un = Un-2 + Un-1 *Con-1 (4.9)
This method is good for calculating the transfer function numerically, and is also very generic. No
matter the order of the filter, this method works to calculate the transfer function. However, it can be
shown that MATLAB can also calculate the transfer function symbolically. The symbolic version follows the
same method as described above, except you have to modify the MATLAB code to handle all of the
functions symbolically. Once this is done, the coefficients of your transfer function are simply the elements
as shown in the previous transfer function calculations (see Table 4-3 for example).
If we consider the same 3rd order low pass filter with ?p=1 dB, ?s=30 dB, ?p=1rad/sec, ?s=2rad/sec,
designed using ideal reactance elements, L1=2.023593, C2=0.994102, L3=2.023593, Rin= Rout=1 and with
lossy elements R1 = G2 = R3 = 0.1 The ideal transfer function calculated in MATLAB is shown in Table 4-4,
and the lossy transfer function calculated in MATLAB is shown in Table 4-5.
S3
L1*C2*L3
S2
L1*C2+C2*L3
S1
L3+L1+C2
S0
2
Table 4-4: 3rd Order ideal low-pass filter with symbolic coefficients ? Calculated in MATLAB
25
S3
L1*C2*L3
S2
R1*C2*L3+L1*G2*L3+L1*C2+L1*C2*R3+C2*L3
S1
L3+R1*G2*L3+R1*C2+R1*C2*R3+L1+L1*G2+L1*G2*R3+G2*L3+C2+C2*R3
S0
2+R3+R1+R1*G2+R1*G2*R3+G2+G2*R3
Table 4-5: 3rd Order lossy low-pass filter with Symbolic coefficients ? Calculated in MATLAB
After plugging in the right values into the correct parameter, it can be shown that both the symbolically
calculated transfer function and the numerically calculated transfer function in MATLAB both match.
MATLAB Symbolically calculated transfer function (Equation 4.10) ? Ideal
g1846g1832 = 1
4.062g1871g2871 + 4.019g1871g2870 + 5.037g1871 + 2 (4.10)
MATLAB Numerically calculated transfer function (Equation 4.11) ? Ideal
g1846g1832 = 1
4.070g1871g2871 + 4.023g1871g2870 + 5.041g1871 + 2.000 (4.11)
MATLAB symbolically calculated transfer function (Equation 4.12) ? Lossy
g1846g1832 = 1
4.062g1871g2871 + 4.829g1871g2870 + 5.69g1871 + 2.321 (4.12)
26
MATLAB numerically calculated transfer function (Equation 4.13) ? Lossy
g1846g1832 = 1
4.070g1871g2871 + 4.835g1871g2870 + 5.695g1871 + 2.321 (4.13)
The MATLAB symbolic method and numerical method are both proven to calculate the transfer function
correctly for any order (N), of a low pass filter.
4.3.2 Calculating the Error Function
The basic theory behind the error function is to reduce the distance between the poles of the ideal
filter and the lossy filter. To calculate the error function, the Error Sum of Squares (SSE) is used, see Equation
4.14.
SSE = g3533g4666g18530 ? g18540g4667^2 + g4666g18531 ? g18541g4667^2 + g4666g1853g1866 ? g1854g1866g4667^2 uni0009g3041
g3036g2880g2869
(4.14)
There are two different ways to calculate the error, one way is to calculate the errors between the poles of
the lossy filter and the ideal filter, and the other way is to calculate the error between the coefficients of the
lossy filter and the ideal filter. Each technique will be presented in the following chapters.
Figure 4-6: Pole locations for 3rd order ideal and lossy low-pass filter
-0.5 -0.4 -0.3-1
-0.5
0
0.5
1
Ideal Roots
Lossy Roots
27
For the 3rd order filter mentioned at the beginning of this chapter, the roots are shown in Table 4-6 and the
pole locations are shown in Figure 4-6. To calculate the error, we substitute the ideal roots and lossy roots
into Equation 4.14.
Ideal Roots Lossy Roots
-0.2537 + 0.9513i
-0.3229 + 0.9583i
-0.2537 - 0.9513i
-0.3229 - 0.9583i
-0.4974
-0.5461
Table 4-6: Ideal Roots and Lossy roots for 3rd order low-pass filter
For example, if we let a0 = -0.1437 + 0.8513i, b0 = -0.2229 + 0.8583i, a1 = -0.1437 - 0.8513i, b2 = -
0.2229 - 0.8583i, a3 = -0.2874, and b3 = -0.3161, and substitute these values in Equation 4.14, the error is
calculated between the lossy roots and the ideal roots. This SSE value is then passed on into the Nelder-
Mead optimization function which is a technique for minimizing the distance between the ideal roots and
the lossy roots. This optimization routine will be explained in the next section.
Another way to calculate the error is to find the difference between the coefficients of the lossy
filter and the ideal filter. Once again, Equation 4.14 is used to calculate the error, however this time, the
coefficients of the transfer function are substituted into a0, b1, etc.
For example, from Equation 4.11, the coefficients for the ideal filer would be a0 = 4.070, a1 = 4.023,
a3 = 5.041, and a4 = 2.000, and the coefficients of the lossy filter would come from Equation 4.13, where b0
= 4.070, b1 = 4.835, b3 = 5.695, b4 =2.321. Once again, this error value is then calculated and passed onto
the optimization routine.
4.3.3 Optimization Routine - Nelder Mead Method
The first optimization routine used to optimize the lossy ladder circuits was the Nelder-Mead
Method. The Nelder-Mead algorithm is designed to solve the classical unconstrained optimization problem
of minimizing a given nonlinear function. This is a fast algorithm which searches for local minimum without
28
the use of derivatives. Using the simplex method, the Nelder-Mead Method finds the minimum point of
error for the lossy ladder filter. The term simplex is a generalized triangle of N dimensions, which is formed
by (N+1) vertices (where N is the number of variables of a function). The worst vertex (which is the largest),
is rejected and replaced with a new vertex, then a new triangle is created and the search continues on. This
process continues on and on and creates a sequence of triangles, and at the same time, the function values
vertices are getting smaller and smaller until the desired error value is found. During the search, the
algorithm performs four routines: Reflection, Expansion, Contraction, and Shrinkage, which is explained
below.
The initial triangle is formed by a initial set of points which represent solutions estimates. If f (x,y) is
the function to be minimized, to start we are given thee vertices of a triangle: Vk = ( Xk, Yk), where k = 1, 2,3.
The function f (x,y) is then evaluated at each of the three points: zk = f (Xk, Yk) for k = 1,2,3. The subscripts are
then reordered so that z1 ? z2 ? z3. Equation 4.14 shows the notation where
B =g4666x1, y1g4667 G = g4666x2, y2g4667 W = g4666x3,y3g4667 (4.14)
B is the best vertex, G is a good vertex (next to best) and W is the worst vertex. The first step in the
algorithm to find the midpoint of the B & G line which is done by averaging the coordinates as shown in
Equation 4.15
g1839 =uni0009g1828 g3397 g1833
2 = g4666
g1876g2869 + g1876g2870
2 ,
g1877g2869 + g1877g2870
2 g4667 (4.15)
Reflection: The function decreases as we move along the side of the triangle from W to B, and it also
decreases as we move from W to G. The test point R is obtained by reflecting the triangle through the side
BG. R is determined by multiplying the midpoint M of the side BG, then drawing the line segment from W to
M and calling this length d. Finally, extend the distance d though M, to locate R (see Figure 4-7).
29
Figure 4-7: The Triangle ABC, the midpoint M and the Reflected point R
Expansion: If the function value R is smaller than the function value at W, then we have moved in the
correct direction towards the minimum. However, if the minimum is father than the point R, the line
segment though M and R is extended to form the new point E (see Figure 4-8). If the function value at E is
less than the function value of R, a better vertex than R has been found.
Figure 4-8: The Triangle BGW and point R with extended point E
Contraction: If the function values at R and W are the same, another point must be tested. Consider the two
midpoints C1 and C2 of the line segments WM and MR (see Figure 4-9). The point with the smaller function
value is called C, and the new simplex (triangle) BGC is created if the function value at C is smaller than the
function value at W
B
G W
R
d
M
d
B
G W
R
d
M
d
d E
30
Figure 4-9: The Contraction point C1 or C2
Shrinkage: If the function values at C is not less than the function values at W, the points G and W will be
shrunk towards B (see Figure 4-9). G will then move toward M, and W is replaced with S, which are
midpoints of the line segments joining B with W.
Figure 4-10: Shrinkage toward the Triangle B
The algorithm will continue to repeat until the desired minimum is found. This Nelder-Mead algorithm is
popular in the fields of chemistry, medicine and engineering and gives significant improvements in the first
few iterations and provides suitable results. However, for larger order filters and for the band-pass filter, the
algorithm does not have the capability to optimize the filters completely. Simulation results using the
Nelder-Mead Algorithim are in the next chapter.
B
G W
R
c1
M
c2
B
G
W
S M
31
5 Nelder-Mead Simulation Results ? Error Calculated From Roots
In this section, simulation results are calculated by trying to reduce the distance between the poles
of the ideal filter and the lossy filter. As explained earlier in the chapter, this error calculation is calculated
from the roots of each transfer function, and then this error value is passed on into the optimization routine.
5.1 3rd Order Chebyshev Low-pass Filter
For our first example let us take into consideration a 3rd order Chebyshev low-pass filter with the
following specifications given in Table 5-1
Element Values
?p 1dB
?s 30dB
?p 1rad/sec
?s 2rad/sec
Table 5-1: 3rd Order Chebyshev low-pass filter specifications
Using the LADDER software, which uses the synthesis algorithm discussed in Chapter 3.1, the
element values are shown in Table 5-2 for the ideal filter, and the ladder circuit for the ideal circuit is shown
in Figure 5-1.
Element Values
C1 2.0236
L2 0.9941
C3 2.0236
R1 = R2 1
32
Table 5-2: 3rd Order Chebyshev low-pass filter element values
Figure 5-1: 3rd order ideal Chebyshev low-pass filter
In order to make the ladder filter lossy, all of the inductors and capacitors in the circuit must be
replaced with their lossy filter counter parts, as explained earlier in the chapter. If we let GC1 = 0.1, RL2 =0.1
and GC3 = 0.1, the lossy filter is shown in Figure 5-2. Using the Nelder-Mead Algorithm as the optimization
routine, the new element values for the lossy filter are shown in Table 5-3, and the simulations results are
shown in the figures below.
Figure 5-2: 3rd order lossy Chebyshev low-pass filter
Element Values
C1 4.0653
L2 0.6163
C3 4.0653
R1 = R2 1
33
Table 5-3: Optimized values for 3rd order Chebyshev
Figure 5-3: 3rd order Chebyshev Pole plot and Frequency Response ? Before Optimization
Figure 5-4: 3rd order Chebyshev Pole plot and Frequency Response ? After Optimization
As you can see from the plots above, after the optimization routine, the roots of the lossy and ideal
filter are a great deal closer than before optimization. However, they are not matched 100%. The frequency
response plots have also been optimized. Apart from the bigger attenuation in the optimized response, the
shape of the filter is different than the lossy response, which is of most importance in this routine. The
-0.5 -0.4 -0.3-1
-0.5
0
0.5
1
Ideal Roots
Lossy Roots
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
-0.5 -0.4 -0.3 -0.2 -0.1-1
-0.5
0
0.5
1
Ideal Roots
Lossy Roots
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
Optimized
34
optimized response has a smoother and more uniform response than the lossy response, and this optimized
response resembles the ideal response better than the lossy response, except for added attenuation. Since
this optimization routine did not optimize the gain of the filter, the optimized response is acceptable with
this optimization scheme.
5.2 5th Order Butterworth Low-pass filter
For our next example let us take into consideration a 5th order Butterworth low-pass filter with the
following specifications given in Table 5-4.
Element Values
?p 1dB
?s 30dB
?p 1rad/sec
?s 2rad/sec
Table 5-4: 5th Order Butterworth low-pass filter specifications
Using the LADDER software, which uses the synthesis algorithm discussed in Section 3.1, the element values
are shown in Table 5-5 for the ideal filter, and the ladder circuit for the ideal circuit shown in Figure 5-5
Element Values
C1 0.617741
L2 1.617266
C3 1.99056
L4 1.617266
C5 0.617741
35
R1 = R2 1
Table 5-5: 5th Order Butterworth low-pass filter element values
Figure 5-5: 5th order ideal Butterworth low-pass filter
If we let GC1 = 0.1, RL2 =0.1, GC3 = 0.1, RL4 = 0.1. and GC5 = 0.1, the lossy filter is shown in Figure 6-5 5-6.
Using the Nelder-Mead Algorithm as the optimization routine, the new element values for the lossy filter are
shown in Table 5-6, and the simulations results are shown below.
Figure 5-6: 5th order lossy Butterworth low-pass filter
Element Values
C1 0.71178
L2 1.59611
C3 1.98569
36
L4 1.59611
C5 0.71178
R1 = R2 1
Table 5-6: Optimized values for 5th order Butterworth
Figure 5-7: 5th order Butterworth Pole plot and Frequency Response ? Before Optimization
Figure 5-8: 5th order Butterworth Pole plot and Frequency Response ? After Optimization
-1.4 -1.2 -1 -0.8 -0.6 -0.4-1
-0.5
0
0.5
1
Ideal Roots
Lossy Roots
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
0.6
Ideal Filter
Lossy Filter
-1.4 -1.2 -1 -0.8 -0.6 -0.4-1
-0.5
0
0.5
1
Ideal Roots
Lossy Roots
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
0.6
Ideal Filter
Lossy Filter
Optimized
37
When it comes to optimization, the Butterworth filter clearly optimizes better than the Chebyshev filter as
show in Figure 5-7 and Figure 5-9. This is due to the fact that the Butterworth filter is flat in the pass-band,
while the Chebyshev has ripples. These ripples make it harder to optimize. After the optimization routine,
the roots of the lossy and ideal filter are well-matched. The optimized response closely resembles the lossy
response in this filter, however this optimized response is still better than the lossy response. This routine
does not match the gain between the optimized filter and the ideal filter, if the gain was matched, both the
ideal response and optimized response would be almost identical.
5.3 8th Order Chebyshev Band-pass Filter
For our next example let us take into consideration a 4th order band-pass low-pass filter with the
following specifications given in Table 5-7
Element Values
?p 1dB
?s 30dB
?p 1rad/sec
?s 2rad/sec
Table 5-7: 8th Order Chebyshev Band-pass low-pass filter specifications
Element Values
C1
L1
0.664475
1.504947
C2
L2
0.992908
1.007143
C3
L3
1.007088
0.992962
38
C4
L4
0.664499
1.504892
R1 = R2 1
Table 5-8: 8th Order Chebyshev Band-pass low-pass filter element values with Q=10%
Using the LADDER software, which uses the synthesis algorithm discussed in Section 3.1, the element values
are shown in Table 5-8 for the ideal filter, and the ladder circuit for the ideal filter is shown in Figure 5-9.
Figure 5-9: 8th order Chebyshev band-pass filter
If we let GC1 = 0.1, RL1 =0.1, GC2 = 0.1, RL2 = 0.1, GC3 = 0.1, RL3 = 0.1, and GC4 = 0.1, RL4 = 0.1, the lossy filter
is shown in Figure 5-10. Using the Nelder-Mead Algorithm as the optimization routine, the new element
values for the lossy filter are shown in Table 5-9, and the simulations results are shown below.
Figure 5-10: 8th order Lossy Chebyshev band-pass filter
Element Values
C1 0.664475
39
L1 1.504947
C2
L2
0.992908
1.007143
C3
L3
1.007088
0.992962
C4
L4
0.664499
1.504892
R1 = R2 1
Table 5-9: Optimized values for 8th order Band-pass filter
40
Figure 5-11: 8th order Band-pass Pole plot and Frequency Response ? Before Optimization
Figure 5-12: 8th order Band-pass Pole plot and Frequency Response ? After Optimization
For this optimization routine, the band-pass filter is harder to optimize than the low-pass filter which is
evident from Figure 5-11 and Figure 5-12. This filter is also harder to optimize because it is an 8th order filter,
so there are more roots to match up. These results are worse than expected, the filter has been optimized,
but another optimization scheme needs to be used for the 8th order band-pass filter. Other methods are
introduced in Chapter 6 and Chapter 7.
-1.5 -1 -0.5 0
-2
-1
0
1
2
Ideal Roots
Lossy Roots
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
-1.5 -1 -0.5 0-3
-2
-1
0
1
2
3
Ideal Roots
Lossy Roots
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
Optimized
41
6 Nelder-Mead Simulation Results ? Error Calculated From Coefficients
In this section, simulation results are calculated by trying to match the coefficients of the lossy transfer
function, with the coefficients of the ideal transfer function.
This procedure is explained earlier in the Chapter 3. Once this Sum of Squares Error is calculated (see
Equation 6.1), it is then passed on into the Nelder-Mean optimization routine (see Chapter 4) which in turn
calculates the optimized element values.
SSE = g3533g4666g18530 ? g18540g4667^2 + g4666g18531 ? g18541g4667^2 + g4666g1853g1866 ? g1854g1866g4667^2 uni0009g3041
g3036g2880g2869
(6.1)
I will re-simulate all of the circuits in Chapter 5, here in chapter 6, using the matching coefficients method.
6.1 3rd Order Chebyshev Low-pass Filter
The 3rd order ideal and lossy Chebyshev filters shown below,
Figure 6-1: 3rd order ideal Chebyshev low-pass filter
Figure 6-2: 3rd order lossy Chebyshev low-pass filter
42
The transfer functions for the ideal circuit is:
g1846g1832 = . 1253
1g1871g2871 + 0.5972 + 0.9284g1871 + .2506 (6.2)
The transfer circuit for the lossy circuit is:
g1846g1832 = . 1253
1g1871g2871 + 0.7975 + 1.0394g1871 + 0.2908 (6.3)
Using the coefficients from the ideal and lossy transfer function, and plugging them into the mean square
value error equation, the error was calculated and then this error was passed on into the optimization
routine.
The transfer circuit for the optimized circuit is:
g1846g1832 = . 0586
1g1871g2871 + 0.5989 + .9278g1871 + 0.1319 (6.4)
The optimized element values for the lossy filter are shown in Table 6-1, and the simulations results are
shown below.
Ideal Values Optimized Values
C1 3.48741 7.16487
L2 .7117 0.34265
C3 3.48729 7.16487
R1 = R2 1 1
Table 6-1: 3rd Order Chebyshev Filter with Ideal and Optimized Values
43
Figure 6-3: 3rd order Chebyshev Coefficient Value Vs Order Plot ? Before Optimization
Figure 6-4: 3rd order Chebyshev Coefficient Value Vs Order Plot ? After Optimization
After the optimization routine, the coefficients of the lossy and ideal filter nearly match, except for
the numerator and the last coefficients value. The frequency responses of the ideal, lossy and optimized
filter are shown below.
1 1.5 2 2.5 3 3.5 40.4
0.6
0.8
1
1.2
1.4
Denominator
Order (sn)
Co
eff
ici
en
t V
alu
e
Ideal Coefficient
Lossy Coefficient
1 1.5 2 2.5 3 3.5 40.4
0.6
0.8
1
1.2
1.4
Denominator
Order (sn)
Co
eff
ici
en
t V
alu
e
Ideal Coefficient
Lossy Coefficient
44
Figure 6-5: 3rd order Chebyshev filter - Before Optimization
Figure 6-6: 3rd order Chebyshev filter- After Optimization
a) Roots Used to calculate error; b) Coefficients used to calculate error
Figure 6-6 shows the frequency response for two types of error calculations. Figure 6-6a shows the
results after optimization, when the error is calculated form the roots (this was presented in section 5.1),
and Figure 6-6b shows frequency response after optimization, when the error has been calculated from the
coefficient difference (explained in this chapter). Both are valid methods, but results are slightly better when
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
Ideal Filter
Optimized Filter
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
Optimized
(b)
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
Optimized
(a)
45
the error is calculated from the roots, which is evident in the optimized response. Calculating the error from
the roots gives a more uniform response for the optimized plot, which closely resembles the ideal filter,
except it has more attenuation. While calculating the error from the coefficients, the optimized plot starts
out just like the roots optimized plot, but the response falls off to some extent when it comes to replicating
the ripple. This routine does not match the gain between the optimized filter and the ideal filter, which is
the cause for a higher attenuation in the optimized plot.
6.2 5th Order Butterworth Low-pass Filter
The 5th order ideal and lossy Butterworth filters shown below,
Figure 6-7: 5th order ideal Butterworth low-pass filter
Figure 6-8: 3rd order lossy Chebyshev low-pass filter
The transfer functions for the ideal circuit is:
g1846g1832 = 0.50121g1871g2873 + 3.237g1871g2872 + 5.241g1871g2871 + 5.243g1871g2870 + 3.2422g1871 + 1.0024 (6.5)
The transfer circuit for the lossy circuit is:
46
g1846g1832 = 0.50121g1871g2873 + 3.735g1871g2872 + 6.419g1871g2871 + 6.1666g1871g2870 + 4.1562g1871 + 1.2857 (6.6)
Using the coefficients from the ideal and lossy transfer function, and plugging them into the mean square
value error equation, the error was calculated and then this error is passed on into the optimization routine.
The transfer circuit for the optimized circuit is:
g1846g1832 = 0.38221g1871g2873 + 3.239g1871g2872 + 5.234g1871g2871 + 5.254g1871g2870 + 3.234g1871 + 0.9805 (5.7)
The new element values for the lossy filter are shown in Table 6-2, and the simulations results are shown
below.
Ideal Values Optimized Values
C1 0.71178 0.71781
L2 1.59611 1.64704
C3 1.98569 1.87059
L4 1.59611 1.64776
C5 0.71178 0.71796
R1 = R2 1 1
Table 6-2: 5th Order Butterworth Filter with Ideal and Optimized Values
47
Figure 6-9: 5th order Butterworth Coefficient Value Vs Order Plot ? Before Optimization
Figure 6-10: 5th order Butterworth Coefficient Value Vs Order Plot ? After Optimization
After the optimization routine, the coefficients of the lossy and ideal filter nearly match, except for
the numerator and the last coefficient value. However, these values are a lot closer to each other than the
results given from the 3rd order Chebyshev filter which, was evaluated in Section 6.1. The frequency
responses of the ideal, lossy and optimized filter are shown below.
1 2 3 4 5 60
2
4
6
8
Denominator
Order (sn)
Co
eff
ici
en
t V
alu
e
Ideal Coefficient
Lossy Coefficient
1 2 3 4 5 60
2
4
6
Denominator
Order (sn)
Co
eff
ici
en
t V
alu
e
Ideal Coefficient
Lossy Coefficient
48
Figure 6-11: 5th order Butterworth filter- Before Optimization
Figure 6-12: 5th order Butterworth filter- After Optimization
Figure 6-12 shows the frequency response after optimization when the error has been calculated
from the coefficient difference. For the Butterworth filter, both error calculating methods have produced
identical results after optimization. This is not the case for Chebyshev filters. The ripples make the
Chebyshev filter harder to optimize. In order to achieve ideal-like results for Chebyshev filters, it is best to
calculate the error from the difference between the roots, and not from the difference between the
coefficients, or use the optimization method presented in Chapter 7.
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
0.6
Ideal Filter
Lossy Filter
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
0.6
Ideal Filter
Lossy Filter
Optimized
49
6.3 8th Order Chebyshev Band-pass Filter
The 4th order ideal and lossy Chebyshev filters shown are below,
Figure 6-13: 8th order Chebyshev Band-pass filter
Figure 6-14: 8th order lossy Chebyshev Band-pass filter
The transfer functions for the ideal circuit is:
g1846g1832 = 0.3941g1871
g2875
1g1871g2869g2869 + 1.019g1871g2869g2868 + 7.0312g1871g2877 + 5.254g1871g2876 + 12.850g1871g2875 + 5.245g1871g2874 +
7.031g1871g2873 + 1.019g1871g2872 + 1.000g1871g2871
(6.8)
The transfer circuit for the lossy circuit is:
g1846g1832
=
0.3941g1871g2869g2870 + 0.6238g1871g2869g2869 + 0.4417g1871g2869g2868 + 0.1845g1871g2877 + 0.0505g1871g2876 + 0.0095g1871g2875 +
0.0012g1871g2874 + 0.0001g1871g2873
1g1871g2869g2873 + 2.950g1871g2869g2872 + 10.663g1871g2869g2871 + 19.88g1871g2869g2870 + 32.142g1871g2869g2869 + 35.715g1871g2869g2868 +
31.702g1871g2877 + 20.812g1871g2876 + 10.354g1871g2875 + 3.870g1871g2874 + 1.028g1871g2873 +. 1848g1871g2872 +. 0215g1871g2871 +uni0009
0.0015g1871g2870 +uni00090.001g1871g2869
(6.9)
50
As you can see from the transfer functions above, the ideal transfer function and the lossy transfer function
are both of different orders even though this is a 8th band-pass order filter. The reason for this difference is
due to the MATLAB algorithm and is discussed in the next section.
Figure 6-15: 8th order Band-pass Coefficient Value Vs Order Plot ? Before Optimization
Figure 6-15 shows graphically how there are a different number of coefficients for the ideal and for the lossy
filter, and the frequency response for the filter in Figure 6-16.
Figure 6-16: 8th order Band-pass Frequency Response ? Before Optimization
10-2 10-1 100 101 102
0
0.1
0.2
0.3
0.4
0.5
Ideal Filter
Lossy Filter
0 5 10 15 200
0.2
0.4
0.6
0.8
Numerator
Order (sn)
Co
eff
ici
en
t V
alu
e
Ideal Filter
Lossy Filter
0 5 10 15 200
5
10
15
20
25
30
35
40
Denominator
Order (sn)
Co
eff
ici
en
t V
alu
e
Ideal Coefficient
Lossy Coefficient
51
6.4 MATLAB Algorithm Limitations
As you can see from the transfer functions and from the plots above, the algorithm has introduced
extra poles and zeroes into both transfer function. This is a 8th order band pass filter which should have a
maximum power of g1871g2876, but yet both transfer functions introduce powers which exceed g1871g2876. This is due to the
way that MATLAB handles rational fractions. In this algorithm, when the impedances is calculated, even
though the denominator for both rational fractions may be the same, when adding the fractions, the
algorithm will multiply the denominator and ignore the common roots. This is the cause for the higher order
roots for the band-pass filter. The algorithm is designed to remove the roots with the small coefficients, but
the order is still higher than the original order. These higher orders of roots do not occur in the low-pass
filters because these type of filters to not have a complex numerator as well as a complex denominator. The
complex numerator and denominator is introduced in the band-pass filter by the series and parallel
combinations formed by the inductors and capacitors show in Figures 6:17-18.
Figure 6-17: Example of Ideal Bandpass Filter Impedance
g1852g3032g3044 = g1871g1838uni0009uni0009uni007Cuni007Cuni0009 g2869g3046g3004uni0009uni0009
52
Figure 6-18: Example of Lossy Bandpass Filter Impedance
To try to eliminate the common roots, I calculated the transfer function symbolically so that like terms/roots
may cancel out, but I still received the same results. The symbolically calculated transfer function for the 8th
order ideal band-pass filter is shown is below.
g17ug143erator
S11
C7*L6^2*C3^3*L2^5
Denog143inator
S15
C7*L8*L6^2*C5*C3^3*L4*L2^5*C1
S14
C7*L8*L6^2*C5*C3^3*L4*L2^5+L6^2*C5*C3^3*L4*L2^5*C1*C7
S13
L6^2*C3^3*L2^5*C5*L4*C1+L6*C3^3*L2^5*C7*L8*L4*C1+L6^2*C3^2*L2^5*C7*L8*C5*C1+
L6^2*C3^3*L2^4*C7*L8*C5*L4+L6^2*C3^3*L2^5*C7*L8*C5+L6^2*C3^3*L2^5*C7*L8*C1+
L6^2*C3^3*L2^5*L4*C1*C7+L6^2*C5*C3^3*L4*L2^5*C7
S12
L6^2*C3^3*L2^5*C5*L4+L6*C3^3*L2^5*C7*L8*L4+L6^2*C3^2*L2^5*C7*L8*C5+L6^2*C3^
3*L2^5*C7*L8+L6^2*C3^3*L2^5*L4*C7+C7*L6*C3^3*L2^5*L4*C1+C7*L6^2*C3^2*L2^5*C
5*C1+C7*L6^2*C3^3*L2^4*C5*L4+C7*L6^2*C3^3*L2^5*C5+C7*L6^2*C3^3*L2^5*C1
S11
L6*C3^3*L2^5*L4*C1+L6^2*C3^2*L2^5*C5*C1+L6^2*C3^3*L2^4*C5*L4+L6^2*C3^3*L2^5
*C5+L6^2*C3^3*L2^5*C1+L6*C3^2*L2^5*C7*L8*C1+L6*C3^3*L2^4*C7*L8*L4+L6*C3^3*L
2^5*C7*L8+L6^2*C3^2*L2^4*C7*L8*C5+L6^2*C3^3*L2^4*C7*L8+L6^2*C3^2*L2^5*C7*C1
+L6^2*C3^3*L2^4*C7*L4+2*C7*L6^2*C3^3*L2^5+C7*L6*C3^3*L2^5*L4+C7*L6^2*C3^2*L
2^5*C5
S10
L6*C3^3*L2^5*L4+L6^2*C3^2*L2^5*C5+L6^2*C3^3*L2^5+L6*C3^2*L2^5*C7*L8+L6^2*C
3^2*L2^5*C7+C7*L6*C3^2*L2^5*C1+C7*L6*C3^3*L2^4*L4+C7*L6*C3^3*L2^5+C7*L6^2*C
3^2*L2^4*C5+C7*L6^2*C3^3*L2^4
S9
g1852g3032g3044 = g4666g1871g1838 + g1844g3039g4667 + g2869g3046g3004g2878g3008
g3278
53
L6*C3^2*L2^5*C1+L6*C3^3*L2^4*L4+L6*C3^3*L2^5+L6^2*C3^2*L2^4*C5+L6^2*C3^3*L
2^4+L6*C3^2*L2^4*C7*L8+L6^2*C3^2*L2^4*C7+L2^5*C7*L6*C3^2
S9
L2^5*C3^2*L6+C7*L6*C3^2*L2^4
S7
C3^2*L2^4*L6
Table 6-3: Coefficients for the Transfer Function for 4th order Ideal Band-pass Filter with Q = 10%
After subbing the ideal component values into the symbolic transfer function above, the ideal transfer
function is:
g1846g1832 = 0.04973g1871
g2869g2869
. 1262g1871g2869g2873 +. 1287g1871g2869g2872 +. 8872g1871g2869g2871 + 0.6619g1871g2869g2870 + 1.621g1871g2869g2869 + 0.6619g1871g2869g2868 +
0.8872g1871g2877 + 0.1287g1871g2876 + 0.1262g1871g2875
(6.10)
When I tried to calculate the lossy transfer function, MATLAB truncated the symbolic results saying that
the ?text exceeds maximum line length?, so I did not include these results, but just like the ideal response,
the lossy response had higher order roots than the order of the filter. Since there is a different number of
coefficients for the ideal and lossy transfer function, this method of optimization is not suitable for complex
filters, another method is introduced in Chapter 7.
7 Levenberg ? Marquardt (LM) Simulation Results
The Levenberg-Marquardt (LM) algorithm, also known as the damped least square method is
another optimization method that provides solutions to the problem of minimizing a non-linear function.
This algorithm is fast and stable, and is used a lot in the field of neural networks. This algorithm is also very
popular when it comes to curve fitting. Curve fitting is the process of constructing a curve that has the best
fit line to a series of data points. The LM algorithm is a combination of two minimization methods: the
Steepest Descent method and the Gauss-Newton method. In the Steepest Decent method, the sum of the
sum of the squared errors is reduced by updating the direction of the greatest reduction of the least squares
objective. This is an iterative method that is given an initial point and follows the negative of the gradient in
order to move the point toward a critical point, which is the desired local minimum. This method is very
popular for optimization problems, but its main disadvantage is that it can take a long time to converge.
There are two main reasons for the slow convergence: the first reason is the step size is small and constant,
so in a place where the gradient is gentle, the training process is very slow. The second reason is that the
54
curvature of the error surface may not be the same in all directions, so the classic ?error valley? problem
may exist.
In the Gauss-Newtown method, the sum of the squared errors is reduced by assuming the least
squares function is locally quadratic, and finding the minimum of the quadratic. The slow convergence of the
steepest descent method is greatly improved in the Gauss-Newton method. Using the second order-
derivatives of the error function to ?naturally? evaluate curvature of the error surface, this algorithm can
find the proper step sizes for each direction and can converge very fast. However, this only happens when
the quadratic approximation of the error function is reasonable. This algorithm does require the calculation
of the second order derivative of the function which can be very complicated, but it uses the Hessian matrix
H, which is equivalent to the second order derivative though. The Hessian matrix H gives the proper
evaluation of the change of the gradient vector g. In order to calculate the Hessian matrix H, the Jacobian
matrix J also needs to be calculated. The Jacobian matrix is a matrix of all first order partial derivatives of the
function. This Jacobian matrix is related to the gradient vector g, and to the error vector e, which is shown in
Equation 7.1
g1859 = g1836g1857 (7.1)
After some integration and substitutions, the relationship between the Hessian matrix H and the Jacobian
matrix J can be written rewritten as the relationship shown in Equation 7.2.
g1834 g3406 g1836g3021g1836 (7.2)
The basic idea of the Levenberg-Marquardt algorithm is that is performs a combined training process
around the area with a complex curvature. The Levenberg-Marquardt algorithm switches to the steepest
descent algorithm, until the local curvature is proper to make a quadratic approximation, this it becomes the
Gauss-Newton method which can speed up the convergence. The update rule for the LM algorithm is shown
in Equation 7.3:
55
g1875g3038g2878g2869 =uni0009g1875g3038 ? (g1836g3038g3021g1836g3038 g3397 g586g1835)g2879g2869g1836g3038g1857g3038 (7.3)
where
w is the weight vector
e is the error vector
g13 is the Jacobian matrix
g961 is the combination coefficient
I is the identity matrix
As the combination coefficient increases the Hessian matrix H will be factored out of the above
equation and the Steepest Descent method is used. As the combination coefficient decreases, the Hessian
matrix H becomes more significant, and the Gauss-Newton method is used. This allows the training
algorithm to interpolate between the Steepest Descent method and the Gauss Newton method. Higher
combination coefficient favors Steepest Descent, and lower combination coefficient favors Gauss-Newton.
This LM Algorithm is used to optimize the amplitude characteristics of the filter. This method uses
the transfer function just like the Nelder-Mead method, however, ?j?? is substituted for ?s? in the transfer
function. Then for different ???, the bode plot will have different amplitudes, and then the LM algorithm is
used to optimize the bode plot characteristics. Pseudocode and a block diagram for the LM algorithm is
shown in Figure 7-1
for iter=1:maxiter
calculate [num_id,den_id] % ideal transfer function
calculate [num_ls,den_ls] % lossy transfer function
for sample w
substitute ?s? with jw for num_id/den_id
substitute ?s? with jw for num_ls/den_ls
calculate error
end
calculate Jacobian matrix with numeric way
gradient=Jacobian?*error
hessian=Jacobian?*Jacobian
del=inv(hessian+mu*I)*gradient % change of elements
use LM algorithm to adjust mu
?
elements=elements+del % update elements
?
end
(a) Pseudocode
56
(b) Block Diagram
Figure 7-1: Block Diagram and Pseudocode for LM Algorithm
The block diagram shows that as the Error Ek+1 decreases, (smaller than the last error), its implied a
quadratic approximation is used on the total error function, and the combination coefficient ? needs to be
reduced, as long as the new error is less than or equal to the max error Emax. (As ? decreases, the Gauss-
Newton method is used) If the error Ek+1 increases, (larger than the last error), this means that it is necessary
to follow the gradient more to look for a quadratic approximation, so the combination component ? needs
to be increased. (As ? increases, the Steepest Descent method is used). The basic steps of the LM Algorithm
are explained in Table 7-1 and the results are in the next section.
wk, m=1
Ek
Jacobian matrix computation
g1823g1811g2878g2778 = uni0009g1823g1811 ? (g1784g1811g1794g1784g1811 g3397 g586g1783)g2879g2778g1784g1811g1805g1811
Ek+1
g2161g2167g2878g2778 ?uni0009g2161g2193 g2161g2167g2878g2778 g3408 uni0009g2161g2193
g2195 ? uni0009g2782 g2195 g3408 uni0009g2782
wk = wk+1 m = m+1
Error Evaluation
Error Evaluation
? = ? ? 10
restore wk
? = ? ? 10
w
k = wk+1
g2161g2167g2878g2778 ? uni0009g2161g2195g2183g2206
End
where
wk is the current weight
wk+1 is the next weight vector
Ek+1 is the total current error
Ek is the last total error
57
1) With the initial weights
(randomly generated)
evaluate the total error (SSE).
3) With new weights evaluate
the total error (SSE)
5. If error is decreased, then accept
step (keep new weight vector) and
decrease g586 by a factor of 10
2) Do an update to adjust
weights:
g1875g3038g2878g2869 = g1875g3038 ? g4666g1836g3038g3021g1836g3038 + g586g1835g4667g2879g2869g1836g3038g1857g3038
4) If error increased, retract
the step (keep new weight
vector as current one) and
increase g586 by a factor of 10.
barb2rightGo to Step 2:
6. barb2rightGo to Step 2 with new weights
until the current total error is
smaller than required value.
Table 7-1 LM Algorithm Basic Steps
7.1 3rd Order Chebyshev Low-pass Filter
Consider the generic 3rd order Chebyshev low-pass filter shown in Figure 7-5 with ?p=1 dB, ?s=30 dB,
?p=1rad/sec, ?s=2rad/sec. The element values for the ideal filter, lossy filter, and optimized filter are shown
in Table 7-3, and the simulation results are shown in the figures below.
Figure 7-2: 3rd Order Chebyshev Low-pass Filter
Ideal Values Lossy Values Optimized Values
C1 3.4823 3.4823 5.4045
L2 0.7714 0.7714 0.5402
C3 3.4823 3.4823 4.4299
R1 1 1 1
R2 1 1 1.5507
Q ? 20 20
Table 7-2: 3rd Order Chebyshev Low-Pass Filter with Ideal, Lossy and Optimized Values
58
Figure 7-3: 3rd Order Chebyshev Low-Pass Filter Frequency Plot ? Before Optimization
Figure 7-4: 3rd Order Chebyshev Low-Pass Filter Frequency Plot ? After Optimization
Figure 7-4 shows how the LM Algorithm has provided better results than the Nelder-Mead method
when optimizing the chebyshev filter. The optimized plot does not match the ideal plot completely, but the
LM Algorithm can reproduce the ripple better than the Nelder-Mead algorithm. The LM Algorithm also
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
ideal
lossy
error
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ideal
lossy
optimized
error
59
matches the gain between the ideal response and the lossy response, which is the reason why both the ideal
plot and lossy plot have equal gain.
7.2 5th Order Butterworth Low-Pass Filter
Consider the generic 5th order Butterworth low-pass filter shown in Figure 7-5 with ?p=1 dB, ?s=30 dB,
?p=1rad/sec, ?s=2rad/sec. The element values for the ideal filter, lossy filter, and optimized filter are shown
in Table 7-3, and the simulation results are shown in the figures below.
Figure 7-5: 5th Order Butterworth Low-pass Filter
Ideal Values Lossy Values Optimized Values
C1 0.617741 0.617741 1.4654
L2 1.617266 1.617266 1.3223
C3 1.99905 1.99905 2.1814
L4 1.617266 1.617266 1.3777
C5 0.617741 0.617741 0.5208
R1 1 1 1
R2 1 1 1.4741
Q ? 20 20
Table 7-3: 5th Order Butterworth Low-Pass Filter with Ideal, Lossy and Optimized Values
60
Figure 7-6: 5th Order Butterworth Low-Pass Filter Frequency Plot ? Before Optimization
Figure 7-7: 5th Order Butterworth Low-Pass Filter Frequency Plot ? After Optimization
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
Qriginal frequency characteristics of 5-th order Butterworth filter
ideal
lossy
error
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ideal
lossy
optimized
error
61
Figure 7-7 shows how the LM Algorithm has provided the best results when it comes to optimizing
lossy ladder filters. The ideal response and the lossy response are pretty much are identical, which is evident
from the ?error? on the plot. The LM Algorithm also matches the gain between the ideal response and the
lossy response, which is the reason why both the ideal plot and lossy plot have equal gain.
7.3 4th Order Chebyshev Low-Pass Filter
Consider the generic 4th order Chebyshev low-pass filter shown in Figure 7-8 with ?p=1 dB, ?s=30 dB,
?p=1rad/sec, ?s=2rad/sec. The element values for the ideal filter, lossy filter, and optimized filter are shown
in Table 7-4, and the simulation results are shown in the figures below.
Figure 7-8: 4th Order Chebyshev Low-Pass Filter
Ideal Values Lossy Values Optimized Values
C1 2.25742 2.25742 3.1759
L2 1.5107 1.5107 1.1393
C3 1.5106 1.5106 1.7863
L4 2.2573 2.2573 2.0249
R1 1 1 1
R2 1 1 1.5702
Q ? 20 20
Table 7-4: 4th Order Chebyshev Low-Pass Filter with Ideal, Lossy and Optimized Values
62
Figure 7-9: 4th Order Chebyshev Low-Pass Filter Frequency Plot ? Before Optimization
Figure 7-10: 4th Order Chebyshev Low-Pass Filter Frequency Plot ? After Optimization
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Qriginal frequency characteristics of 5-th order Butterworth filter
ideal
lossy
error
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ideal
lossy
optimized
error
63
Figure 7-10 shows that the LM Algorithm has provided better results than the Nelder-Mead method when it
comes to optimizing the chebyshev filer. However, better optimized responses can be obtained with
butterworth filters than with chebyshev filters, for both the LM and Nelder-Mead optimization scheme
7.4 8th Order Chebyshev Band-pass Filter
Consider the generic 4th order Chebyshev Bandpass filter shown in Figure 7-11 with ?p=1 dB, ?s=30 dB,
?p=1rad/sec, ?s=2rad/sec. The element values for the ideal filter, lossy filter, and optimized filter are shown
in Table 7-4, and the simulation results are shown in the figures below.
Figure 7-11: 8th Order Chebyshev Band-pass Filter
Element Ideal Values Lossy Values Optimized Values
C1
L1
4.5146
0.2215
4.5146
0.2215
4.0534
0.2467
C2
L2
0.3310
3.0212
0.3310
3.0212
0.2802
3.5694
C3
L3
3.0214
0.3310
3.0214
0.3310
2.2806
0.4385
C4
L4
0.2215
4.5148
0.2215
4.5148
0.1575
6.3476
R1 1 1 1
R2 1 1 1
Q ? 20 20
Table 7-5: 8th Order Chebyshev Band-pass Filter with Ideal, Lossy and Optimized Values
64
Figure 7-12: 8th Order Chebyshev Band-pass Filter Frequency Plot ? Before Optimization
Figure 7-13: 8th Order Chebyshev Band-pass Filter Frequency Plot ? After Optimization
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Qriginal frequency characteristics of of the filter
ideal
lossy
error
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ideal
lossy
optimized
error
65
Once again the LM Algorithm has provided better results, than the Nelder Mead method, as shown in Figure
7-13. The optimized response still has small losses at the peaks of the chebyshev filter which is evident in the
?error? plot, but these results are far superior to any Nelder-Mead method.
7.5 8th Order Butterworth Band-pass Filter
Consider the generic 8th order Butterworth Bandpass filter shown in 13 with ?p=1 dB, ?s=30 dB,
?p=1rad/sec, ?s=2rad/sec. The element values for the ideal filter, lossy filter, and optimized filter are shown
in Table 7-5, and the simulation results are shown in the figures below.
Figure 7-14: 8th Order Butterworth Band-pass Filter
Element Ideal Values Lossy Values Optimized Values
C1
L1
0.8094
1.2355
0.8094
1.2355
0.9639
1.0374
C2
L2
3.2345
0.3092
3.2345
0.3092
2.7620
0.3621
C3
L3
0.2501
3.9981
0.2501
3.9981
0.2297
4.3528
C4
L4
3.2345
0.3092
3.2345
0.3092
2.6492
0.3775,
C5
L5
0.8094
1.2355
0.8094
1.2355
0.3423
2.9210
R1 1 1 1
R2 1 1 1
Q ? 20 20
66
Table 7-6: 8th Order Butterworth Band-pass Filter with Ideal, Lossy and Optimized Values
Figure 7-15: 8th Order Butterworth Band-pass Filter Frequency Plot ? Before Optimization
Figure 7-16: 8th Order Butterworth Band-pass Filter Frequency Plot ? After Optimization
Once again as expected the LM Algorithm has provided better results, than the Nelder-Mead
method, as shown in Figure 7-16. Also since this is a butterworth filter, as expected, better characteristics
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ideal
lossy
optimized
error
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Qriginal frequency characteristics of of the filter
ideal
lossy
error
67
are obtained when compared to the Bandpass filter because this filter does not have ripples, which makes it
easier to optimize. The LM Algorithm has also matched the gain between the ideal response and the
optimized response.
8 Conclusion
Three optimization methods have been presented. The first method used the SSE to minimize the
difference between the poles of the ideal filer and the lossy filter. This procedure used the Nelder-Mead
optimization routine and did produce results which were suitable for the Butterworth response, but for the
Chebyshev filter, the results were impracticable. The second optimization method once again used the
Nelder-Mead optimization routine, however the SSE was calculated from the difference between the
coefficients of the ideal and lossy transfer functions. Both of these methods produced fitting results for low-
pass Butterworth filters, but for the low-pass Chebyshev filter, the ripple was too hard to reproduce for both
of these routines. In the case of more complex designs such as a Band-pass filer, the order of the filters
between the ideal and lossy filters do not match, so these first two optimization methods can only be used
for low-pass filters. The third optimization method uses the LM algorithm for the optimization, and the SSE
was calculated from the difference between the ideal filter and lossy filters magnitude plot. The simulation
results have shown that minimizing the differences between the filters? magnitude plots has been effective
in evaluating the error, when used with the LM algorithm. This method has produced superb results for the
low-pass and band-pass Butterworth filter, and more than adequate results when it comes to band-pass
Chebyshev filters. The LM algorithm also matched the gain between the ideal response and the optimized
response which also has made this algorithm superior than the Nelder-Mead optimization scheme.
In conclusion the filter synthesis algorithm and the optimization methods introduced in this thesis
make it possible for a person to design a filter with non-ideal components, while achieving ideal-like
characteristics. This thesis has also shown another unique synthesis algorithm showing how to design a
ladder filter without the use of look up tables, and also how to calculate the transfer function both
mathematically and symbolically by the use of MATLAB. Since most of the existing ladder synthesis
68
algorithms are based upon ideal components for the design, they are impractical for real world applications.
Using the techniques developed in this thesis allows for one to design a low-pass Butterworth, Chebyshev,
or Band-pass filter, with the use of lossy reactance elements, but yet achieve ideal-like results, which is more
practical for real world applications.
69
9 References
1. A. M. Niknejad, ?Ladder Filter Design, Fabrication, and Measurement?, Berkeley Wireless Research
Center, September 14, 2010
2. Wai-Kai Chen, ?Passive and Active Filters Theory and Implementations? John Wiley and Sons, 1986
3. Rolf Schaumann and Mac E. Van Valkenburg ?Design of Analog Filters? Oxford University Press,
2001
4. M. Jagiela and B.M. Wilamowski, ?A methodology of synthesis of lossy ladder filters?, University of
Information Technology and Management, Autonomous Department of Electronics and
Telecommunications, Rzeszow, Poland
5. John H. Mathews and Kurtis K. Fink, ?Numerical Methods Using Matlab?, 4th Edition, Prentice Hall,
2004
6. Karel Hajek and Jiri Sedlacek, ?GOAL-LOSSY RLC LADDER PROTOTYPES FOR SYNTHESIS OF ARC AND
ASC FILTERS ?Military Academy Brno, Dpt. of Electrical engineering, Kounicova 65, 623 00 Brno,
Czech Rep
7. Fuchang Gao and Lixing Han ?Implementing the Nelder-Mead simplex algorithm with adaptive
parameters?, Received: 13 January 2010 ? Springer Science+Business Media, LLC 2010
8. Pradeep Dandamudi, Timothy A. Brown, Xing Wu, Marcin Jagiela, and Bogdan M. Wilamowski, ?A
Design Methodology of Lossy Transconductance Filters?, IECON 2012 ? 38th Annual Conference on
IEEE Industrial Electronics Society
9. Robert D. Koller and Bogdan Wilamowski, ?LADDER A Microcomputer Tool for Passive Filter Design
and Simulation?, IEEE TRANSACTIONS ON EDUCATION, VOL. 39, NO. 4, NOVEMBER 1996
10. Nam Pham and Bogdan M. Wilamowski, ?Improved Nelder Mead?s Simplex Method and
Applications? Electrical and Computer Engineering, Auburn University, Alabama, US
11. Koller, R. and B. M. Wilamowski, ?A Ladder Prototype Synthesis Algorithm,? proceedings of 35th
IEEE Midwest Symposium on Circuits and Systems, Washington DC, USA, vol. 1, pp. 730-732, August
9-12, 1992
12. Bradley Bell and Alan Schmitzky, ?A generalization of the Gauss-Newton Method that solves
extended Least Square Problems?, Department of Mathematics, University of Southern California,
Los Angeles, CA, and Center for Bioengineering, University of Washington, Seattle, WA 98195
13. Ananth Ranganathan , ?The Levenberg-Marquardt Algorithm?, 8th June 2004
14. Bogdan M. Wilamowski and J. David Irwin ?The Industrial Electronics Handbook ? Intelligent
Systems? CRC Press 2011
15. Alexander J. Casson and Esther Rodriguez-Villegas ?A Review and Modern Approach to LC Ladder
Synthesis?,Circuits and Systems Research Group, Electrical and Electronic Engineering Department,
70
Imperial College London
16. Stefan Niewiadomski, ?Filter Handbook?, Heinwmann Newnes, 1989
17. Bogdan M. Wilamowski and J. David Irwin ?The Industrial Electronics Handbook ? Fundamentals of
Industrial Electronics? CRC Press 2011
18. Henri Gavin, ?The Levenberg-Marquardt method for nonlinear least squares curve-fitting
problems?, Department of Civil and Environmental Engineering Duke University September 28,
2011
19. Gabor C. Themes and Sanjit K. Mitra, ?Modern Filter Theory and Design?, John Wiley and Sons 1973
20. J.A. Nelder and R Mead, ?A simplex method for function minimization?, The Computer Journal
(1965) 7(4): 308-313
71
10 Appendices
A Matlab input function: main.m
%% circuit 1 3rd Order Butterworth
% ideal=[0.999209;1.998418;0.999209];
% lossy=[0.999209;1.998418;0.999209];
% k=ones(1,length(ideal)); k=.1*k;
% perr=cc2zp(ideal,lossy,k);
% [x,fval,exitflag, output]=fminsearch(@(lossy) cc2zp(ideal,lossy,k),lossy,
optimset('TolFun',0.001,'MaxIter',1000))
%% circuit 2 3rd Order Chebyshev
ideal=[3.348729; 0.7117;3.348729];
lossy=[3.348729; 0.7117;3.348729];
k=ones(1,length(ideal)); k=.1*k;
perr=cc2zp(ideal,lossy,k);
[x,fval,exitflag, output]=fminsearch(@(lossy) cc2zp(ideal,lossy,k),lossy,
optimset('TolFun',0.001,'MaxIter',1000))
%% circuit 3 5th Order Chebyshev
% ideal=[3.481289; 0.761919; 4.537546; 0.761919; 3.481287];
% lossy=[3.481289; 0.761919; 4.537546; 0.761919; 3.481287];
% k=ones(1,length(ideal)); k=.1*k;
% perr=cc2zp(ideal,lossy,k);
% [x,fval,exitflag, output]=fminsearch(@(lossy) cc2zp(ideal,lossy,k),lossy,
optimset('TolFun',0.001,'MaxIter',1000))
%% circuit 4 5th Order Butterworth
% ideal=[0.617741; 1.617266; 1.99905; 1.617266; 0.617741];
% lossy=[0.617741; 1.617266; 1.99905; 1.617266; 0.617741];
% k=ones(1,length(ideal)); k=.1*k;
% perr=cc2zp(ideal,lossy,k);
% [x,fval,exitflag, output]=fminsearch(@(lossy)
cc2zp(ideal,lossy,k),lossy,optimset('TolFun',0.001,'MaxIter',1000))
%% circuit 5 4th Order Bandpass
% ideal=[1.961008; 0.509942;1.231108; 0.812276; 0.812279; 1.231108; 0.509942;
1.961008];
% lossy=[1.961008; 0.509942;1.231108; 0.812276; 0.812279; 1.231108; 0.509942;
1.961008];
% k=ones(1,length(ideal)); k=.1*k;
% perr=cc2zp(ideal,lossy,k);
% [x,fval,exitflag, output]=fminsearch(@(lossy)
cc2zp(ideal,lossy,k),lossy,optimset('TolFun',0.1,'MaxIter',1000))
72
B Matlab Impedance Function: impedence.m
function [num,den]=Impedence(param)
type=param(1);
switch type
case 0 % pure resistor
R=param(2);
num=[0 R];
den=[0 1];
case 1 % L with lossy resistor
L=param(2); Rl=param(3);
num=[L Rl];
den=[0 1];
case 2 % C with lossy conductor
C=param(2); Gc=param(3);
num=[0 1];
den=[C Gc];
case 3 % L series with C
L=param(2); Rl=param(3);
C=param(4); Gc=param(5);
Lnum=[L Rl]; Lden=[0 1];
Cnum=[0 1]; Cden=[C Gc];
[num,den]=parallel(Lnum,Lden,Cnum,Cden);
case 4 % L parallel with C
L=param(2); Rl=param(3);
C=param(4); Gc=param(5);
Lnum=[L Rl]; Lden=[0 1];
Cnum=[0 1]; Cden=[C Gc];
[den,num]=parallel(Lden,Lnum,Cden,Cnum);
%%%%% other cases
%%%%% ......
end;
return;
73
C Matlab Error and Plotting Function: error.m
function [err]=cc2zp(ideal,lossy,k)
format compact;
%% 3rd order filter
[num1,den1]=Impedence([1,ideal(1),0]);
[den2,num2]=Impedence([2,ideal(2),0]);
[num3,den3]=Impedence([1,ideal(3),0]);
nX_id=[0 1;num1;num2;num3;0 1];
dX_id=[0 1;den1;den2;den3;0 1];
[num1,den1]=Impedence([1,lossy(1),k(1)]);
[den2,num2]=Impedence([2,lossy(2),k(2)]);
[num3,den3]=Impedence([1,lossy(3),k(3)]);
nX_ls=[0 1;num1;num2;num3;0 1];
dX_ls=[0 1;den1;den2;den3;0 1];
%% 5th order filter
% [num1,den1]=Impedence([1,ideal(1),0]);
% [den2,num2]=Impedence([2,ideal(2),0]);
% [num3,den3]=Impedence([1,ideal(3),0]);
% [den4,num4]=Impedence([2,ideal(4),0]);
% [num5,den5]=Impedence([1,ideal(5),0]);
%
% nX_id=[0 1;num1;num2;num3;num4;num5;0 1];
% dX_id=[0 1;den1;den2;den3;den4;den5;0 1];
%
%
% [num1,den1]=Impedence([1,lossy(1),k(1)]);
% [den2,num2]=Impedence([2,lossy(2),k(2)]);
% [num3,den3]=Impedence([1,lossy(3),k(3)]);
% [den4,num4]=Impedence([2,lossy(4),k(4)]);
% [num5,den5]=Impedence([1,lossy(5),k(5)]);
%
% nX_ls=[0 1;num1;num2;num3;num4;num5;0 1];
% dX_ls=[0 1;den1;den2;den3;den4;den5;0 1];
%% 4th order BandPass
% [den1,num1]=Impedence([4,ideal(2),0,ideal(1),1]);
% [num2,den2]=Impedence([3,ideal(4),0,ideal(3),0]);
% [den3,num3]=Impedence([4,ideal(6),0,ideal(5),0]);
% [num4,den4]=Impedence([3,ideal(8),0,ideal(7),0]);
% nX_id=[num1;num2;num3;num4;0 0 1];
% dX_id=[den1;den2;den3;den4;0 0 1];
%
% [den1,num1]=Impedence([4,lossy(2),k(2),lossy(1),1+k(1)]);
% [num2,den2]=Impedence([3,lossy(4),k(4),lossy(3),k(3)]);
% [den3,num3]=Impedence([4,lossy(6),k(6),lossy(5),k(5)]);
% [num4,den4]=Impedence([3,lossy(8),k(8),lossy(7),k(7)]);
%
% nX_ls=[num1;num2;num3;num4;0 0 1];
% dX_ls=[den1;den2;den3;den4;0 0 1];
%% Plotting Roots
[num_id,den_id]=Ttra_poly(nX_id,dX_id);
[num_ls,den_ls]=Ttra_poly(nX_ls,dX_ls);
%Diplay Numerator and Denminator
74
dispf(num_id,9,4,'num_id');
dispf(den_id,9,4,'den_id');
disp(' ');
dispf(num_ls,9,4,'num_ls');
dispf(den_ls,9,4,'den_ls');
sys_id=tf(num_id,den_id);
sys_ls=tf(num_ls,den_ls);
[z_id,p_id,k_id]=tf2zp(num_id,den_id);
[z_ls,p_ls,k_ls]=tf2zp(num_ls,den_ls);
% Plots
figure(1); clf;
rz_id=real(z_id);rp_id=real(p_id);
rz_ls=real(z_ls);rp_ls=real(p_ls);
iz_id=imag(z_id);ip_id=imag(p_id);
iz_ls=imag(z_ls);ip_ls=imag(p_ls);
plot(rz_id,iz_id,'b*','MarkerSize',12); hold on
plot(rp_id,ip_id,'r*','MarkerSize',12); hold on
plot(rz_ls,iz_ls,'r*','MarkerSize',12); hold on
plot(rp_ls,ip_ls,'b*','MarkerSize',12); hold on
ax=axis; ax(1)=-1.5; axis(ax); grid on
legend ('Ideal Roots', 'Lossy Roots')
res=256;
cof=10^(1/res);
for i=1:4*res+1,
if i==1, w(1)=0.01*j; else w(i)=w(i-1)*cof; end
y_id(i)=polyval(num_id,w(i))/polyval(den_id,w(i));
y_ls(i)=polyval(num_ls,w(i))/polyval(den_ls,w(i));
end;
figure(2); clf;
semilogx(abs(w),abs(y_id),'b','LineWidth',2.2); hold on;
semilogx(abs(w),abs(y_ls),'r','LineWidth',2.2); hold on;
ax=axis; ax(4)=0.55; axis(ax); grid on
legend ('Ideal Filter', 'Lossy Filter')
figure(10); clf;
loglog(abs(w),abs(y_id),'b','LineWidth',2.2); hold on;
loglog(abs(w),abs(y_ls),'r','LineWidth',2.2); hold on;
ax=axis; ax(4)=0.55; axis(ax); grid on
legend ('Ideal Filter', 'Lossy Filter')
% bode(sys_id,'b',sys_ls,'r',P);
[z_id,p_id,k_id]=tf2zp(num_id,den_id);
[z_ls,p_ls,k_ls]=tf2zp(num_ls,den_ls);
nz_id=length(z_id); np_id=length(p_id);
nz_ls=length(z_ls); np_ls=length(p_ls);
if np_id < np_ls
z_id=[zeros(np_ls-np_id,1);z_id];
p_id=[zeros(np_ls-np_id,1);p_id];
end;
75
%% Plot Coefficients
figure(7); clf;
plot(num_id(end:-1:1),'r*','MarkerSize',12); hold on
plot(num_ls(end:-1:1),'*','MarkerSize',12); hold on
title('Numerator');
xlabel('Order (s^n)');
ylabel('Coefficient Value');
legend ('Ideal Filter', 'Lossy Filter')
figure(6); clf;
plot(den_id(end:-1:1),'r*','MarkerSize',12); hold on
plot(den_ls(end:-1:1),'*','MarkerSize',12); hold on
title('Denominator');
xlabel('Order (s^n)');
ylabel('Coefficient Value');
legend ('Ideal Coefficient', 'Lossy Coefficient')
%% Error calculated from Coef
for n=1:nz_ls
err=err+abs(num_id(n)-num_ls(n))^2;
end;
for n=1:np_ls
err=err+abs(den_id(n)-den_ls(n))^2;
end;
err
%% Error calculated from Roots
% for n=1:nz_ls
% err=err+abs(z_id(n)-z_ls(n))^2;
% end;
% for n=1:np_ls
% err=err+abs(p_id(n)-p_ls(n))^2;
% end;
% err
return;
76
D Matlab Transfer Function: Ttra_poly.m
function [num,den]=Ttra_poly(nX,dX)
[ne,k]=size(nX);
nUpp=1; dUpp=1;
[nUp,dUp]=mulRat(nUpp,dUpp,nX(1,:),dX(1,:));
for i=3:ne+1
[ntemp,dtemp]=mulRat(nUp,dUp,nX(i-1,:),dX(i-1,:));
[nU,dU]=addRat(nUpp,dUpp,ntemp,dtemp);
nUpp=nUp;dUpp=dUp;nUp=nU; dUp=dU;
end
num=dU; den=nU;
[num,den]=remrt(num,den);
return;
function [n,d]=mulRat(n1,d1,n2,d2)
n=conv(n1,n2);
d=conv(d1,d2);
d=rem0Poly(d); n=rem0Poly(n);
function [n,d]=addRat(n1,d1,n2,d2)
% add two rational functions
% n=d1*n2+d2*n1 d=d1*d2
% conv - multiplication of two polynomials
n=addPoly(conv(d1,n2),conv(d2,n1));
d=conv(d1,d2);
E 2nd Order low-pass filter symbolic transfer function
g2201g2779 g2201 g2778
L2*C1 L2+C1 2
F 3rd Order low-pass filter symbolic transfer function
g2201g2780 g2201g2779 g2201 g2778
C3*L2*C1 C3*L2+L2*C1 C1+C3+L2 2
77
G 4th Order low-pass filter symbolic transfer function
g2201g2781 g2201g2780 g2201g2779 g2201 g2778
L4*C3*L2*C1 L4*C3*L2+C3*L2*C1 L2*C1+L4*C1+L4*C3+C3*L2 L2+L4+C1+C3 2
H 5th Order low-pass filter symbolic transfer function
g2201g2782 g2201g2781 g2201g2780 g2201g2779 g2201 g2778
C5*L4*C3*
L2*C1
C5*L4*C3*L2+L
4*C3*L2*C1
C3*L2*C1+C5*L2*C1+C5*L4
*C1+C5*L4*C3+L4*C3*L2
C3*L2+C5*L2+C5*L4+L2*
C1+L4*C1+L4*C3
C1+C3+C
5+L2+L4
2
I 6th Order low-pass filter symbolic transfer function
g2201g2783 g2201g2782 g2201g2781 g2201g2780 g2201g2779 g2201 g2778
L6*C5*L4*
C3*L2*C1
L6*C5*L4*
C3*L2+C5*
L4*C3*L2*
C1
L4*C3*L2*C1+L
6*C3*L2*C1+L6
*C5*L2*C1+L6*
C5*L4*C1+L6*C
5*L4*C3+C5*L4
*C3*L2
L4*C3*L2+L6*C3
*L2+L6*C5*L2+L
6*C5*L4+C3*L2*
C1+C5*L2*C1+C5
*L4*C1+C5*L4*C
3
L2*C1+L4*C1+L4*C
3+L6*C1+L6*C3+L6
*C5+C3*L2+C5*L2+
C5*L4
L2+L4+L6+C1+
C3+C5
2
78
J 7th Order low-pass filter symbolic transfer function
g2201g2784 g2201g2783 g2201g2782 g2201g2781 g2201g2780 g2201g2779 g2201 g2778
C7*L6*C5*
L4*C3*L2*
C1
C7*L6*C5*
L4*C3*L2+L
6*C5*L4*C
3*L2*C1
C5*L4*C3*L2*C
1+C7*L4*C3*L2
*C1+C7*L6*C3*
L2*C1+C7*L6*C
5*L2*C1+C7*L6
*C5*L4*C1+C7*
L6*C5*L4*C3+L
6*C5*L4*C3*L2
C5*L4*C3*L2+C
7*L4*C3*L2+C7
*L6*C3*L2+C7*
L6*C5*L2+C7*L
6*C5*L4+L4*C3
*L2*C1+L6*C3*
L2*C1+L6*C5*L
2*C1+L6*C5*L4
*C1+L6*C5*L4*
C3
C3*L2*C1+C5
*L2*C1+C5*L4
*C1+C5*L4*C
3+C7*L2*C1+
C7*L4*C1+C7
*L4*C3+C7*L6
*C1+C7*L6*C
3+C7*L6*C5+L
4*C3*L2+L6*C
3*L2+L6*C5*L
2+L6*C5*L4
C3*L2+C5*L
2+C5*L4+C7
*L2+C7*L4+
C7*L6+L2*C
1+L4*C1+L4
*C3+L6*C1+
L6*C3+L6*C
5
C1+C3+C5+
C7+L2+L4+
L6
2
K 8th Order low-pass filter symbolic transfer function
g2201g2785 g2201g2784 g2201g2783 g2201g2782 g2201g2781 g2201g2780 g2201g2779 g2201 g2778
L8*C7*L
6*C5*L4
*C3*L2*
C1
L8*C7*L6
*C5*L4*C
3*L2+C7*
L6*C5*L4
*C3*L2*C
1
L6*C5*L4*C3*
L2*C1+L8*C5*
L4*C3*L2*C1+
L8*C7*L4*C3*
L2*C1+L8*C7*
L6*C3*L2*C1+
L8*C7*L6*C5*
L2*C1+L8*C7*
L6*C5*L4*C1+
L8*C7*L6*C5*
L4*C3+C7*L6*
C5*L4*C3*L2
L6*C5*L4*C3*L
2+L8*C5*L4*C
3*L2+L8*C7*L4
*C3*L2+L8*C7
*L6*C3*L2+L8*
C7*L6*C5*L2+L
8*C7*L6*C5*L
4+C5*L4*C3*L
2*C1+C7*L4*C
3*L2*C1+C7*L
6*C3*L2*C1+C
7*L6*C5*L2*C
1+C7*L6*C5*L
4*C1+C7*L6*C
5*L4*C3
L4*C3*L2*C1+
L6*C3*L2*C1+
L6*C5*L2*C1+
L6*C5*L4*C1+
L6*C5*L4*C3+
L8*C3*L2*C1+
L8*C5*L2*C1+
L8*C5*L4*C1+
L8*C5*L4*C3+
L8*C7*L2*C1+
L8*C7*L4*C1+
L8*C7*L4*C3+
L8*C7*L6*C1+
L8*C7*L6*C3+
L8*C7*L6*C5+
C5*L4*C3*L2+
C7*L4*C3*L2+
C7*L6*C3*L2+
C7*L6*C5*L2+
C7*L6*C5*L4
L4*C3*L2+L
6*C3*L2+L6
*C5*L2+L6*
C5*L4+L8*C
3*L2+L8*C5
*L2+L8*C5*
L4+L8*C7*L
2+L8*C7*L4
+L8*C7*L6+
C3*L2*C1+C
5*L2*C1+C5
*L4*C1+C5*
L4*C3+C7*L
2*C1+C7*L4
*C1+C7*L4*
C3+C7*L6*C
1+C7*L6*C3
+C7*L6*C5
L2*C1+L4*
C1+L4*C3+
L6*C1+L6*
C3+L6*C5+
L8*C1+L8*
C3+L8*C5+
L8*C7+C3*
L2+C5*L2+
C5*L4+C7*
L2+C7*L4+
C7*L6
L2+L4+L6+L8+
C1+C3+C5+C7
2
79
L 9th Order low-pass filter symbolic transfer function
g2201g2786 g2201g2785 g2201g2784 g2201g2783 g2201g2782 g2201g2781 g2201g2780 g2201g2779 g2201 g2778
C9*L8*C
7*L6*C5
*L4*C3*
L2*C1
C9*L8
*C7*L
6*C5*
L4*C3
*L2+L
8*C7*
L6*C5
*L4*C
3*L2*
C1
C7*L6*C5*
L4*C3*L2*
C1+C9*L6*
C5*L4*C3*
L2*C1+C9*
L8*C5*L4*
C3*L2*C1+
C9*L8*C7*
L4*C3*L2*
C1+C9*L8*
C7*L6*C3*
L2*C1+C9*
L8*C7*L6*
C5*L2*C1+
C9*L8*C7*
L6*C5*L4*
C1+C9*L8*
C7*L6*C5*
L4*C3+L8*
C7*L6*C5*
L4*C3*L2
C7*L6*C5*L4*
C3*L2+C9*L6*
C5*L4*C3*L2+
C9*L8*C5*L4*
C3*L2+C9*L8*
C7*L4*C3*L2+
C9*L8*C7*L6*
C3*L2+C9*L8*
C7*L6*C5*L2+
C9*L8*C7*L6*
C5*L4+L6*C5*L
4*C3*L2*C1+L
8*C5*L4*C3*L
2*C1+L8*C7*L
4*C3*L2*C1+L
8*C7*L6*C3*L
2*C1+L8*C7*L
6*C5*L2*C1+L
8*C7*L6*C5*L
4*C1+L8*C7*L
6*C5*L4*C3
C9*L4*C3*L2*C1
+C9*L6*C3*L2*C
1+C9*L6*C5*L2*
C1+C9*L6*C5*L4
*C1+C9*L6*C5*L
4*C3+C9*L8*C3*
L2*C1+C9*L8*C5
*L2*C1+C9*L8*C
5*L4*C1+C9*L8*
C5*L4*C3+C9*L8
*C7*L2*C1+C9*L
8*C7*L4*C1+C9*
L8*C7*L4*C3+C9
*L8*C7*L6*C1+C
9*L8*C7*L6*C3+
C9*L8*C7*L6*C5
+C7*L4*C3*L2*C
1+C7*L6*C3*L2*
C1+C7*L6*C5*L2
*C1+C7*L6*C5*L
4*C1+C7*L6*C5*
L4*C3+C5*L4*C3
*L2*C1+L8*C5*L
4*C3*L2+L8*C7*
L4*C3*L2+L8*C7
*L6*C3*L2+L8*C
7*L6*C5*L2+L8*
C7*L6*C5*L4+L6
*C5*L4*C3*L2
C5*L4*C3*L2+
L8*C7*L4*C3+
L8*C7*L6*C1+
L8*C7*L6*C3+
L8*C7*L6*C5+
C7*L4*C3*L2+
C7*L6*C3*L2+
C7*L6*C5*L2+
C7*L6*C5*L4+
C9*L4*C3*L2+
C9*L6*C3*L2+
C9*L6*C5*L2+
C9*L6*C5*L4+
C9*L8*C3*L2+
C9*L8*C5*L2+
C9*L8*C5*L4+
C9*L8*C7*L2+
C9*L8*C7*L4+
C9*L8*C7*L6+
L4*C3*L2*C1+
L6*C3*L2*C1+
L6*C5*L2*C1+
L6*C5*L4*C1+
L6*C5*L4*C3+
L8*C3*L2*C1+
L8*C5*L2*C1+
L8*C5*L4*C1+
L8*C5*L4*C3+
L8*C7*L2*C1+
L8*C7*L4*C1
C3*L2*C1+
L4*C3*L2+
L6*C5*L2+
L6*C5*L4+
C9*L2*C1+
C9*L4*C1+
C9*L4*C3+
C9*L6*C1+
C9*L6*C3+
C9*L6*C5+
C9*L8*C1+
C9*L8*C3+
C9*L8*C5+
C9*L8*C7+
L8*C3*L2+
L8*C5*L2+
L8*C5*L4+
L8*C7*L2+
L8*C7*L4+
L8*C7*L6+
L6*C3*L2+
C5*L2*C1+
C5*L4*C1+
C5*L4*C3+
C7*L2*C1+
C7*L4*C1+
C7*L4*C3+
C7*L6*C1+
C7*L6*C3+
C7*L6*C5
C3*L2+C
5*L2+C5
*L4+C7*
L2+C7*L
4+C7*L6
+C9*L2+
C9*L4+C
9*L6+C9
*L8+L2*
C1+L4*C
1+L4*C3
+L6*C1+
L6*C3+L
6*C5+L8
*C1+L8*
C3+L8*C
5+L8*C7
C1+C3+
C5+C7+
C9+L2+
L4+L6+
L8
2