Wide Range Tunable Transconductance Filters
Except where reference is made to the work of others, the work described in this
thesis is my own or was done in collaboration with my advisory committee. This
thesis does not include proprietary or classified information.
Matthew Anderson
Certificate of Approval:
John Y. Hung
Professor
Electrical and Computer Engineering
Bogdan M. Wilamowski, Chair
Professor
Electrical and Computer Engineering
Thaddeus A. Roppel
Associate Professor
Electrical and Computer Engineering
Joe F. Pittman
Interim Dean
Graduate School
Wide Range Tunable Transconductance Filters
Matthew Anderson
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 10, 2008
Wide Range Tunable Transconductance Filters
Matthew Anderson
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Matthew Anderson, the son of Phillip and Cathy Springfield, was born on May
23, 1983. Upon graduating Summa cum Laude in Electrical Engineering at Auburn
University in Spring 2006 he accepted a job at Archangel Systems Inc., also in Auburn.
Since graduation he has worked under his advisor, Dr. Bogdan Wilamowski, to
complete the requirements for the Masters program.
iv
Thesis Abstract
Wide Range Tunable Transconductance Filters
Matthew Anderson
Master of Science, May 10, 2008
(B.S., Auburn University, 2006)
62 Typed Pages
Directed by Bogdan Wilamowski
Generation of filters using operational transconductance amplifiers and capaci-
tors (OTA-C) has become more and more popular over recent years. This is due to
the need for high quality analog filters that can be implemented on a standard CMOS
integrated circuit. While operational amplifier based active filters have served this
purpose in the past, the OTA-C filter has advantages over a classic amplifier design.
A simple yet effective method of analytical synthesis was developed that makes use
of ladder circuits, which are widely understood. Given the ladder circuit of a filter
of any order, an equivalent OTA-C filter can be easily synthesized. The synthesis
process is shown for both a low pass prototype filter, and its band pass transforma-
tion. Higher order filters can easily be synthesized by extending the design process
described. An OTA was developed which was designed specifically to address the
complications involved in implementing filters with high quality factors. The design
of a band pass filter is then implemented in CMOS with the OTA to show the ef-
fect of non-ideal elements. The CMOS filter implementation is capable of achieving
v
a quality factor of 20 up to frequencies of 100 kHz. The reference current can be
used to tune the filter center frequency linearly over three decades. This flexibility
in center frequency tuning can be used to compensate for process variations or for
creating general purpose filters with constant passive element values.
vi
Acknowledgments
I would like to express my deepest appreciation to my advisor, Dr. Wilamowski.
Without his patience and guidance I would not be in the position I am today. From
him I have learned a multitude of things, of which, engineering is only the tip of the
iceberg.
vii
Style manual or journal used Journal of Approximation Theory (together with the
style known as ?aums?). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically
LATEX) together with the departmental style-file aums.sty.
viii
Table of Contents
List of Figures x
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Transconductance Filters 5
2.1 Low Pass Filter Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 LC Ladder Circuit . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 OTA-C Transformation . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Band Pass Filter Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 LC Ladder Circuit . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 OTA-C Transformation . . . . . . . . . . . . . . . . . . . . . . 10
3 OTA?s Implemented in VLSI 14
3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Differential Pair OTA . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Mirrored Cascode OTA . . . . . . . . . . . . . . . . . . . . . . 18
3.2.3 Mirrored Cascode OTA With Improved Current Mirror . . . . 21
4 Tunable Filters in VLSI 27
4.1 Implementation of Low Pass Filter . . . . . . . . . . . . . . . . . . . 27
4.2 Implementation of Band Pass Filter . . . . . . . . . . . . . . . . . . . 30
4.3 Implementation of Higher Order Filter . . . . . . . . . . . . . . . . . 35
5 Conclusion 43
Bibliography 45
Appendices 47
A SPICE level 7 NMOS model for AMIS 0.5 ?m process 48
B SPICE level 7 PMOS model for AMIS 0.5 ?m process 50
ix
List of Figures
2.1 Low pass filter LC implementation . . . . . . . . . . . . . . . . . . . 5
2.2 Frequency response of 3rd order prototype Chebyshev LPF . . . . . . 7
2.3 Low pass filter OTA implementation with input and termination resis-
tances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Low pass filter OTA implementation without input and termination
resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Band pass filter ladder circuit . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Frequency response of 3rd order prototype Chebyshev BPF . . . . . . 13
2.7 3rd order prototype Chebyshev BPF OTA implementation . . . . . . 13
3.1 Simple OTA implementation . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Transconductance vs reference current for the OTA shown in Figure 3.1 16
3.3 Output Resistance vs reference current for the OTA shown in Figure 3.1 16
3.4 OTA with Cascode current mirror . . . . . . . . . . . . . . . . . . . . 18
3.5 Transconductance vs reference current for the OTA shown in Figure 3.4 19
3.6 Output Resistance vs reference current for the OTA shown in Figure 3.4 19
3.7 AC sweep of transconductance for OTA in Figure 3.4 . . . . . . . . . 20
3.8 Two input OTA implementation . . . . . . . . . . . . . . . . . . . . . 23
3.9 Transconductance vs reference current for the OTA shown in Figure 3.8 24
3.10 Output Resistance vs reference current for the OTA shown in Figure 3.8 24
x
3.11 AC sweep of transconductance for OTA in Figure 3.8 . . . . . . . . . 25
3.12 Output current verses differential input voltage for Iref=5 nA . . . . 26
4.1 Third order Chebyshev low pass filter with Fc of 1 kHz . . . . . . . . 28
4.2 Third order Chebyshev low pass filter for five reference currents . . . 29
4.3 Reference current vs LPF critical frequency . . . . . . . . . . . . . . 30
4.4 Third order Chebyshev band pass with Fc = 1 kHz, Q=1 . . . . . . . 32
4.5 Third order Chebyshev band pass with Fc = 1 kHz, Q=20 . . . . . . 33
4.6 Exploded view of third order Chebyshev band pass filter with Fc =
1 kHz, Q=20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.7 Third order Chebyshev band pass filter for four reference currents . . 36
4.8 Reference current vs BPF center frequency . . . . . . . . . . . . . . . 37
4.9 5th order band pass filter ladder circuit . . . . . . . . . . . . . . . . . 38
4.10 5th order prototype Chebyshev BPF OTA implementation . . . . . . 38
4.11 Fifth order Chebyshev band pass filter centered at 10 kHz . . . . . . 40
4.12 Third order Chebyshev band pass filter centered at 10 kHz . . . . . . 41
4.13 Fifth order Chebyshev band pass filter ranging 1 kHz to 10 kHz . . . 42
xi
Chapter 1
Introduction
1.1 Motivation
The Operational Transconductance Amplifier (OTA) finds a wide range of uses in
the field of analog electronics. Over the past two decades, it has become increasingly
popular as a building block in analog filters. These OTA capacitor (OTA-C) filters
have several advantages over the more typical operational amplifier configurations.
These advantages include:
? Fully Integrated Implementation: To create a quality fully integrated fil-
ter a few design constraints must be added. For instance, it is a benefit
to have only grounded capacitors in the filter design. There are two rea-
sons for this design constraint; most importantly, grounded capacitors can
be implemented in a smaller area than their floating counterparts. Also,
grounded capacitors reduce the impact of shunt capacitances that can se-
riously affect the final transfer function.
? Electronic Filter Tuning: Because the transconductance (gm) of an OTA is
used as a design parameter, the filter response can be modified by changing
thegm of various OTA?s. In practice this can be accomplished with a tuning
current or voltage that sets the bias level of the OTA. Electronic tuning is
necessary to overcome process variations. As an added bonus, if the filter
1
center frequency can be varied over a wide band, a generally designed filter
can find use in many applications.
? High Frequency Operation: Although not addressed in detail in this work,
it is a fact that the design of a high frequency OTA is more easily accom-
plished than a high frequency operational amplifier. This is due to the
fact that the OTA does not require the output stage of a typical opera-
tional amplifier. Even with the second order effects that must be consid-
ered, bandpass filters have been reported that can operate in the gigahertz
range.
Despite their advantages, OTA-C filters are often overlooked by design engineers
because they are not well understood. They are perceived as hard to design because
literature on the subject is often either ambiguous or unnecessarily complicated. In
this work, transconductance filters are introduced and explained from the point of
view of the passive ladder filter circuit. The ladder point of view is preferred because
of its popularity and ease of design [2]. Two main techniques exist in the literature to
transform a ladder circuit into its OTA-C implementation and they have been shown
to be theoretically equivalent [4]. The first technique, known as element replacement,
involves using capacitors directly and substituting inductors with an equivalent circuit
that has an inductive input impedance. The other method is referred to as operational
simulation, or signal flow graph method. In operational simulation, the node voltages
and mesh currents of the ladder circuit are simulated by summations and the ladder
2
elements are simulated by integration operators. For the purposes of this paper the
operational simulation method will be used because of its ease of understanding from
a set of state equations or transfer function.
A bulk of research has been dedicated to lower order OTA-C filter structures [5]
- [6]. This research is done with the idea that low order sections can be used to build
high order filters. One problem with this approach is the loss of generality. Once a
second order section is built as a design block, all implementable transfer functions
are based on that block. Another problem is the sensitivity of these blocks to element
values. Since the design assumes each section is identical, any difference in element
values can have serious impacts on the filter response. A better approach is to develop
equations for the circuit to be implemented, and match those equations with only the
OTA and capacitors as building blocks. In this way any order ladder circuit can be
synthesized. Less research is devoted to this concept [8] - [9], and as such it is the
crux of this work.
1.2 Thesis Outline
In the early chapters, ideal OTA?s will be used to support the theory of the filter
synthesis. Chapter 2 will introduce the process and show examples of a low pass and
band pass prototype. Chapter 3 will show the OTA design process. The design will
start with a simple OTA and motivations for improving the OTA will be discussed.
The final OTA design will be developed in SPICE using CMOS 0.5 ?m technology.
Finally, in Chapter 4, the CMOS OTA will be substituted for the ideal one to show
3
the non-ideal effects. In the simpler case of the low pass filter, the non-ideal impacts
will be minor. However, band pass filters with high quality factors can tax even the
most carefully designed amplifiers.
4
Chapter 2
Transconductance Filters
2.1 Low Pass Filter Synthesis
2.1.1 LC Ladder Circuit
Figure 2.1: Low pass filter LC implementation
The implementation of a third order Chebyshev low pass filter is sufficient to
show the design process. The ladder circuit for such a filter, designed for minimum
capacitance, is shown in Figure 2.1. For simplicity, values for the passive elements
are chosen to set the cut-off frequency at 1 Hz and are shown in table 2.1. The
element values were calculated using Matlab software written for ladder circuits [3].
The frequency response of such a circuit is shown in Figure 2.2.
5
Element Capacitance
L1 0.352 F
C2 0.173 F
L3 0.352 F
Table 2.1: Capacitor values for low pass filter prototype
2.1.2 OTA-C Transformation
Using the current and voltage conventions defined in Figure 2.1, one can easily
write the state space equations for this particular circuit. In this case the state
variables have been chosen such that each state variable corresponds to the output
of an OTA.
?i1 = 1
L1(Vin ?Rini1 ?v2)
?v2 = 1C
2
(i1 ?i3) (2.1)
?i3 = 1
L3(v2 ?Routi3)
The transfer function generated from this system of equations is
Vout
Vin =
Rout
L1C2L3s3 + (L1C2Rout +C2L3Rin)s2 + (C2RinRout +L1 +L3)s +Rin +Rout
(2.2)
If the equations shown in equation set (2.1) are used to implement an OTA-C filter,
the circuit would be as shown in Figure 2.4. Analyzing the circuit in Figure 2.4
with the OTA output voltages as state variables, the state equations become those
6
Figure 2.2: Frequency response of 3rd order prototype Chebyshev LPF
shown in equation set (2.3) assuming an equal transconductance for each OTA. The
corresponding transfer function is shown in (2.4).
?v1 = gmL
1
(Vin ?v1 ?v2)
?v2 = gmC
2
(v1 ?v3) (2.3)
?v3 = gmL
3
(v2 ?v3)
7
Vout
Vin =
g3m
L1C2L3s3 + (L1C2 +C2L3)gms2 + (C2 +L1 +L3)g2ms + 2g3m (2.4)
The comparison of equations (2.2) and (2.4) reveals some interesting consequences of
the OTA-C implementation. First, the passive component values (capacitors) must
be scaled by the value of gm to achieve the same filter critical frequency. Secondly,
the transfer function no longer depends on Rin and Rout. This can best be explained
by observing that an OTA with negative feedback works as a resistor. If there is a
resistor in the feedback path, as in Rin and Rout, its value does not impact the output
voltage because the output current will change with the resistance. For a large load,
as would be the case for an operational amplifier input, Rin and Rout can be replaced
with shorts.
Figure 2.3 shows the OTA-C circuit using a four input OTA where necessary.
Using ideal components in SPICE, the frequency response of the circuit is exactly the
same as the LC ladder response shown in Figure 2.2. This is to be expected since, at
least for the ideal case, the two transfer functions are equivalent.
2.2 Band Pass Filter Synthesis
2.2.1 LC Ladder Circuit
Using the standard ladder circuit filter transformations, it is easy to determine
the bandpass filter corresponding to the low pass filter from the previous section.
8
Figure 2.3: Low pass filter OTA implementation with input and termination resis-
tances
This is the circuit shown in Figure 2.5. As before, the values of the elements have
been chosen to obtain a center frequency of 1 Hz. The frequency response for this
filter is shown in Figure 2.6.
Element Capacitance
L1 0.322 F
C1 0.079 F
L2 0.160 F
C2 0.158 F
L3 0.322 F
C3 0.079 F
Table 2.2: Capacitor values for band pass filter prototype
9
Figure 2.4: Low pass filter OTA implementation without input and termination re-
sistances
2.2.2 OTA-C Transformation
One can write the circuit equations as before using the same type of conventions
on Figure 2.5. These results are shown in equation set (2.5).
10
Figure 2.5: Band pass filter ladder circuit
?i1 = 1
L1(Vin ?i1Rin ?v2 ?v4)
?v2 = 1C
1
(i1)
?i3 = 1
L2(v4)
?v4 = 1C
2
(i1 ?i3 ?i5) (2.5)
?i5 = 1
L3(v4 ?v6 ?i5Rout)
?v6 = 1C
3
(i5)
However, the band pass filter presents a slightly more complicated situation than
the low pass. The difficulty arises from the equation for ?i1. Although one can easily
fabricate a four input OTA, it will obviously have two positive inputs and two negative
inputs. Using the equations from equation (2.5) will result in the OTA corresponding
to ?i1 needing three negative inputs. To remedy this situation, it is necessary to
11
manipulate the signs of the variables. It can be clearly seen that if ?v2 were positive
in the equation for ?i1 then a standard four input OTA could be used in the circuit.
This being the case, changing the sign of v2 everywhere it occurs in equation set 2.5
results in equation set 2.6. Clearly, this solution is not unique.
?i1 = 1
L1(Vin ?i1Rin +v2 ?v4)
?v2 = 1C
1
(?i1)
?i3 = 1
L2(v4)
?v4 = 1C
2
(i1 ?i3 ?i5) (2.6)
?i5 = 1
L3(v4 ?v6 ?i5Rout)
?v6 = 1C
3
(i5)
The idealized circuit corresponding to equation set (2.6) is shown in Figure 2.7 with
Rin and Rout replaced with shorts. As in the case of the low pass circuit, the filter
response is exactly the same as show in Figure 2.6.
12
Figure 2.6: Frequency response of 3rd order prototype Chebyshev BPF
Figure 2.7: 3rd order prototype Chebyshev BPF OTA implementation
13
Chapter 3
OTA?s Implemented in VLSI
3.1 Approach
When implementing an OTA in VLSI it is important to consider the end use of
the circuit. This is done in order to decrease the non-ideal characteristics of the OTA
that are most significant to the end application. In the case of the filter application,
the finite output resistance of the OTA can seriously degrade the shape of the desired
filter, especially at higher quality factors [12]. Due to this fact, high output impedance
must be a focus in the OTA design process.
Another necessary consideration that results from the filtering application is the
question of tuning. It is clear from the transfer functions derived in Chapter 2 that the
critical frequency of the filter will change linearly with transconductance. Therefore,
it is advantageous to derive a method of tuning transconductance linearly. If this is
accomplished, adjustments for process variations are much easier to implement.
3.2 Implementation
3.2.1 Differential Pair OTA
The simplest implementation of a two input OTA is a differential pair like the one
shown in Figure 3.1. The current at the output node, Iout, depends on the difference
in the two input voltages and the transconductance is determined by transistor M1 (or
14
+ -
Iref
M2
M1
M4
M3
Vcc
Vee
Vee
V i n +
V i n -
Iout
Vee
Figure 3.1: Simple OTA implementation
15
Figure 3.2: Transconductance vs reference current for the OTA shown in Figure 3.1
Figure 3.3: Output Resistance vs reference current for the OTA shown in Figure 3.1
16
M2). Simulating in SPICE using 0.5 ?m process models, transconductance varies with
current as shown in Figure 3.2. The input transistors are operated in subthreshold
conduction. This is done to keep transconductance linearly tunable and to achieve
larger output resistances. If the amplifier were operated outside the subthreshold
conduction region, the transconductance would be described by equation (3.1).
gm =
radicalbig
2KnId (3.1)
However, in subthreshold conduction the transconductance follows equation (3.2),
where ? in this case is around two.
gm = Id?V
t
(3.2)
The drawback is that the amplifier will not work as well at higher frequencies. How-
ever, to achieve higher quality factors the tradeoff is necessary. For filter design, a
large transconductance value is not as important as a large output resistance. Figure
3.3 shows how the output resistance of the differential pair varies with bias current
which follows equation (3.3).
Rout = ro?pmos bardbl ro?nmos (3.3)
Even for the smallest bias currents, the output resistance is marginal. A more ad-
vanced design must be considered for filtering applications.
17
3.2.2 Mirrored Cascode OTA
M1
M2
M3
M4
M5
M6
M7 M9
M8 M10 M12 M14
M11 M13
+ -
Iref
Vee
Vee
Vcc Vee
V i n -
V i n +
Out
Figure 3.4: OTA with Cascode current mirror
A cascode structure is a good next step since higher output impedances can
obtained without sacrificing transconductance. The OTA structure in Figure 3.4 is a
mirrored cascode [13], with the mirror being a cascode current mirror. The operation
of the circuit is similar as before, but this time the currents produced by the input
transistors are both mirrored to the output. This lets the output resistance of the
18
Figure 3.5: Transconductance vs reference current for the OTA shown in Figure 3.4
Figure 3.6: Output Resistance vs reference current for the OTA shown in Figure 3.4
19
Figure 3.7: AC sweep of transconductance for OTA in Figure 3.4
20
OTA match that of the current mirror which is described by equation (3.4).
Rout = ro?nmos(1 + ro?nmosr
m?noms
) bardbl ro?pmos(1 + ro?pmosr
m?pmos
) (3.4)
Figure 3.5 shows the simulation results for transconductance closely resemble those
shown for the simpler differential pair. However, Figure 3.6 indicates that the output
resistance has increased by a factor of a thousand.
Although the mirrored cascode OTA will perform adequately in a filter applica-
tions [14], improvements can still be made. The drawback of using cascode architec-
tures in OTA design comes from the impact of the non-dominant poles. In the case of
the mirrored cascode, the non-dominant poles of interest are located at the input side
of the current mirrors [13]. If the non-dominant poles are too close to the dominant
pole, the filter transfer function can be distorted at higher quality factors or higher
frequencies [12]. Figure 3.7 shows a plot of the transconductance verses frequency.
The constant roll off shown indicates the non-dominant poles are very close to the
dominant one.
3.2.3 Mirrored Cascode OTA With Improved Current Mirror
To push out the non-dominant poles, an attempt must be made to decrease
capacitances at the nodes of interest. In this case, this occurs at the input sides
of the current mirrors. The cascode current mirrors used in Figure 3.4 have two
gate capacitances connected connected to the input node of the current mirror. The
21
frequency response could be improved if only one gate capacitance was necessary to
achieve the same output resistance. This problem can be solved with an improved
current mirror. The current mirror chosen, presented by Zeki and Kuntman, is known
as IAFCCM [10]. Figure 3.8 shows the mirrored cascode circuit implementing the
IAFCCM. Figures 3.9 and 3.10 show the simulation results for the circuit. The
transconductance is about the same as before, and the output resistance is actually
improved by a factor of ten. However, the main impact of the improved design is
shown in Figure 3.11. The frequency response of the OTA is improved, but more
importantly, the non-dominant pole is pushed to higher frequencies.
22
+ - idc
IrefM1
M2
M3
M4
M5
M6
M7 M9
M11
M13M
15
M17 M
19
M8 M10
M12
M14M16
M18 M20 M22
M21 M
23
M24
M25
M26
M27
M28
M29
Vee
Vee
Vcc
Vee
V c c
V e e
Vcc
V i n +
V i n -
VeeIoutVcc
Vee
Figure 3.8: Two input OTA implementation
23
Figure 3.9: Transconductance vs reference current for the OTA shown in Figure 3.8
Figure 3.10: Output Resistance vs reference current for the OTA shown in Figure 3.8
24
Figure 3.11: AC sweep of transconductance for OTA in Figure 3.8
25
Figure 3.12: Output current verses differential input voltage for Iref=5 nA
26
Chapter 4
Tunable Filters in VLSI
4.1 Implementation of Low Pass Filter
The low pass filter prototype developed in section 2.1 will now be implemented
using the OTA developed in section 3.2.3. The prototype was developed for a critical
frequency of 1 Hz and assuming a transconductance of 1 S. Therefore, the passive
elements must be scaled to account for more realistic OTA transconductances as
well as higher critical frequencies. The values shown in table 4.1 were chosen to
produce a critical frequency of 1 kHz at a reference current of 0.5 nF. These values
were calculated using equation (4.1) with Gm equal to the OTA transconductance at
0.5 nF, 6.92 nS, and Fnew equal to the desired critical frequency, 1 kHz.
Cnew = Cprototype ?GmF
new
(4.1)
The frequency response of this filter can be seen in Figure 4.1. Other than the critical
frequency, the response is identical to the ideal one presented earlier in Figure 2.2.
Element Capacitance
L1 2.43 pF
C2 1.19 pF
L3 2.43 pF
Table 4.1: Capacitor values for low pass filter
27
Figure 4.1: Third order Chebyshev low pass filter with Fc of 1 kHz
Since the transconductance of each OTA helps determine the transfer function of
the filter, the reference currents can be used to change the filter characteristics. There
are many benefits to this flexibility. For one, the individual currents of each OTA
can be adjusted to compensate for errors in the capacitor values caused by process
variations. On the other hand, if all the OTA reference currents are changed together,
the result is a shift in the filter?s critical frequency. Figure 4.2 shows the same filter
for increasing values of reference current.
It is clear that many applications would benefit from the ability to change filter
critical frequencies quickly and easily. For instance, the designed filter can sweep the
28
Figure 4.2: Third order Chebyshev low pass filter for five reference currents
entire range of audio frequencies with a simple change of reference current. Figure 4.2
indicates that the low pass filter critical frequency can be placed anywhere between
10 Hz and 600 kHz without adjusting passive element values. For much of this range,
filter critical frequency depends linearly on the reference current. Figure 4.3 shows a
plot of reference current verses critical frequency for the filters shown in Figure 4.2.
The linearity degrades only slightly at the reference current extremes. Linearity can
only be maintained for certain biasing conditions of the input transistors of the OTA.
In other words, there is a trade-off between the range of the filter and the linearity of
the critical frequency tuning. On the plus side, as reference current increases so does
29
the maximum operating frequency of the OTA. This allows the filter to consume only
the amount of power necessary to operate at a given frequency.
Figure 4.3: Reference current vs LPF critical frequency
4.2 Implementation of Band Pass Filter
Due to its added complexity, the band pass filter implementation will show more
clearly the differences between the VLSI implementation and the ideal. As was the
case with the low pass filter, the elements must once again be scaled based on the
frequency and transconductance. However, with the band pass filter the quality factor
must also be considered when choosing the element values. For a quality factor of 1,
equation (4.1) can still be used. If designed to have a center frequency of 1 kHz at a
30
Element Capacitance
L1 21.13 pF
C1 5.16 pF
L2 10.38 pF
C2 10.50 pF
L3 21.13 pF
C3 5.16 pF
Table 4.2: Capacitor values for band pass filter with Q=1
reference current of 5 nA, equation (4.1) yields the capacitances shown in table 4.2.
Simulating with these capacitances results in the frequency response shown in Figure
4.4.
To design for higher quality factors, the equations for capacitor values must be
revised to those shown in equation set (4.2).
L1new = L1prototype ?Gm ?QF
new
C1new = C1prototype ?GmF
new ?Q
L2new = L2prototype ?GmF
new ?Q
C2new = C2prototype ?Gm ?QF
new
(4.2)
L3new = L3prototype ?Gm ?QF
new
C3new = C3prototype ?GmF
new ?Q
As shown in the equations, the quality factor either multiplies or divides each passive
element in the band pass filter circuit. For a Q of 20, the passive elements become
those shown in table 4.3.
31
Figure 4.4: Third order Chebyshev band pass with Fc = 1 kHz, Q=1
Element Capacitance
L1 422.6 pF
C1 0.258 pF
L2 0.519 pF
C2 210 pF
L3 422.6 pF
C3 0.258 pF
Table 4.3: Capacitor values for band pass filter with Q=20
32
Figure 4.5: Third order Chebyshev band pass with Fc = 1 kHz, Q=20
33
Figure 4.6: Exploded view of third order Chebyshev band pass filter with Fc = 1 kHz,
Q=20
34
As clearly shown in table 4.3, increasing the quality factor means spreading
out the capacitance values required by the circuit. Figure 4.5 shows the frequency
response for the circuit with a quality factor of 20. The smaller bandwidth is very
evident. Figure 4.6 shows the same simulation with an exploded view of the passband.
Here, some of the non-ideal effects begin to show. A close inspection of Figure 4.6
reveals that the passband ripples are not exactly the same. This is caused by the
relatively small values of capacitors needed to achieve a quality factor of 20. At
smaller capacitance levels, the input capacitance of the OTA?s become a factor in the
transfer function. The fact that several OTA inputs are often connected to a single
OTA output only serves to enhance the problem. At a Q of 20, the effect is minor.
However, if the quality factor were increased, or the capacitors decreased the problem
would escalate quickly.
As was the case in the low pass filter, the reference current can be changed to
tune the filter center frequency. Figure 4.7 shows the resulting frequency response of
the high quality factor filter with four different reference currents. Also as before, the
tuning is quite linear. Figure 4.8 shows the tuning linearity by comparing reference
current to center frequency of the band pass filter.
4.3 Implementation of Higher Order Filter
Due to their sharper roll off, higher order filters are desired in many systems.
However, for many circuit topologies their design can be difficult. This often leads
to unnecessarily large and complicated circuits. One of the important benefits of the
35
Figure 4.7: Third order Chebyshev band pass filter for four reference currents
36
Figure 4.8: Reference current vs BPF center frequency
synthesis process presented in this paper is the ease of extension to high order transfer
functions. To emphasize this concept, the previously designed 3rd order Chebyshev
band pass filter will be extended to a 5th order one.
Figure 4.9 shows the minimum capacitance ladder circuit for a 5th order band
pass filter. Keeping the previously defined notation of currents through inductors
and voltages across capacitors, the state equations describing Figure 4.9 are shown
in equation set (4.3). Manipulating the equations to implement them using standard
two and four input OTA designs, the OTA-C circuit is as shown in Figure 4.10.
When compared to the equations and schematic of the 3rd order filter designed in
section 2.2, it is clear that the 5th order design is merely an extension of the 3rd order.
This explains why the first few equations and devices are the same. The design of
37
Chebyshev filter with a quality factor of 10 results in the capacitor values of table
4.4. As mentioned previously, the capacitor values are scaled depending on the desired
frequency range as well as the OTA transconductance.
Figure 4.9: 5th order band pass filter ladder circuit
Figure 4.10: 5th order prototype Chebyshev BPF OTA implementation
38
?i1 = 1
L1(Vin ?i1Rin +v2 ?v4)
?v2 = 1C
1
(?i1)
?i3 = 1
L2(v4)
?v4 = 1C
2
(i1 ?i3 ?i5)
?i5 = 1
L3(v4 ?v6 ?v8) (4.3)
?v6 = 1C
3
(i5)
?i7 = 1
L4(v8)
?v8 = 1C
4
(i5 ?i7 ?i9)
?i9 = 1
L5(v8 ?v10 ?i9Rout)
?v10 = 1C
5
(i9)
L1 339.4 pF
C1 0.745 pF
L2 1.46 pF
C2 173.5 pF
L3 477.1 pF
C3 0.53 pF
L4 1.46 pF
C4 173.5 pF
L5 339.4 pF
C5 0.745 pF
Table 4.4: Capacitor values for 5th order band pass filter with Q=10
Figures 4.11 and 4.12 clearly show the advantages of the higher order design.
With the added roll off, quality factors can be reduced in many applications. This
improvement comes at the expense of only five extra OTAs and their associated
39
Figure 4.11: Fifth order Chebyshev band pass filter centered at 10 kHz
40
Figure 4.12: Third order Chebyshev band pass filter centered at 10 kHz
41
capacitances. Figure 4.13 shows the 5th order filter ranging from 1 kHz to 10 kHz.
As shown, the shape is very consistent through the tuning range.
Figure 4.13: Fifth order Chebyshev band pass filter ranging 1 kHz to 10 kHz
42
Chapter 5
Conclusion
In this thesis, a convenient method of synthesizing OTA-C filters is presented.
Any transfer function that can be implemented in a simple ladder circuit can be
transformed into its OTA-C equivalent. Unlike other synthesis methods, it does not
depend on a second order section or any other type of building block. This fact gives
the designer more freedom as well as decreases the filter?s sensitivity to component
values. To emphasize this point, a 3rd order Chebyshev low pass filter and its bandpass
transform were synthesized to their equivalent OTA-C circuit.
Using SPICE a prototype OTA is developed in 0.5 ?m technology. Special care
is taken to address those issues most important to filtering applications. Foremost,
maintaining a large output resistance throughout the operating region is accomplished
through the use of a mirrored cascode structure. To make it possible to tune the
transconductance linearly, the biasing currents are kept low. Finally, the frequency
response of the amplifier is improved by improving the current mirror used in the
mirrored cascode design. The resulting amplifier is appropriate for use in filtering
applications requiring high quality factors.
The prototype OTA developed throughout Chapter 3 was then substituted into
the circuits from Chapter 2. In the case of the low pass filter, the results were almost
identical to the ideal case. The implemented filter was capable of being tuned linearly
from a critical frequency of 10 Hz to around 500 kHz using reference currents from
43
0.05 nA to 500 nA. Even higher critical frequencies can be obtained from the circuit,
but the critical frequency tuning becomes less linear at higher reference currents. For
a quality factor of 1, the band pass filter can have of a response as ideal as the low
pass. However, things become more interesting as the quality factor increases. The
filter implemented in section 4.2 can achieve a quality factor of 20 while the center
frequency of the filter can be tuned linearly from 100 Hz to around 50 kHz. Although
higher quality factors can be reached, trade offs must be made between capacitor
sizes, frequency, and Q. As an example of the ease with which higher order filters
can be implemented, a 5th order Chebyshev band pass filter was designed. Because
of the much steeper roll off, quality factor constraints can often be reduced in filters
of higher order. The results show a very versatile filter structure that can find use in
many applications.
44
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46
Appendices
47
Appendix A
SPICE level 7 NMOS model for AMIS 0.5 ?m process
.MODEL CMOSN NMOS ( LEVEL = 7
+VERSION = 3.1 TNOM = 27 TOX = 1.41E-8
+XJ = 1.5E-7 NCH = 1.7E17 VTH0 = 0.6394078
+K1 = 0.8797763 K2 = -0.101648 K3 = 22.7282152
+K3B = -9.4213761 W0 = 1E-8 NLX = 1E-9
+DVT0W = 0 DVT1W = 0 DVT2W = 0
+DVT0 = 2.7334674 DVT1 = 0.419695 DVT2 = -0.1437711
+U0 = 479.4186448 UA = 2.410464E-13 UB = 2.100688E-18
+UC = 2.161768E-12 VSAT = 1.549123E5 A0 = 0.593756
+AGS = 0.138164 B0 = 2.476574E-6 B1 = 5E-6
+KETA = -3.000619E-3 A1 = 1.257922E-4 A2 = 0.3989121
+RDSW = 1.447499E3 PRWG = 0.014654 PRWB = 0.0320564
+WR = 1 WINT = 2.627592E-7 LINT = 7.265689E-8
+XL = 1E-7 XW = 0 DWG = -2.390541E-8
+DWB = 1.036659E-9 VOFF = 0 NFACTOR = 1.1157256
+CIT = 0 CDSC = 2.4E-4 CDSCD = 0
+CDSCB = 0 ETA0 = 2.616252E-3 ETAB = -9.856187E-5
+DSUB = 0.0879772 PCLM = 1.4276648 PDIBLC1 = 1
+PDIBLC2 = 2.890383E-3 PDIBLCB = -0.0207482 DROUT = 0.9774041
48
+PSCBE1 = 6.226031E8 PSCBE2 = 1.380813E-4 PVAG = 0
+DELTA = 0.01 RSH = 81.8 MOBMOD = 1
+PRT = 0 UTE = -1.5 KT1 = -0.11
+KT1L = 0 KT2 = 0.022 UA1 = 4.31E-9
+UB1 = -7.61E-18 UC1 = -5.6E-11 AT = 3.3E4
+WL = 0 WLN = 1 WW = 0
+WWN = 1 WWL = 0 LL = 0
+LLN = 1 LW = 0 LWN = 1
+LWL = 0 CAPMOD = 2 XPART = 0.5
+CGDO = 1.99E-10 CGSO = 1.99E-10 CGBO = 1E-9
+CJ = 4.270492E-4 PB = 0.9112977 MJ = 0.4304805
+CJSW = 3.220481E-10 PBSW = 0.8 MJSW = 0.1979137
+CJSWG = 1.64E-10 PBSWG = 0.8 MJSWG = 0.1979137
+CF = 0 PVTH0 = -0.051744 PRDSW = 346.3408616
+PK2 = -0.0305149 WKETA = -0.0230394 LKETA = 4.606854E-3 )
49
Appendix B
SPICE level 7 PMOS model for AMIS 0.5 ?m process
.MODEL CMOSP PMOS ( LEVEL = 7
+VERSION = 3.1 TNOM = 27 TOX = 1.41E-8
+XJ = 1.5E-7 NCH = 1.7E17 VTH0 = -0.9362224
+K1 = 0.55078 K2 = 4.146308E-3 K3 = 9.6961154
+K3B = -0.3221286 W0 = 1E-8 NLX = 3.858149E-8
+DVT0W = 0 DVT1W = 0 DVT2W = 0
+DVT0 = 2.273307 DVT1 = 0.46069 DVT2 = -0.081477
+U0 = 233.5715825 UA = 3.503118E-9 UB = 2.45786E-21
+UC = -4.49043E-11 VSAT = 2E5 A0 = 0.8357554
+AGS = 0.1592968 B0 = 8.972676E-7 B1 = 5E-6
+KETA = -1.126827E-3 A1 = 1.166646E-4 A2 = 0.3423346
+RDSW = 3E3 PRWG = -0.030906 PRWB = -0.0122687
+WR = 1 WINT = 2.418604E-7 LINT = 9.180742E-8
+XL = 1E-7 XW = 0 DWG = -1.773591E-8
+DWB = 2.843285E-8 VOFF = -0.0798059 NFACTOR = 0.7194077
+CIT = 0 CDSC = 2.4E-4 CDSCD = 0
+CDSCB = 0 ETA0 = 6.759168E-3 ETAB = -0.120809
+DSUB = 1 PCLM = 1.9549807 PDIBLC1 = 0.0840943
+PDIBLC2 = 4.767307E-3 PDIBLCB = -0.0470009 DROUT = 0.2679617
50
+PSCBE1 = 5.610155E9 PSCBE2 = 5.295303E-10 PVAG = 0.1243898
+DELTA = 0.01 RSH = 105.3 MOBMOD = 1
+PRT = 0 UTE = -1.5 KT1 = -0.11
+KT1L = 0 KT2 = 0.022 UA1 = 4.31E-9
+UB1 = -7.61E-18 UC1 = -5.6E-11 AT = 3.3E4
+WL = 0 WLN = 1 WW = 0
+WWN = 1 WWL = 0 LL = 0
+LLN = 1 LW = 0 LWN = 1
+LWL = 0 CAPMOD = 2 XPART = 0.5
+CGDO = 2.28E-10 CGSO = 2.28E-10 CGBO = 1E-9
+CJ = 7.130689E-4 PB = 0.9618253 MJ = 0.4965243
+CJSW = 3.009822E-10 PBSW = 0.99 MJSW = 0.3379563
+CJSWG = 6.4E-11 PBSWG = 0.99 MJSWG = 0.3379563
+CF = 0 PVTH0 = 5.98016E-3 PRDSW = 14.8598424
+PK2 = 3.73981E-3 WKETA = -2.023716E-3 LKETA = -6.019389E-3 )
51