chapter 2 realization of some novel transconductance...
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16
CHAPTER 2
REALIZATION OF SOME NOVEL TRANSCONDUCTANCE FILTERS
This chapter is devoted to the realization of some novel active circuits by using
transconductance amplifiers. The transconductance amplifier can be realized by the widely
used device Op-Amp and specially designed operational transconductance amplifier (OTA). By
using single Op-Amp in one circuit itself we can get two filter responses. The first circuit
realizes first order low pass – high pass response and the second one realizes second order high
pass-band pass filter response. The quality factor Q of the second filter realization is low. The
low value of Q is used in systems for which damping is important such as image frequency
rejection and for lower bass in audio systems. It can also be used at the first stage of a cascaded
filter. All the proposed realizations have low sensitivity to parameter variations.
Applications of Op-Amps as transconductance amplifier are limited. Electronically controlled
applications, variable frequency oscillators, filters and variable gain amplifier stages, are more
difficult to implement with standard Op-Amps. In view of inherent tuning capability, the
operational transconductance amplifier (OTA) is extensively used as a basic active device in
many applications as compared to conventional Op-Amps. The internal circuit diagram of
OTA is simpler than operational amplifier and therefore higher bandwidth can be obtained.
Subsequently the chapter presents the basic operation of OTA along with its CMOS model,
realization of the waveform generator and low-pass elliptical filter circuits using OTA.
2.1 APPLICATION OF OP-AMPS FOR FILTER REALIZATION
As discussed in Section 1.4 several contributions have been reported for realization of active -
RC filters employing Op-Amps. Notable amongst them are Sallen- Key [5], Deliyannis-Friend
[7-9], Moschytz [10], KHN [12], Tow-Thomas [13-14], Ackerberg-Moserberg [15], Tarmy-
Ghausi [16], etc.
Since the early stage of technological development the components R and C could not be
realized with precision, the active-RC circuits were therefore not found suitable for precision
17
monolithic design at voice frequencies. To overcome this problem biquad active-R circuits were
proposed. These realizations were found to be suitable for the monolithic implementation due to
elimination of external capacitors. A simple active-R biquad realized by Rao and Srinivasan
[20] is shown in Figure 2.1.
Figure 2.1: Active-R biquad realized by Rao and Srinivasan
The single pole model of the operational amplifier can be expressed as :
where,ωc is the cutoff frequency and A0 is the DC open-loop gain of the operational amplifier. If
the frequency range of interest is connected to the region where cs ω>> equation 1 reduces to
where B is gain-bandwidth product of an operational amplifier.
c
c
sAsA
ωω
+= 0)(
(2.1)
sB
sAsA c ==ω0)( (2.2)
Vin Vout
R2 R1
18
The transfer function of circuit shown in Figure 2.1 can be obtained as
1211
1
02
2
0
+⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++⎥⎦
⎤⎢⎣⎡
⎥⎦⎤
⎢⎣⎡+⎥
⎦
⎤⎢⎣
⎡+
=
AK
Bs
BKs
BKs
AK
VV
in
out (2.3)
Assuming the parameter K of the circuit given by 1
21RRK += with the constraint 10 <<AK ,
the transfer function (2.3) can be expressed as follows:
11
1
22 +⎥⎦
⎤⎢⎣⎡+⎥⎦
⎤⎢⎣⎡
⎥⎦⎤
⎢⎣⎡+
=
Bs
BKs
BKs
VV
in
out
(2.4)
The transfer function thus obtained realizes bandpass filter. The center frequency ω0 and the
quality factor Q of the realization depend on the gain-bandwidth product B and K.
Several active-R filter realizations using Op-Amps have been reported in the literature.
However, these circuits suffered from the following limitations:
• Temperature dependence of ω0: The gain bandwidth product of the Op-Amps B is
given by Cm CgB = , where gm is the transconductance of the first input stage of the Op-
Amp and CC is the internal compensated capacitance. Since gm is inversely proportional
to the temperature, the cutoff frequency of the filter ω0 depends on the temperature.
• Dynamic range limitation: The maximum distortion less output signal is determined
by the slew rate of the Op-Amp. Since the transfer characteristics of an Op-Amp is
nonlinear before the onset of saturation, the output of the amplifier shows distortions at
the level well below the one determined by the slew rate value.
• Parasitic pole: The parasitic capacitance within the Op-Amp causes additional phase
shift in the circuit response. Schaumann and Brand [21] have shown the effect of
additional parasitic capacitance on ω0 and Q.
19
With the advent of MOS technology, a new approach was adopted to design filters by replacing
altogether integrators and resistors by the only passive components capacitors. One such filter
realization, proposed by Schaumann and Band [22] employing only ratioed capacitors as
passive components is shown in Figure 2.2. They employed extra resistance of high value in
parallel with C3 for DC biasing. The circuit realizes bandpass and lowpass filter responses
simultaneously with outputs V01 and V02.
The transfer function of the bandpass and lowpass filters realized with this circuit are
respectively given by equations (2.5) and (2.6)
321
32
321
22
321
1
01
CCCCB
CCCCsBs
CCCCsB
VV
in
+++
+++
++−= (2.5)
321
32
321
22
321
12
02
CCCCB
CCCCsBs
CCCCB
VV
in
+++
+++
++−= (2.6)
Figure 2.2: Band pass and low pass filters proposed by Schaumann and Band
The cutoff frequency and the quality factor are same for equation (2.5) and (2.6) and expressed
as follows:
R
C1 C2
C3
Vo1 Vo2
Vin
20
321
30 CCC
CB++
=ω (2.7)
)(13213
2
CCCCBC
Q ++= (2.8)
The orthogonality between the cutoff frequency and the quality factor could not be achieved
with the active-R and active-C filter realizations reported in the literature. The second order
lowpass filter realized by Xiao [23] with one pole model of Op-Amp provides orthogonality
between cutoff frequency and quality factor. Xiao’s lowpass filter having low sensitivity to
parameter variation is shown in Figure 2.3.
Figure 2.3: Xiao’s low pass filter
The transfer function, the quality factor Q and cutoff frequency ω0 of the filter are respectively
given by the following expressions:
41
4312
431
431
12
4310
11111
111111
111
RRRRRBR
RRBRs
RRRBCs
RRRVV
i
+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+++
⎟⎟⎠
⎞⎜⎜⎝
⎛+
+⎟⎟⎠
⎞⎜⎜⎝
⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
(2.9)
R1
R2
R3R4
C1
Vi
Vo
21
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
++=
3
4
1
4110
1
1
RR
RRRC
Bω (2.10)
⎟⎟⎠
⎞⎜⎜⎝
⎛++++
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++
=
3
4
1
4
2
1
3
4
3
4
1
411
11
1
RR
RR
RR
RR
RR
RRRBC
Q (2.11)
The expression (2.10) and (2.11) exhibit orthogonality and hence the quality factor can be tuned
independently of cutoff frequency by varying R2.
The circuits discussed so far realized voltage mode transfer functions. Several other voltage
mode transfer function realizations using Op-Amp pole have been published [24-26].
Higashimura first proposed current mode realization of transfer function using one pole model
of the Op-Amp [27-28]. His circuit [27], shown in Figure 2.4, realizes highpass and bandpass
filters with transfer function respectively given by equation (2.12) and (2.13).
Figure 2.4: Higashimura’s proposed current mode high pass and band pass filters
112111
2
2
11CRB
RCRCss
sII
IN
HP
+⎟⎟⎠
⎞⎜⎜⎝
⎛++
= (2.12)
C1
R2 R1
IIN
IBP
IHP
22
112111
2
11
11
1
CRB
RCRCss
CRs
II
IN
BP
+⎟⎟⎠
⎞⎜⎜⎝
⎛++
= (2.13)
As evident from the expression given in equations (2.14) and (2.15) for the quality factor Q and
ωo for the quality factor can be tuned independently of cutoff / central frequency.
110 CR
B=ω (2.14)
1
1
21
11R
BCRR
Q ⎟⎟⎠
⎞⎜⎜⎝
⎛+= (2.15)
As stated earlier several voltage mode filters and current mode filters have been published. A
number of filter realizations using one pole model of operational amplifier operating in the
voltage mode (VM) and current mode (CM) have also been proposed in the literature [15-28].
Voltage mode filter are not suitable for small impedance load. A current mode filter circuit is
suitable for all types of load. Since it offers low input impedance, a current mode filter is not
suitable in many applications. Voltage-current converters, which overcome the difficulties
which arise in voltage-mode and current- mode circuits, can therefore be employed in most
applications. Further, these transconductance filters meet the demanding specifications in
several signal processing applications in different bands such as given in [6].
A notable contribution of realizations of lowpass and bandpass transconductance filter using
single pole model of Op-Amp is due to Shah et al. [29]. The proposed circuit also offers less
sensitivity to parameter variation as well as orthogonality between the cutoff frequency and the
quality factor. The realizations of the circuits using one-pole Op-Amp model suffer from the
limitation due to absolute value of gain-bandwidth product that is likely to vary with process
tolerances and as a result change in the cutoff frequency can take place [21].
In the following section realization of two transconductance filters are proposed using zero-pole
model of Op-Amp. These circuits offer high input impedance and low sensitivity figures.
Nonlinear analysis of the proposed transconductance filters is also presented.
23
2.2 PROPOSED TRANSCONDUCTANCE FILTER REALIZATIONS
USING OP-AMP
Two proposed transconductance active filters employing single operational amplifier have been
proposed in this section. The first circuit configuration realizes a first-order lowpass-highpass
filter, whereas a second-order highpass-bandpass filter responses is realized using another
configuration. Both the circuits employ passive R and C components with spread not more than
1:10.
2.2.1 REALIZATION OF FIRST ORDER TRANSCONDUCTANCE LOW PASS AND
HIGH PASS FILTERS
Considering the zero-pole model of operational amplifier a first order transconductance low
pass and high pass filter can be realized as shown in Figure 2.5.
Figure 2.5: Proposed low pass-high pass transconductance filter
Two transfer functions of the filter circuit of Figure (2.5) are given by equation (2.16) and (2.17)
)1()1( 21121 ARRACRsR
AVI
in
LP
++++= (2.16)
)1()1( 21121
11
ARRACRsRARsC
VI
in
HP
++++= (2.17)
where A is the open loop gain of the operational amplifier. These equations respectively
represent the transconductance functions for LP and HP responses with same denominator. The
cutoff frequency ω0 of the responses is given by:
R1
C1
R2
Vin ILP+
-
IHP
24
111112121
210
11)1(
1)1()1(
CRCRACRACRRARR
≈++
=+++
=ω (2.18)
The cutoff frequency can therefore be controlled by either R1 or C1. The sensitivities of ω0 due to variation in active gain-bandwidth product A and passive components are given by 00 ≅ω
AS (2.19) 10
1≅ω
RS (2.20)
00
2≅ω
RS (2.21)
10
1−=ω
CS (2.22)
These sensitivities are small and not more than one. The magnitude and phase responses of the
LP and HP filters are shown in Figure 2.6. These match with the designed specifications.
Figure 2.6(a): High pass filter magnitude response of proposed filter
25
Figure 2.6(b): High pass filter phase response of proposed filter
Figure 2.6(c): Low pass filter magnitude response of proposed filter
26
Figure 2.6(d): Low pass filter phase pass filter phase response of proposed filter
2.2.2 REALIZATION OF SECOND ORDER TRANSCONDUCTANCE HIGH PASS
AND BAND PASS FILTERS
Considering the zero-pole model of the operational amplifier, the circuit configuration shown in
Figure 2.7 can be used to realize the high pass (HP) and band pass (BP) filter responses.
Figure 2.7: Proposed high pass and band pass transconductance filter
The following second order transconductance functions for HP and BP responses can be
obtained for this realization:
R1
C1
R2
C2
Vin IBP+
IHP
27
012
1212
αα ++=
ssARCCs
VI
in
HP (2.23)
012
2
αα ++=
ssAsC
VI
in
BP (2.24)
where the parameters α1 and α0 are given by:
)1()1()1(
2121
2122111 ACCRR
CRACRACR+
++++=α (2.25)
)1(
1
21210 ACCRR
A+
+=α
(2.26)
The cutoff frequency (ω0) and Q of the above circuit are:
21210
1CCRR
=ω (2.27)
2211
2121
CRCRCCRR
Q+
= (2.28)
The active and passive sensitivities of ω0 and Q are small. By taking R1 = R2 value of Q is found
to be 0.5 and the sensitivities with respect to various parameters are obtained as follows:
00 =ωAS (2.29)
21
0
2
0
1
0
2
0
1−==== ωωωω
CCRR SSSS (2.30)
21
2121==== Q
CQC
QR
QR SSSS (2.31)
The magnitude and phase responses of the HP and BP filters are shown in Figure 2.8 match
with the designed specifications.
28
Figure 2.8(a): High pass filter magnitude response of proposed filter
Figure 2.8(b): High pass filter phase response of proposed filter
29
Figure 2.8(c): Band pass filter magnitude response of proposed filter
Figure 2.8(d): Band pass filter phase response of proposed filter
30
RESPONSE CONSIDERING ONE-POLE MODEL OF OP-AMP
Practical Op-Amps have finite (but large) input impedance, non zero (but small) output
impedance and finite (but large) differential gain. The dependence of the differential gain of the
Op-Amps on frequency is considered in the AC circuit model of the Op-Amps. This approach is
based on the first order approximation of the Op-Amp, which is reasonable to determine the
frequency response. Using first-order model of the Op-Amp in place of zero-order model, the
circuits shown in Figure 2.5 yield the following transfer function:
3131311312 )( BRBCRRRRsCRRs
BVI
in
LP
++++= (2.32)
3131311312
11
)( BRBCRRRRsCRRsCsBR
VI
in
HP
++++= (2.33)
With the first order model of the Op-Amp following transfer function can be realized for the
circuit shown in Figure 2.7
( )( )( ) 21
22211
2112
11 CRsBsCsRCsRCCBRs
VI
in
HP
++++= (2.34)
( )( )( ) 21
22211
2
11 CRsBsCsRCsRsBC
VI
in
BP
++++= (2.35)
It is thus observed that equations (2.16), (2.17), (2.23) and (2.24) obtained with zero-order
model of the Op-Amp get transformed to equations (2.32), (2.33), (2.34) and (2.35) respectively
for the first-order model. Comparing the above equations it is observed that the order of the
denominator gets increased and in some cases both the orders of denominator and numerator
increase in one pole model. The frequency responses of the filters using zero and first order
models of the Op-Amp have been shown in Figure 2.9-2.10.
31
Figure 2.9 (a): Low pass filter magnitude responses of circuit shown in Figure 2.5
Figure 2.9 (b): Low pass filter phase responses of circuit shown in Figure 2.5
32
Figure 2.9(c): Magnitude responses of high pass filter of circuit shown in Figure 2.5
Figure 2.9(d): Phase responses of high pass filter of circuit shown in Figure 2.5
33
Figure 2.10(a): Magnitude responses of high pass filter of circuit shown in Figure 2.7
Figure 2.10(b): Phase responses of high pass filter of circuit shown in Figure 2.7
34
Figure 2.10(c): Magnitude responses of band pass filter of circuit shown in Figure 2.7
Figure 2.10(d): Phase responses of band pass filter of circuit shown in Figure 2.7
35
It is observed that there is no major difference in the frequency response of the filters for two
models up to frequency of 100 kHz (approx). The cutoff frequency of the lowpass and highpass
and center frequency of the bandpass filters remain unaltered in both the models (zero order as
well as first order) of Op-Amp. Thus, the zero-order model realizes the desired filter response.
Due to high open loop gain of the operational amplifier, the first order model does not alter the
response in terms of the standard definition of the filter.
Since, the consideration of the pole of the Op-Amp does not play significant role in the low
frequency response. The proposed circuit has been designed using zero-pole model of the Op-
Amp.
2.3 PERFORMANCE OF THE PROPOSED TRANSCONDUCTANCE
FILTERS USING OP-AMP
The performance of the circuits shown in Figure 2.5 and 2.7 employing operational amplifier
LM741, for realizing lowpass-highpass combination and highpass-bandpass combination
respectively, have been verified experimentally as well as by PSPICE simulation. These
realizations have been obtained with R1 = 1kΩ, R2 = 1kΩ and C1 = 100nF. Figure 2.6 (a) and
2.6(c) show the magnitude response of the low pass and high pass filters, each having cutoff
frequency 1.5 kHz. The phase responses of these filters have been shown in Figure 2.6(b) and
2.6(d) respectively.
The highpass and bandpass filter circuits employ R1 = 1kΩ, R2 = 1kΩ, C1 = 100nF and C2 =
100nF. The magnitude and phase responses of these filters have respectively been shown in
Figure 2.8(a), 2.8(c), 2.8(b) and 28(d). Each of these filters has the same cutoff frequency and
center frequency 1.5 kHz.
2.4 CHARACTERISTICS OF OPERATIONAL TRANSCONDUCTANCE
AMPLIFIER (OTA)
Transconductance is defined as the ratio of the current change at the output port with respect
to the change in voltage at the input port. It is normally denoted by gm and mathematically is
defined as:
36
inm V
IgΔΔ
= 0 (2.36)
A transconductance amplifier is an example of voltage controlled current source (VCCS) that
provides output current proportional to the input voltage. An ideal transconductance amplifier
is characterized by: infinite input impedance, infinite output impedance and infinite
bandwidth.
A transistor can be used to provide the characteristics of a transconductance amplifier as its
collector current is proportional to the input voltage applied at the base terminal. Further, the
requirements of a transconductance amplifier are also met with the impedances of a transistor
being high as seen at the base terminal, very high at the collector terminal, and low at the
emitter terminal. A NPN or a PNP transistor does not provide both positive and negative
currents. Since a transconductance amplifier can provide both positive and negative currents,
it can be used as a current source as well as a current sink.
Operational amplifier has one more input terminal to control the transconductance of the
amplifier, externally. Transconductance amplifier is called an operational transconductance
amplifier, if the output current is made proportional to the differential input voltage applied at
the input port.
The symbolic representation of the transconductance amplifier and its small signal equivalent
circuit representation are respectively shown in Figure 2.11(a) and (b).
Figure 2.11(a): Ideal model of OTA
Figure 2.11(b): Equivalent circuit of OTA
37
The output current I0 of the ideal OTA can be expressed by equation (2.37). )(0 NPm VVgI −= (2.37)
where gm, the transconductance can be expressed in terms of bias current (Ibias), charge (q),
Boltzmann constant (k) and temperature (T) in Kelvin, as follows:
kT
qIg bias
m 2= (2.38)
Since the output of an OTA is derived as the current, the output impedance of the OTA is very
high (ideally infinity). In view of low output conductance of the OTA, it is suitable as an
ideal current generator. An OTA can be made to work as an operational amplifier, if a
resistance is connected at its output terminals. Since gm of the OTA is dependent on the Ibias
current, the output characteristics of the OTA may be controlled externally by the bias current
(Ibias). It adds new dimension to design and applications of OTA circuit.
Operational transconductance amplifier is a versatile building block that intrinsically offers
wider bandwidth. The principal difference between OTA and Op-Amp are as follows:
The output of OTA is current where as the output of the Op-Amp is voltage.
In linear applications, OTA is mainly used in the open loop mode while the Op-
Amp is used in close loop mode with negative feedback.
The load impedance decides the output voltage of the OTA. Therefore the output
voltage of the OTA is controllable by the load impedance.
The first stage of the Op-Amp may be considered as a voltage to current convertor; hence
OTA can be seen as an integral part of an Op-Amp. Thus the OTA can be realized by a
differential pair followed by the current mirror load. In 1969, single output OTA was made
commercially available by RCA. During 1980-1990 many papers were reported for OTA
design and its application. In 1985 Gieger and Sanches reported the possible synthesis of
biquad and controllable impedance circuits using OTA [35].
38
The basic CMOS model of OTA consists of eight MOS transistors as shown in Figure 2.12.
The MOS transistors M1 and M2 constitute a differential input voltage pair with M5 and M6
transistors acting as active load of the differential pair. The differential transistors are biased
by the current source (Ibias). Remaining transistors M8, M7, M3 and M4 act as current mirrors.
Figure 2.12: Circuit diagram of CMOS OTA
Commercially available OTAs are CA3080, CA3080A, LM13600, LM13700 etc. Among
them LM13700 is more commonly used in view of its low leakage current and good control
on tansconductance and wider input voltage range. The internal circuit diagram of LM13700
is shown in Figure 2.13 [32-33].
39
Figure 2.13: Circuit Diagram of LM13700
The transistor Q4 and Q5 are the differential input pair of the OTA. The ratio of the collector
current flowing through Q4 and Q5 is proportional to the differential input voltage Vin as
shown in equation (2.39)
4
5lnII
qKTVin = (2.39)
where KT/q is approximately 26mV at 25oC and I5 and I4 are the collector currents of the
transistor Q5 and Q4 respectively. Transistors Q1, Q2 and diode D1 form a current mirror for
the externally applied bias current Ibias such that bias current is the summation of the collector
current I4 and I5.
54 IIIbias += (2.40)
Q6, Q7 and D4 form a current mirror for I4 which is again followed by a current mirror
constituted with Q8, Q9 and D5. Similarly Q10, Q11 and D6 form a current mirror for I5. The
difference current is obtained from the output terminal. For DC analysis (no signal condition)
40
the currents supplied by the matched transistor Q4 and Q5 are equal and depend on the bias
current as shown in equation (2.41).
254biasI
II == (2.41)
For small differential input voltage Vin, equation (2.39) can be expressed as:
4
45
4
5lnI
IIq
KTII
qKTVin
−×≈= (2.42)
KTqIV
II in 445 =− (2.43)
⎟⎠⎞
⎜⎝⎛=−=
KTqI
VIII biasin 2450 (2.44)
The transistor Q12 and Q13 are connected as Darlington pair, to work as voltage buffer. For
very small input voltage (order of mV) the equation (2.39) may be considered to be linear and
be approximated as equation (2.41). The diodes D2 and D3 are used to increase the linear
range of operation of the differential amplifier provided that the signal current Is is less than
diode current ID/2.
OTA is versatile building block with on chip tuning capability. It can be employed for the
realization of basic building blocks such as floating resistor, grounded resistor, negative
resistance and inductor. Gieger and Sanchez [35] have reported various applications of OTA
in the synthesis of first order and second order filters. OTA circuits employing capacitor as
load (OTA-C) are employed for realization of various linear and non-linear circuits [36-57].
They can be also used for implementing voltage controlled oscillators (VCO) and voltage
controlled filters (VCF) for analog synthesizer. Further, OTA can be used to operate as a two
quadrant multiplier, driver circuit for light emitting diode (LED), for realization of automatic
gain control (AGC) amplifier, pulse integrators, control loops for capacitive sensors, active
filters and oscillators.
In this chapter a new waveform generator using OTA has been proposed with special feature
of generating three waveforms square, triangular and sinusoidal simultaneously.
41
2.5 WAVEFORM GENERATOR CIRCUITS USING OTA
A waveform generator with current/voltage mode has wide range of application in the
instrumentation, communication systems and signal processing. The sinusoidal, triangular and
square waveforms can be realized by using Op-Amp or OTA. Op-Amps are very well known
active device, which are mainly used with negative feedback for introducing some pole
constraint in the design. Op-Amps are also not gain programmable and therefore can not be
compensated for drift errors. The frequency of the circuit can not be changed without
changing the passive components. Due to these limitations OTA is preferred over the Op-
Amp as a basic active element. It works in open loop mode, which does not add any constraint
on the frequency response to compensate for local feedback introduced pole [37] and the
transconductance of the OTA is electrically tunable. The circuits realized with OTA have
simple configuration and wide frequency sweep. Several circuits using OTA have been
reported in the literature. Barranco et al. [38-40] have proposed the sinusoidal oscillator
employing two OTA and three capacitors as shown in Figure 2.14.
Figure 2.14: Barranco’s two OTA and three capacitor oscillator
The frequency of oscillation 0ω of the circuit may be obtained as
C3
C1 C2
+ -
+ -
gm1 gm2
42
133221
210 CCCCCC
gg mm
++=ω (2.45)
The transconductance gm1 and gm2 are used for the trimming of the waveform. In order to
place the poles in the right half of s-plane nearer to origin, the transconductance gm1 is kept
larger than gm2. Subsequently a sinusoidal oscillator has been realized using three OTA with
two capacitors. The circuit diagram is shown in Figure 2.15
Figure 2.15: Three OTA and two capacitor oscillator
The frequency of this oscillator circuit is given by equation (2.46).
21
31
21
CCgg
f mmo π= (2.46)
The second OTA employed in the circuit, is used to keep the poles of characteristic equation
in the right half of s-plane, close to the origin.
Senani [41] proposed tunable OTA-C filter employing three OTA and two capacitors as
shown in Figure 2.16.
+ -
C1 C2
gm1 gm3
V01 V02
+ -
gm2
+ -
43
Figure 2.16: Sinusoidal oscillator proposed by Senani.
Senani’s circuit works as an oscillator under the constraint 32 mm gg = . The frequency of
oscillation is given by equation (2.47).
21
310 2
1CCgg
f mm
π= (2.47)
Abuelma’atti [52] realized sinusoidal oscillator employing two OTAs and two capacitors as
shown in Figure 2.17.
Figure 2.17: Sinusoidal Oscillator proposed by Abuelma’atti
The frequency of oscillator is dependent on the condition mmm ggg == 21 and is given by
equation (2.48).
+ - gm1
C1
+ - gm2C2
+ - gm1
+- gm2
C1
+- gm3
C2
44
2121
210
12
.21
CCg
CCgg mmm
ππω == (2.48)
.A number of sinusoidal oscillators employing other active devices such as Op-Amp, current
conveyor, etc has been reported in the literature.
The conventional circuit for generating square waveforms using operational amplifier is
shown in Figure 2.18.
Figure 2.18: Square waveform generator using one Op-Amp
This circuit suffers from slew rate limitation and the frequency of oscillation can be controlled
by passive components only. Though square wave is obtainable at the output terminals, the
triangular waveform across the capacitor is exponential in nature. The modified version of the
above circuit is shown in Figure 2.19.
Figure 2.19: Square and triangular wave generator using two Op-Amps
R1
R2 R3
R4 C
+
-
R2
R3
R1
C
Vout
45
Haslett [42] proposed a square wave generator employing one OTA, one comparator and one
Op-Amp as shown in Figure 2.20. The frequency of oscillation of the circuit proposed by him
can be controlled by the externally applied voltage.
Figure 2.20: Square waveform generator proposed by Haslett
This circuit also suffers from slew rate limitation as it employs Op-Amp along with other
active and passive components. Further, secondary source of error arises due to current
imbalances between the positive and negative terminal of OTA. Chung et al. [43] proposed
temperature-stable voltage controlled oscillator using two OTAs and operational amplifiers as
shown in Figure 2.21.
RB1
RB2 R1
R2
C1
VCC
Vin
1 2
46
Figure 2.21: Temperature insensitive voltage controlled oscillator
Their circuit employs more active and passive components and slew rate limitations are not
overcome.
Many square wave generator circuits have been reported in the literature [44-46]. Later,
Chung et al. [46] realized square and triangular waveform generators using three OTA as
shown in Figure 2.22.
R2 R1
RB1
R3
RB2
C
VB1
VB2
VO1
VO2
47
Figure 2.22: Triangular and square wave generator proposed by Chung et al
The amplitude and frequency of the oscillation of this circuit can be controlled independently.
All these circuits discussed above may be used to generate either the sinusoidal waveform or
square and triangular waveforms. None of these circuits produce square, triangular and
sinusoidal wave form simultaneously.
2.5.1 PROPOSED WAVEFORM GENERATOR CIRCUIT
The proposed circuit generates square, triangular and sinusoidal signal simultaneously. It
employs three OTAs, two resistors and two capacitors as shown in Figure 2.23. The amplifier
bias current Ibias which may or may not be same in all three OTAs.
C
R1 R2
48
Figure 2.23: Circuit Diagram of proposed waveform generator
The OTAs of the proposed circuit employs positive feedback. The output of the OTA1 is
IB1/C. Since the positive feedback of the OTA2 makes it saturated, its output will be ±IB2R1.
With the output of OTA2 being +IB2R1, the output of OTA1 across the capacitor C1 tries
linearly to build up the voltage and when it reaches this value the output of OTA2 drops to -
IB2R1. As a result the capacitor C1 starts loosing the voltage linearly till it reaches to this
negative voltage. When the voltage across the capacitor becomes -IB2R1, the voltage goes
down and the output of OTA2 changes to +IB2R1. This process repeats in generation of ramp
output across C1 and square waveform across R1. The ramp signal is again integrated in the
circuit to obtain the sinusoidal waveform. The time period T1 during which the output voltage
of OTA changes from IB2R1 to -IB2R1 is given by equation (2.49).
1
1
1
2121 )(CI
TIRIR BBB =
−− (2.49)
Rearranging equation (2.49), we obtain
C1
C2
R1
R2
gm1
gm2
gm3
IB1 IB2
IB3
49
1
2111 2
B
B
IIRCT = (2.50)
Since the charging and discharging path for the capacitor is same, the frequency of the
waveform generator may be expressed by equation (2.51)
211
1
1 4211
B
B
IRCI
TTf === (2.51)
The desired shape of the sinusoidal is obtained across R2 by integrating the ramp signal in the
circuit. The frequency of the waveform generator can be changed by changing the bias current
of OTA1 and OTA2 for the given R1 and C1 from the above equation.
The Figure 2.24 shows the various waveform responses of the frequency 4.4 kHz by selecting
IB1=950.35μA, IB2 =559.55μA, IB3 =950.35μA and R1 =10 kΩ, R2=10 kΩ, C1 =10nF and
C2=10nF. The main feature of the proposed circuit is its linearity. Figure 2.25 shows the graph
between the bias current of the first OTA and frequency of the waveform generator.
Figure 2.24(a): Sinusoidal, triangular and square waveform of proposed waveform generator
50
Figure 2.24(b): Waveforms in CRO
Figure 2.25: Graph between bias current IB1 and oscillation frequency of the proposed waveform generator
51
2.6 ELLIPTICAL FILTER REALIZATION USING OTA
Among the various filter approximations, the most generalized filters approximations are
Butterworth, Chebyshev and Cauer. The Butterworth filter responses are also known as
maximally flat response. It provides maximally flat band response in the pass band and stop
band. The roll-off is smooth and monotonic with a rate of 20dB/ pole. The transition band of
this filter response is much higher than others. Another approximation of the ideal filter is
Chebyshev response or equal ripple response in the pass band. It has a steeper roll rate near the
cutoff frequency as compared to the Butterworth filter and poorer transient response.
Cauer filters are also called elliptical filters. The elliptical filter has ripples in both in the pass
band and stop band. It has a very narrow transition band as compared to other approximations. It
is also an optimal realization of ideal filter response.
Higher order filters are needed in order to satisfy the selectivity requirement of the
telecommunication systems and many other applications. These filters are realized by following
approaches:
i) Cascade second order stages without negative and with negative feedback.
ii) Simulation of passive LC filters.
Among these two approaches the first approach cascade second order stages without negative
takes more area and active/passive components for the realizations. Its design is simple and
easily tunable but it offers bad sensitivity. Cascade stages with negative feedback circuits have
low sensitivity to parameter variations but their design is complex. The second approach based
on the simulation of the passive LC filters is more attractive due to the simplicity in design and
extremely low sensitivity. Earlier grounded and floating inductors were simulated using Op-
Amps, Deboo’s gyrator, Riordan gyrators and other circuits. With the advent of OTA and other
novel active devices it has been possible to realize filters by using second approach mentioned
above.
In this section 7th order elliptical low pass filter is realized for the cutoff frequency 1MHz by
employing OTA. The normalized 7th order doubly terminated (for improved sensitivity) LC
network is shown in Figure 2.26.
52
Figure 2.26: Normalised 7th order doubly terminated LC filter
The ladder circuit designed for frequency and impedance denormalized for the cutoff frequency
of 1MHz is shown in Figure 2.27.
Figure 2.27:7th order LC filter for cutoff frequency 1MHz
The state equation of the circuit shown in Figure by virtually eliminating C2, C4 and C6 are
given in the following Table 2.1.
Table 2.1: The state equations for the circuit shown in Figure 2.27
1
1 )(R
VVI in
i−
=
1
11 sC
IV =
2
22 sL
VI =
3
33 sC
IV =
R1
C1
L2
C2
C3
L4
C4
C5
L6 C6
C7 Vin
1KΩ 187.2pF
30pF240.5pF 203.3pF 133pF
1KΩ
115.22µH
160.9pF 113.3pF
133µH 191.1µH
RL
R1
C1
L2 C2
C3
L4
C4
C5
L6
C6
C7 Vin
1Ω 1.176F
193.93mF1.5113F 1.2776F 835.97mF
1Ω
723.98mH
1.01098F 712.11mF
801.65mH 1.19393H
53
4
44 sL
VI =
5
55 sC
IV =
6
66 sL
VI =
7
77 sC
IV =
2RV
I outout =
The circuit diagram shown in Figure 2.27 can be realized by using OTA-C filters with lossy
integrators with finite poles describing the equations in Table 2.1.
The realized 7th order elliptical filter using OTA and capacitors with component values is shown
in Figure 2.28
Figure 2.28: 7th order elliptical filter using OTA
C1= 187.2pF, C2 = 30pF, C3 = 240.7pF, C4 = 160.9pF, C5 = 203.3pF, C6 = 113.3pF,
C7 = 133pF, CL1 = 6pF, CL2 = 20pF, CL3 = 41.6pF
C1
C2 C3
C4
C5
C6 C7
Vin CL1 CL2 CL3
- +
+ - + - + -+ -
+ -+ - + -
Vout
54
The frequency response of the OTA-C filter is shown in Figure 2.29. The response matches the
one available with LC ladder circuit of the figure.
Figure 2.29(a): Magnitude response of the 7th order elliptical filter
Figure 2.29(b): Phase response of the 7th order elliptical filter
55
2.7 CONCLUSION This chapter presents various developments in the realizations of the active filters employing
Op-Amps, the performance of the filter realizations employing zero-pole model and one-pole
model of Op-Amp. Next, the chapter focuses realization of two transconductance filters using
single Op-Amp. One transconductance filter employing three passive components- one
capacitor and two resistors simultaneously realizes low pass-high pass transconductance filter
responses. Whereas another transconductance filter realizes high pass-band pass
transconductance filter with two resistors and two capacitors. The cutoff /center frequency ω0 is
to be tunable by the changing the values of the passive components and is independent from the
open loop gain of the operational amplifier. The realizations have low sensitivity to variable and
sensitive parameter A0 (gain –bandwidth product) and employ minimum number of passive
components.
Subsequently the chapter is devoted to realization of waveform generators using OTA
LM13700. Various waveform generators employing OTA, reported in the literature, realize
individual waveforms such as sinusoidal waveform, square and triangular waveforms. In the
literature no single wave-shaping circuit appears to have been reported for realization of more
than two different types of waveforms at a time. The proposed single waveform generator
circuit realizes square, triangular and sinusoidal waveforms simultaneously.
Further, the 7th order elliptical filters realization based LM13700 has been simulated using
PSPICE and compared with LC network. It can be used for video signal processing in TV.