CIRCUIT THEORY A N D APPLICATIONS, VOL. 2, 13-22 (1974)
ANALYSIS OF NONLINEAR MODEL FOR OPERATIONAL
AMPLIFIERS IN ACTIVE RC NETWORKS
ANDERS FORSEN AND LARS KRISTIANSSON
Telecommunication Theory, Royal Institute of Technology, Stockholm, Sweden
SUMMARY
In active RC networks certain nonlinear phenomena may occur due to the nonlinear properties of the amplifiers. It is
shown that one of these, the slew rate limitation, may give rise to the jump phenomenon. The class of networks to be
studied are described by a third order nonlinear differential equation. The theoretical and experimental results are found
to correspond well.
INTRODUCTION
While studying active filters in the higher part of the LF spectrum and with moderate Q ( ‘Y 5 ) certain phenomena arose which led to a study of methods of calculation for nonlinear analysis of active filters.
A mathematical model for slew rate limited amplifiers in RC networks is derived and analyzed.
The model is sufficiently accurate for study of one of the phenomena, the sudden jumps in the output
amplitude which may occur when a sinusoidal input slowly varies in frequency or amplitude. This so-called
jump phenomenon is known in other contexts. For instance a second order filter with nonlinear inductance
permits an amplitude jump in the voltage across the inductance. The phenomenon is called in this context
ferroresonance and is described by a second order nonlinear differential equation, Duffing’s equation.
In our case the nonlinearity consists of the limitation of slew rate in the operational amplifier, which,
with the other components of the active filter, leads to a higher order nonlinear differential equation.
The following notation will be used throughout the analysis:
Lower case letters : time functions
Upper case letters : Laplace transforms
In the figures upper case letters will be used when Laplace expressions are present.
OPERATIONAL AMPLIFIER MODEL
By using the Miller effect one can compensate an operational amplifier with only one capacitor. Using this
method the first differential stage exhibits a dominant pole, so that the gain vs. frequency characteristic
will have a slope which does not exceed 6 dB/octave down to 0 dB. In Figure 1 is depicted a simplified
model of a typical Miller-compensated amplifier.2
We now seek a relation between u , and i which permits us to replace the first differential stage by a voltage
controlled current generator.
From Figure 2 we find by means of well known current-voltage relations for the transistor
. - I. N i,(exp (nuBl)
- 1)
to
i
uB1-
N
i,(exp (nuBz)1)
uBz + R,(io - 2i)
Received 9 April 1973
Revised 12 June 1973
0 1974 by John Wiley & Sons, Ltd.
13
= u1
(3)
14
ANDERS FORSEN AND LARS KRISTIANSSON
t
t
OR
C
Figure 1. Miller-compensatedamplifier (simplifiedmodel)
t io-i
ti
Figure 2. Emitter-degenerated differential stage
where
R
4
kT
= - z 40 volt-'
is
N
at room temperature.
0.2.
A
(4)
(5)
Neglecting is equation (6) may easily be derived
I =
10
1 +exp(Ro,C,)
(6)
If we let Re + 0, i.e. C1-, 1, equation (6) changes into the known expression for the simple differential stage :
I =
i0
1 +exp (nu,)
We now insert the nonlinear current generator in our model and obtain the diagram in Figure 3.
We use Thevenin's theorem and obtain for the equivalent voltage source
ueq = Ri = ioR
1
1 +exp(Ru,C,)
(7)
ANALYSIS OF NONLINEAR MODEL FOR OPERATIONAL AMPLIFIERS IN ACTIVE RC NETWORKS
15
t
Figure 3. Amplifier model with nonlinear current generator
This means that, for the input signal ul = 0, ues = (i0R/2),which leads to u2 # 0. In actual fact this DC
voltage is eliminated from the circuitry aspect. As equivalent voltage source u (Figure 4), accordingly,
we have
Figure 4. Voltage model for nonlinear amplifier
The relation between V,(s) and V ( s )is obtained from the Laplace relation
or
Now it is simple to divide the model into a linear and a nonlinear block. Let Au be the output voltage of the
memoryless nonlinearity. This dummy variable is named u ; . The nonlinearity is followed by a one pole
low-pass filter. See Figure 5, where the following relations prevail :
A=
1
( A 1)RC
+
16
ANDERS FORSEN AND LARS KRISTIANSSON
Figure 5. Block diagram for nonlinear amplifier
PARAMETER IDENTIFICATION
Usually, the parameters used in the preceding analysis are not available. We shall now identify and express
these parameters in well known quantities-pen-loop
gain, bandwidth and slew rate.
By defining Fo as the small-signal open loop gain we may easily derive the relation
Fo=--
ioRA QC,
2
2
or
2F0
-ioRA
=-
nc,
2
From the small-signal behaviour of the model we deduce
I
=0
(16)
1
where o1equals the small-signal 3 dB bandwidth.
Owing to interior time constants in the amplifier the rate of change of the output voltage is limited. When
the amplifier is largely overdriven at high frequencies this phenomenon causes a high distortion. In extreme
cases the output signal will be a triangular wave.
This maximum rate of change of the output voltage is often called the slew rate ( p ) of the amplifier.
If we examine our model under this high frequency overdriven conditions we may easily derive
i.e. the relation linking u1 and u; in the model will be
Thus we can identify and express the parameters i. and (i0RA/2) in terms of the 'black box' parameters:
Fo = small-signal open-loop gain
o1= small-signal 3 dB bandwidth
p = slew rate
As we shall later use the describing function technique, we must calculate the describing function for the
n~nlinearity.~
We assume
u,(t) = B cos ot
(19)
u;(t) =
BF(B) cos at
(20)
This procedure implies that we disregard the generation of harmonics in the nonlinearity. u; can now be
easily calculated by Fourier series expansion :
BF(B) = - A o1
n/20
exp [(2F00,/p)B cos or]- 1
cos at dt
exp [(2F001/p)B cos ot] 1
+
ANALYSIS OF NONLINEAR MODEL FOR OPERATIONAL AMPLIFIERS IN ACTIVE RC NETWORKS
17
or
*i2
exp [(2F00,/p)Bcos ot]- 1
cos wt dt
exp [(2F0w,/p)Bcos or] 1
+
This integral, together with the expression for the linear portion of the amplifier, now permits us to make
simple nonlinear analyses of active filters with slew-rate-limited amplifiers.
EXAMPLE OF NONLINEAR ANALYSIS
To check the validity of the amplifier model, a study was made of the behaviour of a simple active lowpass
filter.
The filter in Figure 6 is recognized as the well known UG c~nfiguration.~
-'invout- -
-
e =
1
29
f5L
s2+sl+l
29
a
Figure 6. UG low pass network
We shall use in the calculation a normalized cut-off frequency of 1 rad/s. Proceeding from Figure 6 we
can draw a corresponding block diagram, where N is a memoryless nonlinearity.
Figure 7. Block schematic for UG network
The two transfer functions H , ( s )and H 2 ( s )are easily calculated from Figure 6 by means of the superposition
theorem
1
I
H,(s) =
s2
+ s(2Q+ 1/Q)+ 1
= s2
+ s(2Q + 1/Q) + 1
2Qs
H2(S)
We now make the simplified assumption that
w 1
-w1
s+w,
within the frequency range of interest.
s
18
ANDERS FORSEN AND LARS KRISTIANSSON
From Figure 7 we can calculate the relation between the three voltages &(s), V2(s)and V3(s).
w1
K,,(s)Hl(s)= V2(s)+ V3(s)(l -Hi+))S
The corresponding differential equation is
We now assume according to the describing function technique
u,,(t)
=
fincos(wt + 4)
u2(t) = B C O S W ~
u3(t) = BF(B) cos mt
Insertion of these time functions in the differential equation (27) gives us
- fi,w
sin ( w t + 4 ) = Bw3 sin wt - B
W1
- Bw sin wt - w 1BF(B)w2cos wt -BF(B)w sin a t + wlBF(B)cos wt
Q
(31)
By splitting of the equation into sine and cosine parts we obtain the two corresponding equations (32) and
(33).
0 1
- finw cos 4 = B o 3 - BO - -BF(B)w
(32)
Q
- P,,,wsin
4
=
-B
w2-o,BF(B)w2+wlBF(B)
(33)
Squaring of the equations and summation gives
{
pin=
(34)
This equation relates the amplitude pinof the input signal to the input amplitude B of the nonlinearity N.
To complete the analysis we must find a relation between the amplitude of the output signal,
and the
output amplitude of the nonlinearity and an expression for the phase shift in the filter.
From the block diagram in Figure 7 with the simplification (25) we obtain directly the relation
.
tu,,
As the signal after the nonlinearity N passes an integrator, the phase shift between the output signal and the
signal after N will be - 4 2 rad. From equation (32) we obtain
We can now summarize the preceding results :
where
Po,,,is calculated from equation (34) and (35).
arg H(jw) =
II
---4
2
ANALYSIS OF NONLINEAR MODEL FOR OPERATIONAL AMPLIFIERS IN ACTIVE RC NETWORKS
19
As it is difficult to form a notion of the significance of these relations, a computer programme was written
for graphic presentation of the equations. In Figures 8(a) and 8(b) we see the result of a run with amplifier
parameters according to the nominal values for the operational amplifier 741.
A
vOUT
10
5-
0
0
5
10
15
4
N
Figure 8(a). Relation between input and output amplitudes for UG network with Q-value = 5 and resonant frequency = 15.9 kHz
PHASE
-;1
Figure 8(b). Dependence of the phase shift on the input amplitude for the filter in Figure 8(a)
20
ANDERS FORSEN AND LARS KRISTIANSSON
These parameters are :
DC gain: 100,OOO
Gain-bandwidth product : 1,OOOkHz
Slew rate: 0-5 V / p s
The input frequency in this simulation is 0.8 when normalized to the resonant frequency.
We can draw an interesting conclusion from the curves. For certain values of the input signal amplitude
there is more than one value of the output signal amplitude and of the phase shift in the filter. The possibility
of more than one output signal for a given input signal is known in the literature. The transfer from one type
of output signal to another takes place in jumps and this phenomenon is called 'the jump phenomenon'.
A good description of this phenomenon can be found in Reference 1 pp. 130-132.
MODEL VERSUS REALITY
A comparison was made between a series of measurements of an active UG network and corresponding
theoretical calculations. The parameters of the amplifier in question were measured as accurately as possible
and used as input data for the programme. Figures 9-12 show the results of this comparison both for amplitude and phase displacement. These parameters have been common for all the figures :
Amplifier : DC gain: 100,OOO
Gain-bandwidth product : 1,350 kHz
Slew rate: 0-54VIps
Filter : Resonant frequency : 15.9 kHz
Q-value: 5
The input frequency normalized to the resonant frequency is given in the figure legends. In all of these
figures the calculated results are represented by fully drawn and the measured results by dashed lines.
!5'
1
Figure 9. Input frequency (normalized):0.6
ANALYSIS OF NONLINEAR MODEL FOR OPERATIONAL AMPLIFIERS IN ACTIVE RC NETWORKS
A
-270
-180-
Figure 10. Input frequency (normalized):0.6
Figure 1 1 . Input frequency (normalized):1.2
21
22
ANDERS FORSEN AND LARS KRISTIANSSON
A
-
-270
-180'
c
- 90
I
1
%
00
5
10
15
Figure 12. Input frequency (normalized): 1.2
As will be seen, there is close correspondence and we note that we cannot verify the parts of the calculated
curves which have a negative slope. We see from the curves that a jump can take place only when the input
signal frequency is lower than the resonance frequency for the filter. This property, that a jump can occur
only at frequencies either lower or higher than the resonance frequency, is typical of the jump phenomenon.
In the literature the phenomenon is customarily ascribed to soft or hard spring force.
CONCLUSIONS
A simple but useful model of a slew rate limited operational amplifier has been achieved by considering
the saturation of the input differential stage. The model gives a good description of the jump phenomenon
in a simple active network. More complicated networks should also be possible to study with the applied
technique.
REFERENCES
1 . Stoker, Nonlinear Vibrations, Interscience, New York, 1970.
2. W. E. Hearn, 'Fast slewing monolithic operational amplifiers', IEEE J . of Solid-Slate Circuifs,SC-6, 20-24 (1971).
3. J. C. West. 'Analytical techniques for nonlinear control systems, 101-1 32, The English Universities Press Ltd., London, 1960.
4. R. P. Sallen and E. L. Key, 'A practical method of designing RC active filters', IRE Trans. Circuit Theory, CT-2, 7 4 8 5 (1955).