EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
PART I
1
Experimental characterization of operational amplifiers: a system
identification approach - Part I: theory and simulations
R. Pintelon(1), G. Vandersteen(2), L. De Locht(1, 2), Y. Rolain(1) and J. Schoukens(1)
(1) Vrije Universiteit Brussel, dept. ELEC, Pleinlaan 2, 1050 Brussels, BELGIUM
Tel. (+32-2) 629.29.44, Fax. (+32-2) 629.28.50, E-mail: Rik.Pintelon@vub.ac.be
(2) IMEC, dev. DESICS, Kapeldreef 75, 3001 Heverlee, BELGIUM
Keywords - operational amplifier, system identification, linear characteristics, nonlinear
distortions, open loop gain, common mode rejection, power supply rejection.
Abstract - Using special designed broadband periodic random excitation signals, the
open loop, the common mode, and the power supply gains of operational
amplifiers are measured and modelled. The proposed modelling technique (i) takes
into account the measurement uncertainty and the nonlinear distortions, (ii) gives
information about possible unmodelled dynamics, (iii) detects, quantifies and
classifies the nonlinear distortions, and (iv) provides opamp parameters (time
constants, gain-bandwidth product, …) with confidence bounds. The approach is
suitable for the experimental characterization of operational amplifiers (see Part II)
as well as the fast evaluation of new operational amplifiers designs using network
simulators (see Part I). Part I describes the modeling approach and illustrates the
theory on simulations.
I. INTRODUCTION
An abundant literature exists on the measurement and/or simulation of the frequency
dependent operational amplifiers (opamp) characteristics such as the open loop gain A( f )
([1-6]),
the
common
mode
gain
A CM( f )
or
common
mode
rejection
ratio
+ ( f ) , A - ( f ) or
CMRR( f ) = A( f ) ⁄ A CM( f ) ([1], [7-9]), and the power supply gains A PS
PS
+ ( f ) , PSRR -( f ) = A( f ) ⁄ A - ( f ) ([10,
power supply rejection ratios PSRR +( f ) = A( f ) ⁄ A PS
PS
11]). In all cases single sine measurements and/or simulations are performed to obtain the
operational amplifier characteristics. Except in [1], the calculation of the opamp parameters
(poles, gain-bandwidth product, …) is based on a fixed first or second order model. The
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
PART I
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proposed estimation procedures (i) do not take into account the measurement uncertainty;
(ii) give no information about the possible unmodelled dynamics (a first or second order
+ ( f ) or A - ( f ) ) and the nonlinear
model may be insufficient to explain A( f ) , A CM( f ) , A PS
PS
distortions (an operational amplifier behaves nonlinearly for “sufficiently large” inputs); and
(iii) provide no confidence bounds on the estimated opamp parameters.
In this series of two papers we present a system identification approach to the experimental
characterization of operational amplifiers. The proposed method uses the matched resistor
circuit shown in Figure 1 in combination with special designed broadband periodic random
excitations signals (multisines [12]) for measuring the open loop gain A( f ) , the common
+ ( f ) and A - ( f ) . Following the
mode gain A CM( f ) , and the power supply gains A PS
PS
measurement procedure of [13, 14], the periodic random excitation signal allows to gather in
one single measurement (i) the frequency response functions in a broad frequency band, (ii)
the noise levels, and (iii) the class (even and/or odd degree) and the level of the nonlinear
distortions in the output spectrum. Averaging of the FRF measurements over different
realizations of the periodic random excitation gives the variance information needed in the
parameter estimation procedure. Next a parametric model is identified (estimation of the
model parameters and model order selection), resulting in a best linear approximation of the
characteristic together with its uncertainty (due to measurement noise and/or nonlinear
distortions). Note that no prior first or second order model is imposed in the procedure.
Before going to silicon the performance of new operational amplifier designs is verified via
simulations. The number of transistors in such circuits is usually that large that the
calculation time of network simulators is a non negligible part of the design loop. Hence,
there is a need for a fast evaluation of the linear and the nonlinear behavior of operational
amplifiers. In Part I a method is proposed that allows to evaluate simultaneously the linear
opamp characteristics, the level of the nonlinear distortions, and the type of nonlinearity
(even and/or odd degree distortions).
Since the measurement of the common mode rejection ratio is (very) sensitive to the resistor
mismatch R 2 ⁄ R 1 ≠ R 4 ⁄ R 3 in Figure 1, a simple calibration procedure is proposed in Part II
to eliminate this error. It consists in making a bridge measurement with the four resistances
when the opamp is removed. Note that the calibration procedure also accounts for the nonzero output impedance of the voltage buffer in the feedback loop, and for the possible
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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frequency dependent behavior of the resistances (MHz range).
The contributions of this series of two papers hence are:
(i) (simultaneous) measurement of the opamp characteristics, the noise levels, and the
levels of the nonlinear distortions using special designed multisines,
(ii) classification of the nonlinear distortions of the opamp in odd and even degree
contributions,
(iii) parametric identification of the opamp characteristics taking into account the
disturbing noise and the nonlinear distortions,
(iv) calculation of opamp parameters (time constants, gain-bandwidth product, …) with
uncertainty bounds,
(v) a simple calibration procedure to eliminate the resistor mismatch in the common
mode rejection ratio measurement,
(vi) fast evaluation of the performance of a new opamp design using a network simulator
in combination with special designed multisines.
Part I elaborates the measurement/identification procedure and illustrates the theory on a
network simulation of an LM741 operational amplifier. Part II discusses the calibration of the
experimental set up and illustrates the whole identification approach on real measurements
of two UA741’s.
II. BASIC MEASUREMENT SET UP
In this section we study the basic matched resistor circuit used for measuring the open loop
+ ( f ) , A - ( f ) gains (see Figure
A( f ) , the common mode A CM( f ) , and the power supply A PS
PS
1). The circuit consists of four resistances, a voltage buffer, and the device under test (DUT).
The voltage buffer in series with the resistor R 2 (ideally, infinite input impedance, zero
output impedance) suppresses the effect of the non-zero output impedance of the
operational amplifier on the measurements. The analysis of the circuit starts from the basic
open loop equation of the unloaded DUT
A CM( f )
+ ( f )x +( f ) + A - ( f )x -( f )
- ( v +( f ) + v -( f ) ) + A PS
v out( f ) = A( f ) ( v +( f ) – v -( f ) ) + -----------------PS
2
(1)
with v + and v - the voltages of the + and - inputs of the DUT; and x + and x - the varying
signals at the positive ( V cc ) and negative ( V ee ) power supply terminals. The calculations
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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made in the sequel of this section assume that the buffer in the feedback loop is an ideal
voltage buffer (infinite input impedance, zero output impedance) with gain H ( f ) , and that
the resistors are not matched ( R 2 ⁄ R 1 ≠ R 4 ⁄ R 3 ). Proceeding in this way R 2 also accounts for
the non-zero output impedance of the buffer.
A. Open loop gain
The switch in Figure 1 is in position 1, v g(t) ≠ 0 , V cc = + 15 V , V ee = - 15 V , and v out , v are measured. Using v + = 0 , and assuming that A CM « A , it follows from (1) that
v out( f )
A( f ) = – --------------v -( f )
(2)
B. Common mode rejection ratio
The switch in Figure 1 is in position 2, v g(t) ≠ 0 , V cc = + 15 V , V ee = - 15 V , and v out , v in
are measured. Using
v +( f ) = α 43 v in( f )
Ri
with α ij = ---------------Ri + R j
v -( f ) = α 12 H ( f )v out( f ) + α 21 v in( f )
(3)
it follows from (1) that
T ( f ) ( 1 + α 12 H ( f ) A( f ) ) + β A( f )
v out( f )
- with T ( f ) = --------------- and β = α 21 – α 43
A CM( f ) = 2 ------------------------------------------------------------------------------α 12 H ( f )T ( f ) + α 21 + α 43
v in( f )
(4)
(see [1]). Hence, the common mode rejection ratio (CMRR) equals
A( f ) ( α 12 H ( f )T ( f ) + α 21 + α 43 )
A( f )
CMRR( f ) = ------------------- = ----------------------------------------------------------------------------------------------------------------A CM( f )
2 [ T ( f ) ( 1 + α 12 H ( f ) A( f ) ) + ( α 21 – α 43 ) A( f ) ]
(5)
C. Power supply rejection ratio
The switch in Figure 1 is in position 2, v g(t) = 0 , Z g = 0 , either V cc = + 15 V + x +(t) and
V ee = - 15 V , or V cc = + 15 V and V ee = - 15 V + x -(t) , and v out and x + or x - are
measured. Using v + = 0 and v - = α 12 H ( f )v out , and assuming that A CM « A , it follows
from (1) that
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
PART I
v out( f )
± ( f ) = ( 1 + α H ( f ) A( f ) )T ±( f ) with T ±( f ) = --------------A PS
12
x ±( f )
5
(6)
where superscript ± means that the signal source is connected to either the positive (+) or
negative (-) power supply terminal. Hence, the power supply rejection ratios (PSRR) equal
A( f )
A( f )
PSRR ±( f ) = ----------------- = -------------------------------------------------------------±
A PS ( f )
( 1 + α 12 H ( f ) A( f ) )T ±( f )
(7)
III. MEASUREMENT PROCEDURE
A. Introduction
The operational amplifier characteristics (2-7) are measured using random phase multisines.
These are periodic signals consisting of the sum of N harmonically related sine waves with
user defined amplitudes R̂ k and random phases ϕ k
s(t) = N – 1 / 2 ∑
N
k = –N, k ≠ 0
R̂ k e
j ( 2πtkf max ⁄ N + ϕ k )
(8)
where s = v g , x + , or x - (see [13, 14]). The phases ϕ k are randomly chosen such that
E{ e
j ϕk
} = 0 , with E{ } the expected value; for example, ϕ k is uniformly distributed in
[ 0, 2π ) . In [14, 15 and 16] it has been shown that the frequency response function (FRF) of
a wide class of nonlinear systems, obtained using a random phase multisine (8) with N
sufficiently large, can be written as
G( f k) = G 0( f k) + G B( f k) + G S( f k) + N G( f k)
(9)
with G 0 the FRF of the true underlying linear system, N G the measurement noise, and
f k = k f max ⁄ N . G B is the bias or deterministic nonlinear contribution which depends on
the odd degree nonlinear distortions and the power spectrum of the input only, and G S is
the zero mean stochastic nonlinear contribution
2 ( f ), E{ G ( f )G ( f )} = O(N – 1)
E{ G S( f k)} = 0, E{ G S( f k) 2 } = σ G
k
S k S l
E{ G S( f k)G S( f l)} = O(N – 1) for ( k ≠ l )
(10)
where the overline denotes the complex conjugate, and where the expected values are
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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taken w.r.t. the different random phase realizations of the excitation (8). Due to property
(10), the stochastic nonlinear contributions G S act as circular complex noise for N
sufficiently large. Hence, over different realizations of the random phase multisine, G S
cannot be distinguished from the measurement noise N G . The sum
G R( f k) = G 0( f k) + G B( f k)
(11)
is the best linear approximation to the nonlinear system for the class of Gaussian excitation
signals (normally distributed noise, periodic Gaussian noise, and random phase multisines)
with a given power spectrum (see [14, 15 and 16] for the details). It can be approximated
arbitrarily well by a rational form in the Laplace variable s .
Since the even degree nonlinear distortions do not affect the bias term G B while they
increase the variance of G S , the variability of the FRF measurement (9) can be reduced by
using random phase multisines which excite the odd harmonics only [12, 13]: (8) with
R̂ 2k = 0 . These so called odd random phase multisines allow to detect the presence and the
level of even degree nonlinear distortions by looking at the even harmonics in the output
spectrum. To detect the presence and the level of the odd degree nonlinear distortions one
should leave out some of the odd harmonics in the odd random phase multisine. The
optimal strategy consists in splitting the odd harmonics in groups of equal number of
consecutive lines, and eliminating randomly one line out of each group. This can be done
for a linear as well as a logarithmic frequency distribution (see [12] for the linear case). The
resulting excitation, an odd random phase multisine with random harmonic grid (linear or
logarithmic frequency distribution), is used throughout the paper.
Two measurement strategies are proposed in the sequel of this section (see Figure 2): the
first uses one phase realization of the odd random phase multisine with random harmonic
grid and is suitable for A( f ) only, while the second uses multiple phase realizations (each
± (f).
time with the same random harmonic grid) and is suitable for A( f ) , A CM( f ) , and A PS
B. First measurement strategy
The first strategy for measuring the open loop gain A( f ) (2) and its uncertainty consists of
the following steps:
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CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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1. Choose the amplitude spectrum R̂ 2k – 1 ( k = 1, 2, …, N ⁄ 2 ) and the frequency
resolution f max ⁄ N of the odd random phase multisine (8) ( R̂ 2k = 0 ).
2. Split the excited odd harmonics (linear or logarithmic frequency distribution) in
groups of equal number of consecutive lines, and eliminate randomly one odd
harmonic out of each group (for example, hundred excited odd harmonics are split in
twenty five groups of four consecutive excited odd harmonics, and one out of the
four odd harmonics is randomly eliminated in each group).
3. Make a random choice of the phases ϕ k of the non-zero harmonics of the random
phase multisine (8), and calculate the corresponding time signal v g(t) ( s = v g ).
4. Apply the excitation v g(t) to the circuit (see Figure 1) and measure P ≥ 6 consecutive
periods1 of the steady state input v -(t) and output v out(t) (see Figure 2, one
horizontal line).
[ p ] , p = 1, 2, …, P , one can calculate the
From the P noisy input/output spectra v - [ p ] , v out
average open loop gain and its sample variance
1
 = --- ∑P
A [ p ] , σ̂ 2 =
Â
P p=1
A [ p ] – Â 2
P
-------------------------p = 1 P(P – 1)
∑
[ p ] ( f ) ⁄ v - [ p ]( f )
with A [ p ]( f ) = – v out
(12)
Since the sample variance σ̂ 2 is calculated over P consecutive periods of one particular
Â
realization of the random phase multisine, it is clear that it only contains the contribution of
the measurement noise to the open loop gain measurement (E{ σ̂ 2 } = var(N G) ⁄ P ).
Â
The presence and the level of the odd and even degree nonlinear distortions is revealed by
analyzing the non-excited frequencies (= missing harmonics in v g(t) ) in the output spectrum
[ p ]( f ) . However, straightforward interpretation of the output spectrum is impossible.
v out
Indeed, due to the feedback loop in the set up for measuring A( f ) (see Figure 1, feedback
resistor R 2 ), the input spectrum v - [ p ]( f ) is also contaminated by the nonlinear distortions of
[ p ]( f ) are partially
the device under test (DUT) and, hence, the non-excited frequencies in v out
due to the linear feed through of the distorted input v - [ p ]( f ) . Since the DUT is dominantly
1. At least six periods are needed to preserve the properties of the maximum likelihood estimator used in the
parametric modeling step [14].
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
PART I
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c [ p ]( f ) is obtained by subtracting the linear contribution of
linear, a first order correction v out
the DUT from the output at the non-excited harmonics
c [ p ]( f )
v out
[ p]
 v out ( f )
= 
[ p ]( f ) – A [ p ]( f ) ( – v - [ p ]( f ) )
 v out
excited harmonic in v g
(13)
non-excited harmonic in v g
where, according to the frequency resolution, A [ p ]( f ) at the non-excited frequencies is
obtained through linear or cubic interpolation of the values at the excited frequencies.
c [ p ]( f )
Further, the sample mean and sample variance of the corrected output spectrum v out
1 P
c = -v c [ p ] , σ̂ v̂2c =
v̂ out
P ∑ p = 1 out
out
c [ p ] – v̂ c 2
v out
out
P
--------------------------------p = 1 P(P – 1)
∑
(14)
are calculated. Within the measurement uncertainty σ̂ v̂ c , the presence and the level of the
out
c
even and non-excited odd harmonics in v̂ out
reveals the presence and the level of
respectively the even and odd degree nonlinear distortions at the non-excited frequencies
c ? For odd
[12, 13]. What about the non-linear distortions at the excited odd harmonics in v̂ out
random phase multisines with random harmonic grid (linear or logarithmic distribution), the
level of the stochastic non-linear contributions σ voutS at the excited odd harmonics is
obtained by linear or cubic interpolation of the level at the non-excited odd harmonics [12].
c is bounded by
The bias contribution v outB at the excited odd harmonics in v̂ out
1
--- σ voutS ≤ v outB ≤ γ σ voutS
γ
(15)
where γ is a heuristic factor depending on the power spectrum of the excitation and the
system (see [12, 13] for the linear frequency distribution). Typical values for γ lie between
two and ten. Finally, calculating
σ AS( f ) = σ voutS( f ) ⁄ v -( f ) and A B( f ) = v outB( f ) ⁄ v -( f )
(16)
at the excited odd harmonics gives the level of the stochastic σ AS( f ) and the bias A B( f )
contributions on the open loop gain measurement A( f ) .
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CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
PART I
9
C. Second measurement strategy
The first three steps of the second strategy for measuring the operational amplifier
± ( f ) (6) are identical to the first strategy for
characteristics A( f ) (2), A CM( f ) (4) and A PS
measuring A( f ) (see Section III.B). In addition to steps 1-3 of Section III.B we have:
4. Apply the excitation s(t) ( s = v g , x + , or x - ) to the circuit in Figure 1 and measure P
± ) from P ≥ 2 consecutive
frequency response functions G( f ) ( G = A , A CM , or A PS
periods of the steady state response.1
5. Repeat steps 3 and 4 M ≥ 6 times2 (see Figure 2, the M horizontal lines).
From the M × P noisy frequency response functions G [ m, p ] , m = 1, 2, …, M and
p = 1, 2, …, P , one can calculate for each experiment the average frequency response
function (FRF) Ĝ [ m ] and its sample variance σ̂ 2 [ m ]
Ĝ
1
Ĝ [ m ] = --- ∑P
G [ m, p ] , σ̂ 2 [ m ] =
Ĝ
P p=1
G [ m, p ] – Ĝ [ m ] 2
P
----------------------------------------p=1
P(P – 1)
∑
(17)
An additional averaging over m gives the final FRF Ĝ of the whole measurement procedure
Ĝ =
Ĝ [ m ]
M
------------ ,
m=1 M
∑
σ̂ 2 =
Ĝ
Ĝ [ m ] – Ĝ 2
M
---------------------------m = 1 M (M – 1)
∑
(18)
together with its sample variance σ̂ 2 .
Ĝ
From (9), (17) and (18) it follows that
E{ σ̂ 2 [ m ] } = var(N G) ⁄ P and E{ σ̂ 2 } = ( var(G S) + var(N G) ⁄ P ) ⁄ M
Ĝ
Ĝ
(19)
Hence, if the system is linear ( G S = 0 ), then σ̂ 2 should be approximately equal to the
Ĝ
mean value of σ̂ 2 [ m ] divided by M
Ĝ
1. At least two periods are needed to calculate a sample variance.
2. See footnote 1 on page 7.
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
σ̂ 2
Ĝ n
If σ̂ 2 (18) is larger than σ̂ 2
Ĝ
Ĝ n
PART I
10
1
= -------- ∑M σ̂ 2 [ m ]
M 2 m = 1 Ĝ
(20)
(20), then this is an indication that the systems behaves
nonlinearly ( G S ≠ 0 ), and
2 – σ̂ 2 )
 M ( σ̂ Ĝ
Ĝ n
= 
ĜS
0
σ̂ 2
σ̂ 2 > σ̂ 2
Ĝ
Ĝ n
(21)
otherwise
is an estimate of var(G S) . Using (15), (16) and (21), the nonlinear bias contribution G B in
(9) can be bounded by
1
--- σ̂ ≤ G B ≤ γ σ̂
ĜS
γ ĜS
(22)
where γ is defined in (15) (see [12, 13] for the details).
D. Discussion
The advantages of the first measurement strategy are (i) the reduced measurement time (all
information is gathered in one single experiment); and (ii) the classification in odd and even
degree nonlinear distortions. Its disadvantages are (i) the variance of the FRF measurement
only accounts for the measurement noise; and (ii) an approximation (extrapolation) is
needed to characterize the nonlinear stochastic contributions on the FRF measurement.
The advantages of the second measurement strategy are (i) the contribution of the stochastic
nonlinear distortions to the FRF measurement are obtained without any approximation
(extrapolation); and (ii) the variance of the FRF measurement accounts for the measurement
noise as well as the stochastic nonlinear distortions. Its disadvantages are (i) the increased
measurement time (several experiments are needed); and (ii) except for the open loop gain,
no classification in odd and even degree nonlinear distortions can be made.
If the input signal-to-noise ratio is smaller than 6 dB, then the relative bias on the FRF
measurement can no longer be neglected [14, 17]. It can be reduced by appropriate
averaging of the M × P noisy input/output spectra before calculating the FRF. At the cost of
knowing exactly a reference signal (typically the signal stored in the arbitrary waveform
generator), the second measurement strategy can be generalized to handle input/output
spectra (see [18] for the details).
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
IV. MODELING
PART I
11
PROCEDURE
± ) is modelled by a
The measured frequency response function G( f ) ( G = A , A CM , A PS
rational form in the Laplace variable s
n
G(s, θ) =
with
b
b sn
∑
n=0 n
---------------------------------
na
∑m = 0 am
(23)
sm
θ T = [ a 0, a 1, …, a n , b 0, b 1, …, b n ] . The model parameters are obtained by
a
b
minimizing the sample maximum likelihood cost function
F
V SML(θ, Z ) =
Ĝ( f k) – G(s k, θ) 2
-------------------------------------------∑
2(f )
σ̂
k
k=1
(24)
Ĝ
w.r.t. θ , where s k = j2π f k and where the summation index k runs over the excited
harmonics [14, 19]. The values of n a and n b are obtained by minimizing the minimum
description length (MDL) model selection criterion
log(2F )
V SML(θ̂, Z )  1 + n θ ------------------

2F 
(25)
with θ̂ the minimizer of (24), n θ the number of free parameters in (23), and F the number
of frequencies in (24) (see [20]). The result is an estimate θ̂ together with its covariance
matrix Cov(θ̂) . Finally, from θ̂ and Cov(θ̂) the poles, the zeros, the gain and their
uncertainties are calculated (see [14] for the theoretical background and [21] for the software
implementation). Finally, the value of cost function at its minimum V SML(θ̂, Z ) is compared
to the 95% uncertainty interval of the cost function constructed under the assumption that no
modeling errors are present
[ γ 1 V noise – 2 γ 2 V noise, γ 1 V noise + 2 γ 2 V noise ]
where
M–1
( M – 1 )3
V noise = F – n θ ⁄ 2 , γ 1 = -------------- , and γ 2 = ----------------------------------------M–2
( M – 2 )2( M – 3 )
(26)
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(see [14]). This hypothesis test allows to verify whether or not residual model errors are
present.
V. SIMULATION RESULTS
A. Introduction
The whole measurement/modeling procedure is illustrated on a network simulation (Spectre
RF® circuit simulator version 4.4.6 of Cadence Design Systems Inc.) of an LM741 operational
amplifier (see [22], p. 424 for the circuit). For the open loop and common mode gain
simulations the voltage source v g(t) with impedance Z g in Figure 1 can be implemented in
two ways in the circuit simulator: either as one voltage source where the random phase
multisine signal is defined by an array of numbers, or as the series connection of F (=
number of excited harmonics in the multisine) ideal sinewave voltage sources with
appropriate amplitude and phase in series with the impedance Z g . The first solution has the
disadvantage that the time derivatives of the simulated random phase multisine signal are
not exact. The time derivatives of the second solution are exact, however, it has the
disadvantage that the number of nodes of the circuit increases with the number of excited
harmonics F . Replacing the F series connected ideal voltage sources in series with Z g by
F parallel connected ideal current sources in parallel with Z g , removes the drawback of the
second solution. The solution with current sources is used in all simulations.
For the power supply gain simulations an ideal voltage source x ±(t) in series with an ideal
DC source must be implemented ( V cc = + 15 V + x +(t) and V ee = – 15 V or V cc = + 15 V
and V ee = – 15 V + x -(t) ). The ideal voltage source x ±(t) is realized by F parallel
connected ideal sinewave current sources and one ideal current controlled voltage source.
The goal of the simulation is the validation of the first measurement strategy. Therefore, the
open loop gain is measured using the first and second measurement strategy. These results
are reported in Section V.B.
Although no disturbing noise is added in the simulations, the two measurement strategies of
Section III are applied with P = 2 . It allows to verify the level of the arithmetic noise of the
simulator (= variability of the steady state response from one period to the other), but gives
no information about the systematic error of the integration method used (here the
EXPERIMENTAL
CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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13
trapezium rule). For linear circuits the systematic error of the trapezium rule is eliminated by
a bilinear warping of the frequency axis
T
2
ω → -----tan(ω -----s)
2
Ts
(27)
where T s is the sample period of the simulator. For nonlinear circuits with a dominantly
linear behavior the frequency warping is a first order correction. It is applied to the
simulation data for the parametric modeling of the operational amplifier characteristics.
B. Simulation of the operational amplifier characteristics
The set up of Figure 1 is used with Z g = 50 Ω , R 1 = R 3 = 300 Ω , R 2 = R 4 = 12 kΩ ,
V cc = – V ee = 15 V , ideal voltage buffers, and odd random phase multisine excitations
with random harmonic grid (see Section III). Of each signal, P = 2 periods and
N = 2 16 = 65536 points per period are calculated at the sampling rate f s = 62.5 kHz . The
frequencies of the odd random phase multisines are logarithmically distributed between
f min = f s ⁄ N ≈ 0.95 Hz
and
f max = 9999 f s ⁄ N ≈ 9.5 kHz . Of each group of three
consecutive odd harmonics, one odd harmonic is randomly eliminated. The resulting odd
random phase multisines with random harmonic grid contain F = 299 odd excited
harmonics. All frequency response function calculations are performed M = 25 times with
different phase realizations of the odd random phase multisines.
For the open loop gain simulation the source level v g(t) is such that v -(t) = 6.1 mVrms .
Figures 3 and 4 show the results. From Figure 3 it can be seen that v -( f ) is contaminated by
the nonlinear distortions: the non-excited harmonics of v g( f ) (‘o’ and ‘*’) are well above the
arithmetic noise level (solid line) in v -( f ) . After correction as in (13), it follows from the
c ( f ) that the odd degree nonlinear distortions are dominant (the level of
output spectrum v out
the even harmonics ‘*’ is 20 dB below the level of the non-excited odd harmonics ‘o’), and
that the nonlinear distortions are quite large in the band [1 Hz, 30 Hz] and decrease with
increasing frequency. The latter is not visible in the original (uncorrected) output spectrum
because of the linearizing effect of the feedback loop (resistor R 2 in Figure 1).
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CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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Figure 4 compares the open loop gain obtained from one realization of the multisine
excitation (first measurement strategy, see Section III.B) to that obtained from M = 25
multisine realizations (second measurement strategy, see Section III.C). It can be seen that
the second measurement strategy leads to a less noisy FRF (see the phase characteristic),
and a lower level of the stochastic nonlinear distortions. Both effects are explained by the
averaging of the FRF over the M = 25 multisine realizations leading to a reduction of the
stochastic nonlinear distortions A S in (9) by a factor
M = 5 = 14 dB . Since the prediction
2 (16) in the first measurement strategy
of the level of the stochastic nonlinear distortions σ AS
(gray +) is based on a single experiment, it is much noisier than σ̂ 2 (18) obtained by the
Â
2 , averaged
second measurement strategy (gray bold line). From Figure 5 it follows that σ AS
over the M = 25 multisine experiments, equals σ̂ 2
ÂS
= Mσ̂ 2 (eq. (21) with σ̂ 2
Ĝn
Â
= 0)
within a few dB.
The second measurement strategy has also been applied to the common mode and power
supply gain measurements. It turns out that the level of the nonlinear distortions on the
common mode and power supply gains is much smaller than that on the open loop gain.
C. Modeling of the operational amplifier characteristics
To suppress the systematic error of the trapezium integration rule, the frequency axis is
warped
according
to
(27).
It
transforms
the
original
digital
frequency
band
[ 0.95 Hz, 9.5 kHz ] to the analog frequency band [ 0.95 Hz, 10.3 kHz ] . The open loop gain
measurements (18) obtained by the second measurement strategy are modelled by a rational
form (23). Using the modeling procedure of Section IV it follows that a model of order
n b = 4 , n a = 5 is necessary to explain the data: the minimum of the cost function equals
V SML(θ̂, Z ) = 273.7 , and lies in the 95% confidence region [ 269.4, 344.2 ] constructed
under the hypothesis that no model errors are present (26). A correlation analysis of the
residuals shows that the residuals are white, which confirms that all dynamics are captured
by the model. As could be expected for a nonlinear RC-network, all poles and zeros of the
identified model lie on the real axis. This is not the case when no frequency warping (27) is
applied to the simulation data.
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Table 1 gives the estimated DC open loop gain Â(0) and the cut-off frequency fˆ0 . Although
the influence of the stochastic nonlinear distortions on the estimate θ̂ tends to zero as either
F → ∞ or M → ∞ or F , M → ∞ , θ̂ ( Â(0) , fˆ0 , …) still depends on the nonlinear distortions
through the bias contribution G B in (9). Hence, the values given in Table 1 depend on the
particular power spectrum of the excitation. However, they can be used to predict the
response to Gaussian noise, periodic noise and random phase multisines excitations [16].
VI. CONCLUSIONS
A system identification approach for modeling the linear operational amplifier characteristics
has been presented. It includes the choice of the excitation; the detection, qualification and
quantification of the nonlinear distortions; and the parametric modeling. Two measurement
procedures have been proposed that allow to measure (simultaneously) the linear
operational amplifier characteristics, the noise level, and the level of the odd and even
nonlinear distortions. The first measurement approach is especially useful for the fast
evaluation of (large scale) nonlinear circuits using network simulators. In Part I of this series
of two papers the theory has been validated via simulations on an LM741 operational
amplifier.
VII. ACKNOWLEDGEMENT
This work is sponsored by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish
Government (GOA-IMMI) and the Belgian Program on Interuniversity Poles of Attraction
initiated by the Belgian State, Prime Minister’s Office, Science Policy programming (IUAP 5).
REFERENCES
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amplifier characteristics in the frequency domain,” IEEE Trans. Instrum. Meas., vol. 34,
no. 1, pp. 59-64, 1985.
[2] K. Higuchi and H. Shintani, “New measurement methods of dominant-pole-type
operational amplifier parameters,” IEEE Trans. Industr. Electr., vol. 34, no. 3, pp. 357365, 1987.
[3] S. S. Awad, “A simple method to estimate the ratio of the second pole to the gainbandwidth product of matched operational amplifiers,” IEEE Trans. Instrum. Meas., vol.
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CHARACTERIZATION OF OPERATIONAL AMPLIFIERS:
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39, no. 2, pp. 429-432, 1990.
[4] S. Natarajan, “A simple method to estimate gain-bandwidth product and the second pole
of the operational amplifiers,” IEEE Trans. Instrum. Meas., vol. 40, no. 1, pp. 43-45, 1991.
[5] S. Porta and A. Carlosena, “On the experimental methods to characterize the opamp
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[6] G. Giustolisi and G. Palumbo, “An approach to test open-loop parameters of feedback
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[7] R. Pallás-Areny and J. G. Webster, “Common mode rejection ratio in differential
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[8] M. E. Brinson, D. J. Faulkner, “New approaches to measurement of operational amplifier
common-mode rejection ratio in the frequency domain,” IEE proc. Circuits Devices Syst.,
vol. 142, no. 4, pp. 247-253, 1995.
[9] G. Giustolisi, G. Palmisano, and G. Palumbo, “CMRR frequency response of CMOS
operational transconductance amplifiers,” IEEE Trans. Instrum. Meas., vol. 49, no. 1, pp.
137-143, 200.
[10] M. S. J. Steyaert, and W. M. C. Sansen, “Power supply rejection ratio in operational
transconductance amplifiers,” IEEE Trans. Circuits Syst., vol. 37, no. 9, pp. 1077-1084,
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[11] M. E. Brinson and D. J. Faulkner, “Measurement and modelling of operational amplifier
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[12] K. Vanhoenacker, T. Dobrowiecki and J. Schoukens, “Design of multisine excitations to
characterize the nonlinear distortions during FRF-measurements,” IEEE Trans. Instrum.
and Meas., vol. 50, no. 5, pp. 1097-1102, 2001.
[13] J. Schoukens, R. Pintelon and T. Dobrowiecki, “Linear modeling in the presence of
nonlinear distortions,” IEEE Trans. Instrum. and Meas., vol. 51, no. 4, pp. 786-792, 2002.
[14] R. Pintelon and J. Schoukens, System Identification: A Frequency Domain Approach,
IEEE Press, New York (USA), 2001.
[15] J. Schoukens, T. Dobrowiecki, and R. Pintelon, “Parametric identification of linear
systems in the presence of nonlinear distortions. A frequency domain approach,” IEEE
Trans. Autom. Contr., vol. AC-43, no. 2, pp. 176-190, 1998.
[16] R. Pintelon and J. Schoukens, “Measurement and modeling of linear systems in the
presence of non-linear distortions,” Mechanical Systems and Signal Processing, vol. 16,
no. 5, pp. 785-801, 2002.
[17] R. Pintelon and J. Schoukens, “Measurement of Frequency Response Functions using
EXPERIMENTAL
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Periodic Excitations, Corrputed by Correlated Input/Output Errors,” IEEE Trans. Instrum.
and Meas., vol. 50, no. 6, pp. 1753-1760, 2001.
[18] R. Pintelon, P. Guillaume, S. Vanlanduit, K. De Belder and Y. Rolain, “Identification of
Young’s modulus from broadband modal analysis experiments,” Mechanical Systems
and Signal Processing, vol. ?, no. ?, pp. ?-?, 2003.
[19] R. Pintelon, J. Schoukens, W. Van Moer and Y. Rolain, “Identification of linear systems in
the presence of nonlinear distortions,” IEEE Trans. Instrum. and Meas., vol. 50, no. 4,
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[20] Schoukens J., Y. Rolain and R. Pintelon, “Modified AIC rule for model selection in
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[21] I. Kollár, J. Schoukens, R. Pintelon, G. Simon, and G. Román, “Extension for the
Frequency Domain System Identification toolbox for Matlab: Graphical user interface,
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System Identification, Santa Barbara (USA), June 21-23, 2000.
[22] P. R. Gray, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, John
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EXPERIMENTAL
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Figure and Table captions
Figure 1: Basic scheme for measuring/simulating the characteristics of an operational
amplifier ( R 1 = R 3 and R 2 = R 4 ).
Figure 2: Frequency response function G( f ) measurement: applying M different random
phase realizations of the excitation, and measuring each time P periods of length T 0 after a
waiting time T W .
Figure 3: Spectral content of the signals of one experiment (realization excitation) of the
open loop gain simulation. Top row: minus terminal DUT v -( f ) , middle row: output DUT
c ( f ) . Legend: ‘+’ excited odd harmonics,
v out( f ) , and bottom row: corrected output DUT v out
‘o’ odd non-excited harmonics, ‘gray *’ even non-excited harmonics, ‘__’ standard deviation
excited harmonics, ‘....’ standard deviation non-excited harmonics.
Figure 4: Open loop gain simulation. Top row: first measurement strategy, and bottom row:
second measurement strategy. Legend: ‘bold black line’ frequency response function, ‘solid
line’ standard deviation measurement noise, ‘bold gray lines’ total sample variance, and ‘gray
+’: predicted level of the stochastic nonlinear distortions.
Figure 5: Comparison of the predicted levels of the stochastic nonlinear distortions A S on
the open loop gain obtained by the first (gray +) and second (solid line) measurement
strategy.
Table 1: Some open loop gain model parameters and their uncertainty.
EXPERIMENTAL
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Figure 1: Basic scheme for measuring the characteristics of an operational amplifier
( R 1 = R 3 and R 2 = R 4 ).
v-
v in
R2
1
Vcc
R1
Zg
(1)
vg
vout
v+
(2)
R3
R4
DUT
Vee
EXPERIMENTAL
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Figure 2: Frequency response function G( f ) measurement: applying M different random
phase realizations of the excitation, and measuring each time P periods of length T 0 after a
waiting time T W .
P
transient
G [1,1] G [1,2]
T0
T0
G [2,1] G [2,2]
M
TW
T0
T0
G [M,1] G [M,2]
TW
T0
T0
G [1,P]
T0
...
...
TW
...
...
G [2,P]
T0
G [M,P]
T0
EXPERIMENTAL
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Figure 3: Spectral content of the signals of one experiment (realization excitation) of the
open loop gain simulation. Top row: minus terminal DUT v -( f ) , middle row: output DUT
c ( f ) . Legend: ‘+’ excited odd harmonics,
v out( f ) , and bottom row: corrected output DUT v out
‘o’ odd non-excited harmonics, ‘gray *’ even non-excited harmonics, ‘__’ standard deviation
excited harmonics, ‘....’ standard deviation non-excited harmonics.
Input spectrum
-60
Amplitude (dBV)
-100
-140
-180
-220 0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
Output spectrum
-20
Amplitude (dBV)
-60
-100
-140
-180 0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
Corrected output spectrum
-20
Amplitude (dBV)
-60
-100
-140
-180 0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
EXPERIMENTAL
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Figure 4: Open loop gain simulation. Top row: first measurement strategy, and bottom row:
second measurement strategy. Legend: ‘bold black line’ frequency response function, ‘solid
line’ standard deviation measurement noise, ‘bold gray lines’ total sample variance, and ‘gray
+’: predicted level of the stochastic nonlinear distortions.
Open loop gain
Open loop gain
100
20
-20
20
Phase (°)
Amplitude (dB)
60
-20
-60
-60
-100 0
10
1
10
2
10
Frequency (Hz)
3
10
-100 0
10
4
10
1
10
Open loop gain
2
10
Frequency (Hz)
3
10
4
10
Open loop gain
100
20
-20
20
Phase (°)
Amplitude (dB)
60
-20
-60
-60
-100 0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
-100 0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
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Figure 5: Comparison of the predicted levels of the stochastic nonlinear distortions A S on
the open loop gain obtained by the first (gray +) and second (black solid line) measurement
strategy.
Variance stochastic nonlinear distortions
100
Amplitude (dB)
60
20
-20
-60 0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
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Table 1: Some open loop gain model parameters and their uncertainty.
A(0)
f 0 (Hz)
estimate
standard deviation
66145
(96.4 dB)
19
11.813
0.003