Op-Amp With Complex Impedance
ZF
Z1
Vin
Vo
+
ZL
Inverting Configuration
•
•
•
•
•
•
•
Av = - (ZF/Z1)
“-” : 180° phase shift
Z=a±jb
Z = M <θ (polar form)
M = Sqrt(a2 + b2)
θ = tan-1 (b/a)
Z = M Cosθ + j M Sinθ
Op-Amp With Complex Impedance
ZF
Z1
Vo
+
ZL
Vin
Noninverting Configuration
• Av = 1+ (ZF/Z1)
• Av = (Z1+ZF)/Z1
Differentiator
• Differentiator: circuit whose output is proportional to
the derivative of its input
• Derivative of a function is the instantaneous slope or
rate of change of function
• Output of differentiator is proportional to the rate of
change of input signal, with respect to time
• Output of op amp differentiator will always lag input by
90° (inversion of true derivative)
V(t)
dv/d
t
V’(t)
Differentiator
Operational Amplifiers and Linear
Integrated Circuits: Theory and Applications
by Denton J. Dailey
Differentiator
• l Av l = l R/(1/jωC) l = l jωRC l = ωRC
• Av = -ωRC <90 = ωRC <-90
R
C
-
Vin
Vo
+
RL
Differentiator
• Main problem with op
amp differentiator is noise
sensitivity
• Gain of ideal differentiator
is zero at dc, and increases
with frequency at a rate of
20 dB/decade
• High frequency noise will
tend to be amplified
greatly
electronics-tutorials.ws
Practical Differentiator
RF
R1
C
-
Vin
Vo
+
RL
• To reduce gain to high frequency noise, a resistor is
placed in series with the input resistor
Practical Differentiator
• Problem: noise at high frequency
• To reduce noise at high frequency a resistor is placed in
series with the input capacitor
• To reduce noise, R1 < RF
• R1 may be chosen such that 10R1< RF to reduce high
frequency gain and noise
• Before adding R1: Gain characteristics of unmodified
differentiator is superimposed on a typical op-amp
open-loop Bode plot; differentiator will act correctly up
to f0
• After adding R1: differentiator gain levels off at f1
l A l (dB)
Practical Differentiator
Before adding R1
l A l (dB)
AOL
AOL
f0
Log f
f1 = 1/(2πR1C)
After adding R1
f1
Log f
Differentiation of Nonsinusoidal Inputs
• Linear ramp input:
V0 = -RCk
K: function slope (V/s)
• Triangular input:
V0 = -RCkn
Kn: function slope (V/s)
Operational Amplifiers and Linear
Integrated Circuits: Theory and Applications
by Denton J. Dailey
Integrator
• Process of integration is complementary to that
of differentiation
• Relationship is analogous to that between
multiplication and division
• Function being integrated is called integrand, and
dt is called the differential
• Integration produces equivalent of the
continuous sum of values of function at infinitely
many infinitesimally small increments of t
• Output of integrator will maintain 90° phase lead,
regardless of frequency
Integrator
V(t)
∫ V(t) dt + C
∫
C1
R1
Vin
Vo
+
RL
Operational Amplifiers and Linear
Integrated Circuits: Theory and Applications
by Denton J. Dailey
Integrator
•
•
•
•
Av = - (ZF/Z1) = - 1/(jωCR)
l Av l = l 1/(ωRC) l and phase = 90
Av = 1/(ωRC) <90
f  0 (dc), Av  ∞ R1
Vin
Reset switch added to force
integrator initial conditions
to zero
Reset
C1
Vo
+
RL
Integration of Nonsinusoidal Inputs
•
•
Constant voltage:
V0 = -Vin t / RC
V0 = 0 at starting
Ramp input:
V0 = - kt2 / 2R
k is rate of change of
Vin (V/s)
Operational Amplifiers and Linear
Integrated Circuits: Theory and Applications
by Denton J. Dailey
Integrator (Square Wave Input)
• V1 = - (Vmt(+))/RC
• t(+) = t – t0
• V2 = V1 + [-(Vmt(-))/RC]
Operational Amplifiers and Linear
Integrated Circuits: Theory and Applications
by Denton J. Dailey
Integrator
• Integrator effectively accumulates voltage over time;
presence of input offset voltage will cause capacitor to
charge up producing error in output
• Smaller the capacitor, more quickly offset error builds
up with time
• Solutions
–
–
–
–
–
Use of larger capacitor
Use of low-offset op amps
Bias compensation resistor RB on noninverting terminal
Use of resistor RC in parallel with feedback capacitor
RC ≥ 10R1
Practical Integrator
RC
C
R1
Vin
Vo
+
RL
RB = R1 ll RC