Lecture 15
Integrator and Differentiator
Objective: To present an op amp integrator and differentiator. The circuit techniques
applied for resistive op amp circuits can be extended to study op amp circuits
with both resistors and capacitors. The most fundamental RC op amp circuits
are the integrator and the differentiator.
At the end of this class you should be able to:
* Understand the operation of op amp integrator and differentiator
* Analysis the integrator and differentiator circuits assuming ideal and non-ideal
op amps.
Inverting Integrator:
In this configuration:
„ Feedback resistor R2 in the inverting amplifier is replaced by capacitor C.
„ The circuit uses frequency-dependent feedback.
Fig. 1: Integrator: (a) circuit (b) response to step input
•
In analyzing RC op amp circuit, you may use time domain or frequency domain
method.
Using time domain analysis:
v
is = s
R
ic = −C
dvo
dt
Since ic= is
1
vs dτ
RC
t
∴vo (t ) = − 1 ∫ vs (τ )dτ + vo (0)
RC 0
vo (0) = −Vc (0)
∫ dvo = ∫ −
* Voltage at the circuit’s output at time t is given by the initial capacitor voltage plus the
integral of the input signal from start of integration interval, here, t=0.
* Integration of an input step signal results in a ramp at the output as shown in Fig. 7(b)
Example 1:
Draw the output waveform of the integrator shown in Fig. 1(a) in response to the input
shown in Fig. 2. Assume R=10kΩ, C=10nF and vo(0)=0.
Fig. 2: Step input
Solution:
t
1
1
vo(t ) = −
t for 0 ≤ t ≤ 1ms
1 × dτ = −
∫
CR 0
1 × 10 − 4
Thus, the output voltage will decrease linearly with time from 0V at t=0 to –10V at
t=1ms as shown in Fig. 3.
Fig. 3: Response of the step input
* Frequency domain analysis of the integrator is very simple. By writing the KCL
equation at v- and solving for vo(s) results in:
vo ( s )
1
=−
vi ( s )
sRC
For physical frequency s → jω :
v o ( jω )
1
=−
v i ( jω )
jωRC
Therefore, the magnitude and phase responses are as follows:
vo
1
=
Magnitude:
vi
ωCR
Phase:
φ = 180 − 90 = 90 o constant
The Bode plot of the magnitude response of the integrator is shown in Fig. 4.
Fig. 4: Bode plot of the integrator transfer function
Problem: Note that this means any small dc component of vi(t) results in ∞ output. In
practice, the op amp will saturates at a voltage close to V+ or V- depending on
input voltage polarity.
Example 2: Determine the output voltage due to both offset voltage and biasing current
for the integrator circuit shown in Fig. 5
Fig. 5: DC errors in the integrator
At t<0, reset switch is closed, circuit becomes a voltage-follower
Vo = V
OS
At t=0, reset switch is opened, circuit starts integrating its own offset voltage and bias
current. Using superposition analysis, output due to VOS can be found as follows.
Assume the initial capacitor voltage is zero.
t
1
By KVL: Vo (t ) = VOS + ∫ iC dt
C0
But ic =
VOS
R
Vo (t ) = VOS
t
V
1 VOS
+ ∫
dτ = VOS + OS t
C0 R
CR
(1)
This means that Vo(t) increases linearly with time until op amp saturates. Therefore, the
offset voltage is very dangerous for the integrator operation.
When VOS is set to zero, the output due to IB2 will be:
I
vo (t ) = B2 t
C
Thus the total output is given by:
V
I
voT (t ) = V + OS t + B2 t
OS RC
C
This means that the output becomes ramp with slope determined by VOS and IB2 and
saturates at one of the power supplies.
Differentiator:
Fig. 6: Differentiator
In this configuration:
„ Input resistor R1 in the inverting amplifier is replaced by capacitor C.
„ Derivative operation emphasizes high-frequency components of input signal,
hence is less often used than the integrator.
* Using time domain analysis:
v
i =− o
R
R
is = C
dvs
dt
Since iR= is
vo = −RC
dvs
dt
Output is scaled version of derivative of input voltage.
Problem: the differentiator is noise magnifier (i.e. spikes are produced at output due to
sharp changing in vi(t). Therefore, it is usually avoided in practice.
In frequency domain, writing KCL at v- and solving for transfer function yields:
vo (s)
v ( jω )
= − sCR ⇒ o
= − j ω CR
vi (s)
v i ( jω )
⇒
vo
= ω CR
vi
⇒ φ = − 90 D
The Bode plot of the magnitude response of the differentiator is shown in Fig. 7.
1
CR
Fig. 7: Bode plot of the differentiator transfer function