DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS 02139
DIFFERENTIATOR / INTEGRATOR INSIGHTS
For the differentiator:
− R2
1
R1 +
sC
Av =
Av
− sCR2
; s = jω ; at frequencies 1/10 x fLO, sCR1 << 1 , so
sCR1 + 1
=
− sCR2
; multiplying by s equals differentiation.
1
=
For the integrator:
R2
Av
= −
R1
; s = jω ; at frequencies 10 x fHI, sCR2 >> 1 , so
1 + sCR2
R2
Av
=
−
R1
sCR2
=
−
1
sCR1
; dividing by s equals integration.
C
R2
R2
+15
+15
vin
0.1μF
vin
0.1μF
R1
2
-
6
3
4
+
vout
R3
-15
[a] Integrator/Low Pass Filter
Differentiator / Integrator
2
-
7
6
LF356
0.1μF
R3
C
7
LF356
3
R1
1
+
4
0.1μF
-15
[b] Differentiator/High Pass Filter
10/04/06
Cite as: Ron Roscoe, course materials for 6.101 Introductory Analog Electronics Laboratory, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
vout
AV (dB)
0
-3dB
slope = 6 dB / octave
slope = 20 dB / decade
log f
fLO or f-3dB
Degrees
PHASE LEAD
90o
45o
0o
-45o
log f
fLO or f-3dB
[c] High Pass Filter/Differentiator Bode plot: differentiation works only at f ≤ 1/10 fLO
AV (dB)
0
-3dB
slope = -6 dB / octave
slope = -20 dB / decade
log f
fHI or f-3dB
Degrees
PHASE LAG
0o
-45o
-90o
log f
fHI or f-3dB
Differentiator / Integrator
2
10/04/06
Cite as: Ron Roscoe, course materials for 6.101 Introductory Analog Electronics Laboratory, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
[d] Low Pass Filter/Integrator Bode plot: integration works only at f ≥ 10 fHI
Amplitude
It’s easy to understand that an integrator will turn a square wave into a triangle wave
and a differentiator will turn a triangle wave into a square wave because we all know the
results of integrating or differentiating these simple functions. However, we get
additional insight into how these circuits work their magic if we look at the amplitudes of
the harmonics of these common waveforms:
Time
Cycle
4
PR = π
1
3
1
5
Response DB
1
0
-10
3
5
7
-20
-30
100
9
11
200
400
800 1000
Frequency in cycles per second
13
15
17 19
2000
A rectangular wave, the equation of the wave, and the spectrum for a fundamental frequency
of 100 cycles, that is, t = 1/100 second for a complete cycle.
Figure by MIT OpenCourseWare.
If we apply the above square wave to the input of an integrator, consider that the falling
frequency response of the integrator above the corner frequency, fHI, will attenuate the
upper harmonics of the square wave relative to the lower harmonics and the
fundamental. Since the only difference between a square wave and a triangle wave is
the relative amplitudes of their harmonics, as well as phase shift, rolling off the
harmonics of the square wave and phase shifting them creates a triangle wave. [Next
page.]
Conversely, if we apply the triangle wave on the next page to the input of a
differentiator, with the fundamental frequency at 1/10 of fLO, the rising frequency
response below the corner frequency fLO will amplify the upper harmonics of the triangle
wave relative to the fundamental and the lower harmonics and phase shift them, thus
changing the triangle wave into a square wave.
Differentiator / Integrator
3
10/04/06
Cite as: Ron Roscoe, course materials for 6.101 Introductory Analog Electronics Laboratory, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Amplitude
Time
Cycle
PR =
0
8
p2
cos wt +
1
1
cos 3 wt + cos 5 wt ----9
25
1
Response DB
-10
-20
-30
3
5
7
9
-40
11
-50
-60
100
200
400
800 1000
Frequency in cycles per second
13
15
17
19
2000
A triangular wave, the equation of the wave, and the spectrum for a fundamental frequency
of 100 cycles, that is, t = 1/100 second for a complete cycle.
Figure by MIT OpenCourseWare.
Differentiator / Integrator
4
10/04/06
Cite as: Ron Roscoe, course materials for 6.101 Introductory Analog Electronics Laboratory, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].