Transcript Cascaded Op Amps

ECE 222
Electric Circuit Analysis II
Chapter 11
Op Amp Cascade
Herbert G. Mayer, PSU
Status 5/10/2016
For use at CCUT Spring 2016
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Syllabus
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Review
Goal
Op Amp Cascade
Example 1
Cascaded Integrating Op Amps
Bibliography
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Review Ideal Op Amp
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Focus and limitation of ideal Op Amp:
In ideal Op Amp, no currents flowing into --or out of-inputs pins: ip = in = 0 [A]
Voltages vn and vp at negative and positive input
terminals are identical: vn = vp
Hence if one input pin is connected to ground, both
voltages are zero: vn = vp = 0 [V]
Holds even if ground connection is via resistor R, as
no current flows through R, hence no voltage drop!
Op Amp easily saturates, unless feedback resistor is
included from output vo to one of the input pins
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Review Ideal Op Amp
in and ip are 0 [A], and vn = vp in Ideal Op Amp
if
in
Ra
+
va
ip
ia
Rb
-
vp
Rf
+VCC
-
vn
vo
+
ib
+
vb
-
-VCc
Rp
Ip, in = 0 [A], for Ideal Op Amp; Here Difference Op Amp
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Goal to Integrate
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Use cascade of Op Amps to generate vo as a function
of signal input voltage vg
vo = f( vg )
Doing so in cascade of 2 stages!
Goal to have f( vg ) be the integral function! How can
we do this?
And if done twice, then we solve Second-Order DE
Other circuit parameters: resistors R1 and R2 at Op
Amp input pins, and C1 and C2 capacitors at output
pin, to feed back signal
Note that voltage function with capacitor includes first
derivative, this is how integral function can be built!
Hence the name: integrating Op Amp!
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Op Amp Cascade
Circuit and Equations
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Op Amp Cascade – The Circuit
Below cascade of 2 Op Amps: The output of the first
Op Amp becomes input to second Op Amp
In both cases, use feedback via capacitor: first
derivative function
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C1
C2
iR1
R1
+
vg
-
iC1
in
vn1
-
+VCC
iR2
R2
+
ip v
+
p1
-VCC
vo1
iC2
vn2
vp2
-
+VCC
+
+
vo
-VCC
-
Cascading OpAms, Integrating vg to Generate vo1 then vo
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-
Op Amp Cascade – Current Equation
 In left Op Amp of The Circuit, with ip = in = 0 from KCL
iR1 + iC1 = 0
vg / R1 + C1 dvo1 / dt = 0
dvo1 / dt = - vg / (R1 C1)
 Similarly in the right Op Amp, using KCL:
vo1 / R2 + C2 dvo / dt = 0
dvo / dt = - vo1 / (R2 C2)
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Op Amp Cascade – Current Equation
 Differentiating dvo/dt = -vo1/(R2 C2) one time yields:
d2vo / dt2 = -dvo1 / dt * 1/(R2 C2)
 And substituting dvo1/dt = -vg/(R1 C1) yields:
d2vo / dt2 = vg / (R1 C1) * 1/(R2 C2)
 That is the key, second order differential equation, for
the step response of circuit with two cascading Op
Amps
 Each of these two is an integrating Op Amp
 vo is a second order DE of vg, with other electric
parameters, such as: R1, C1, R2, C2
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Example 1
Cascaded Integrating
Op Amps
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Example 1
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Example 1, a cascade of 2 integrating Op Amps, with
a step function for vg
When t < 0, no energy is stored in capacitors vg = 0 V
At t = 0, vg jumps from 0 V to vg = 25 mV
R1 = 250 kΩ, R2 = 500 kΩ, C1 = 0.1 μF, C2 = 1 μF
Now compute:
1.
Intermediate unit: 1 / R1 C1
2.
Intermediate unit: 1 / R2 C2
3.
Formula for: d2vo/dt2
4.
Formula for vo(t)
5.
Time ts to saturation
6.
Which Op Amp saturates first
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Example 1 – Integrating Circuit
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Apply voltage step function vg = 25 mV at time t = 0
Express vo as function of vg
0.1 µF
1 µF
iC1
+5 V
250 kΩ
-
500 kΩ
+
-
iC2
+9 V
-
+
+
vg
iR2
-5 V
vo1
-
+
+
-9 V
vo
-
Cascading OpAms, Integrating Sudden Step Voltage vg = 25 mV, at t=0
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Example 1
1. 1 / R1 C1 = 1 / ( 250 * 103 * 0.1 * 10-6 ) = 40 [s-1]
2. 1 / R2 C2 = 1 / ( 500 * 103 * 1 * 10-6 )
3. d2vo / dt2
d2vo / dt2
d2vo / dt2
d2vo / dt2
=
=
=
=
vg / (R1 C1) * 1/(R2 C2)
vg / ( 40 * 2 )
25 * 10-3 * 80
2
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= 2 [s-1]
Example 1
4. Now integrate twice, given that d2vo / dt2 = 2
dvo / dt = 2 t
vo
= 2 * 1/2 t2 + vo
-- vo at t = 0, thus vo (0) = 0
V
vo
= t2
-- max vo = 9 V
5. Compute time tS to saturation:
tS
= 3 [s]
-- at 3 s Op Amp saturates
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Example 1
6. Analyze, if first Op Amp saturates earlier:
dvo1 / dt = - vg / (R1 C1)
dvo1 / dt = - 25 10-3 * 40 = -1
Now integrate:
vo1
= -1 * t
= -t
But for t = tS = 3 s, we see that vo1 = -3 V
This is legal range, hence the first Op Amp does NOT
saturate, and the second Op Amp is the one saturating
first, at time tS = 3 [s]
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Bibliography
1.
2.
3.
4.
Nilsson, James W., and Susan A. Riedel, Electric
Circuits, © 2015 Pearson Education Inc., ISBN 13:
9780-13-376003-3
Differentiation rules:
http://www.codeproject.com/KB/recipes/Differentiati
on.aspx
Euler’s Identities:
https://en.wikipedia.org/wiki/Euler%27s_identity
Table of integrals: http://integraltable.com/downloads/single-page-integral-table.pdf
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