A single-formula approach for designing
positive summing amplifiers
This circuit-theory approach on op-amp design and analysis has two benefits: You
can use it on all op-amp designs without learning special formulas or cases. And it
makes possible a rigorous method for designing positive summing amplifiers.
By Max Bernhardt, Lange Sales
After a discussion on the general theory of converting an op-amp design into circuit
theory with an emphasis on creating one formula, I’ll give examples of simple circuits to
prove the theory behind this method. Finally, I’ll present a simple positive summing
amplifier, which was developed with conversations with Dieter Knollman (Reference 1).
Because op amps are linear devices in the s domain, circuit theory lends itself well to
using these devices. By using circuit theory, you can reduce many complex systems to
basic blocks that can be easily evaluated using advanced computer software. The more
masochistic among us can readily use pen and paper. To use this theory, start with a
generalized op-amp circuit:
VP1
ZP1
VP2
ZP2
VPm
ZPm
VN1
ZN1
.
.
.
V+
VVN2
.
.
.
VNi
+
Vout
-
ZN2
Zf
ZNi
Where the variables are defined in the following manner:
VPm = voltage input to the positive side of the amplifier at any particular mth location.
VNi = voltage input to the negative side of the amplifier at any particular ith location.
ZPm = input impedance seen at any particular mth location.
ZNi = input impedance seen at any particular ith location.
Zf = Feedback impedance.
Now, convert the op-amp circuit into a circuit system by grounding all the voltage inputs
and voltage outputs and applying a voltage source at the input terminals of the op amp.
Looking exclusively at the negative input to the op amp, the circuit looks like the
following:
+
V-
ZN1
ZN2 ... ZNi
Zf
From this circuit, you can write an equation for the parallel input impedance seen by the
negative input to the op amp of this form:
ZN − = ZN f || ZN 1|| ZN 2||...|| ZN i
Where ZN- is the parallel input impedance seen by the negative input of the operational
amplifier.
Switching to the positive side of the op amp, you can write the equation for the positive
input impedance (ZP+):
ZP + = ZP1 || ZP 2 || ... || ZP m
From basic circuit theory and throwing in offset voltage and input bias current, we can
then show that the operational amplifier circuit then becomes the following block
diagram:
Ib
VP1
ZP+
ZP1
VP2
ZP+
ZP2
VPm
ZP+
ZPm
ZP+
Σ
1
Σ
Vos
VN1
A
Ts+1
Σ
Vo
ZNZN1
-1
VN2
ZNZN2
VNi
ZNZNi
ZNZf
Σ
ZNIb
When designing your system, you can set ZN- and ZP+ to be approximately equal, which
allows you to ignore the input bias current Ib, because it will sum to zero at your
summing node. In addition, assume that the pole (open loop –3 dB corner set by Ts+1)
caused by the gain stage of the operational amplifier is much smaller than the pole in the
rest of the system and can also be ignored. Finally, let Vos be very small in order to have
this technique follow classical analysis and you get the following block diagram:
VP1
ZP+
ZP1
VP2
ZP+
ZP2
VPm
ZP+
ZPm
Σ
1
Σ
A
Σ
Vo
ZNZN1
VN1
-1
VN2
ZNZN2
VNi
ZNZNi
Σ
ZNZf
Writing the loop equations starting from VP1, you get:
AZP +
Vo
ZP1
=
VP1 1+ AZN −
Zf
Then, take the limit as A gets very large:
AZP +
Vo
ZP1
=
Lim
A→ ∞ VP1 1+ AZN −
Zf
Because you’ve set ZP+ equal to ZN-, the equation for the input voltage 1 (VP1) comes
out to:
V o=
VP1*Z f
ZP1
Because the system is linear, you can combine all the input equations and get the
generalized case for the positive inputs to be:
∞ VP m* Z f
Vo = ∑
m =1 ZP m
Do the same exercise for the negative inputs:
AZN −
Vo
ZN 1
=
VN 1 1+ AZN −
Zf
AZN −
Vo
ZN 1
=
Lim
A→ ∞ VN 1 1+ AZN −
Zf
Vo =
VN 1*Z f
ZP1
∞ VN i *Z f
Vo = ∑
i =1 ZN i
Combining the equations for both sides and realizing that the gain from the negative input
is multiplied by negative one gives you the generalized formula for all op-amp circuits:
∞ VP m*Z f
∞ VN i*Z f
− ∑
Vo = ∑
i =1 ZN i
m =1 ZP m
This formula verifies the derivation “Designing with op amps: Single formula technique
keeps it simple” (Dieter Knollman, EDN, March 2, 1998) using a circuit theory
approach.
Some examples that prove this equation too be correct are:
Negative input gain:
V+
4R
5
VN1
R
V-
+
-
Vout
4R
From classical theory, the equation to use is:
Vout = -4VN1.
Using our equation, you get:
Vo =
0*4 R VN 1*4 R
−
4R
R
5
Vo = -4VN1
The example of a positive amplifier is:
VP1
V+
4R
5
R
V-
+
-
Vout
4R
From classical theory, the equation that you would use is:
Vout = VP1(1+(4R/R) = 5VP1.
Using our equation, you get:
Vo =
VP1*4 R 0*4 R
−
4R
R
5
Vo = 5VP1
The example of a negative summing amplifier is:
2R
3
VN1
V+
R
V-
VN2
4R
+
Vout
4R
From classical theory, the equation that you would use is:
Vout = -4*VN1 – VN2.
Using our equation, you get:
Vo =
0*4 R VN 1*4 R VN 2*4 R
−
−
4R
R
4R
5
Vo = -4*VN1 – VN2
Now, derive the positive summing amplifier using this circuit-design method. To do this
circuit, balance the Ib to be as equal as possible, therefore, ZP+ = ZN-. When you know
that the input impedances are equal, the design of this circuit is straightforward. First,
pick your gains on ZP1, ZP2, etc. Then, set Zf to meet those gains. Finally, select Zi to
the Zf circuit to match the input impedance (ZP+) seen by the positive side of the
amplifier. Thus, on the following circuit, ZP+ should equal 4R/5.
VP1
R
VP2
4R
V+
V-
R
+
-
Vout
4R
Using our equation, you get:
Vo = −
0*4 R VP1*4 R VP 2*4 R
+
+
4R
R
R
Vo = VP2 + 4*VP1
To show this circuit in Spice, the following circuit was generated.
R2
4K
V0
2
R
1K
XOp_amp
V1
0
IVm2
+2.00
IVm
+2.00
IVm1
0
R0
1K
R1
4K
If you hold V0 constant at –2V and sweep V1 from –1.5V and 1.5V, you get the
following graph:
PosSumOpAmp-DC Transfer-10
(V)
-1.500
-1.000
-500.000m
+0.000e+000
V1
+500.000m
+1.000
+1.500
+4.000
+2.000
+0.000e+000
-2.000
-4.000
-6.000
-8.000
V(IVM)
V(IVM2)
V(IVM1)
You can use this design method to solve complex difficult problems that would be hard
to solve using other classical methods. This method can also be used with tools like
Matlab to bridge the gap between physical systems and electrical design.
Reference
1. Dieter Knollman, PhD, is a distinquished member of the technical staff of Lucent
Technologies (Denver), where he has worked for 33 years. In his current position, he
designs PBX port circuits. Knollman earned a BSEE from the Virginia Polytechnic
Institute and State University (Blacksburg, VA) an MSEE from the University of
Illinois—Urbana/Champaign, and a PhD from the New York University (New York).
Author’s biography
Max Bernhardt is a field application engineer for Lange Sales. He works with key
leaders in the industry developing and new technologies for the market. Max holds a BS
in math education and a BSEE from the University of Wyoming.