Plato’s Gain Formula
For Single Op-Amp Circuits
V(out)/V(in) = p Zf/Zi
Plato’s Gain
Formula is the unified gain formula for a single op-amp circuit,
the gain is a constant p times the Feedback Impedance (Zf)
divided by the Input Impedance (Zi). For a
circuit with equal Impedance at the (+) and (-) op-amp inputs, the gain
magnitude is simply Zf/Zi, or Rf/Ri
for resistors. The gain is positive for inputs connected to the (+) op-amp
input, and negative for inputs connected to (-) op-amp input.
Plato’s formula
implies Superposition. For multiple inputs, the gain is the gain for a
single input with the other inputs equal to zero. The output formula will
contain a term for each input. Just add the terms to form the output
equation.
Plato says “Use Rf/Ri for all gains, not just negative gain”. Let’s
look at the details.
Legacy circuit analysis has separate circuits and formulas for inverting amplifiers, non-inverting amplifiers, integrators, differentiators, differential amplifiers, and summing amplifiers. All of these circuits are a subset of the general summing amplifier shown below.
General Summing Amplifier
Where:
Vp1 to Vpn are positive inputs. You need at least one.
Vn1 to Vnm are negative inputs. You may have none.
Zpi and Zni are input impedances. Zpi may be zero. Zni may
not be zero.
Zf is the feedback impedance. You need Zf. Zf may be
zero.
Vxy is an ideal voltage source. It may be zero.
The op-amp is ideal, infinite gain, zero
output impedance, and infinite input impedance.
Non-ideal input sources are allowed, if the source impedance is included in the input impedance. The gain will be the gain from the ideal source to the op-amp output, not the gain from the circuit input to the output.
Z is not restricted
to ideal components. It can be a two terminal network of components.
Plato’s Gain
formula specifies the gain from the input, V(in), to
the op-amp output, V(out).
V(out)/V(in) = p
Zf/Zi
p = ZP+ / ZP- for inputs connected to the (+) non-inverting op-amp input
p = -1 for inputs connected to the (-) inverting op-amp input
Zf = Feedback Impedance
Zi = Input Impedance
ZP+ = Parallel combination of impedances connected to the non-inverting op-amp input
ZP- = Parallel combination of impedances connected to the
inverting op-amp input
Plato’s Gain
formula states that the op-amp circuit gain is equal to a circuit constant (p)
times the Feedback Impedance, Zf, divided by the Input Impedance, Zi. For
inputs connected to the (+) op-amp input, p is equal to ZP+/ZP-. For
inputs connected to the (-) op-amp input, p is equal to –1.
Plato’s Gain
Formula is the same as the Legacy Inverting Amplifier Formula for negative
gains. For positive gains, Plato's formula appears to differ from the Legacy 1
+ RF/RI formula. For a non-inverting amplifier, ZP+ = Rp1. If you
cancel these terms in Plato’s gain formula you get the Legacy formula after
changing Rn1 to RI to match the naming. A detailed discussion is
presented in Analysis Examples .
Proof
The analysis of
the General Summing Amplifier is a derivation of Plato’s Gain Formula. The op-amp gain is set to infinity to obtain Plato’s
Gain formula.
Interpretation
The simplest interpretation is that the gain
is Zf/ Zi times p. For resistors, the gain is Rf/Ri times p,
and if p=1, than the gain is simply Rf/Ri. Can Single Op-amp circuits
really be that simple? YES !
Plato states that the Rf/Ri formula not only applies for negative gains, but also for positive gains, and any combination of gains. If the circuit is poorly designed you need to calculate p and apply it to positive gains.
Circuits that present the same impedance to
the op-amp at the (+) and (-) input terminals will have ZP+ equal to ZP- and p
= 1. These balanced circuits have minimum bias current error. Since
modern op-amps have very low bias current, designers often don't add the
additional resistor needed to balance the design. In the K9
Design procedure, Shadow always creates balanced
circuits.
Plato says “ Single
op-amp circuits are easy. The gain is Rf/Ri.
If the input connects to the (-) terminal, the gain is negative. If the
input connects to the (+) terminal the gain is positive and may require a
correction”.
There's more to Plato's formula. Good
circuits need finite gain. The equation tells you when the gain becomes
infinite. Let's look at the terms in the equation.
Zf
Zf is the feedback impedance. You need Zf, it may not be an open (infinite impedance). If Zf is a capacitor, you need a parallel resistor to avoid
infinite gain at DC. A two terminal network is allowed for Zf. Just use the Zf
impedance value when calculating the gain. T networks are not allowed.
You need to transform them to an equivalent delta circuit, see Y Delta transform.
ZP+/ZP-
This term contains all the circuit
components. Can I avoid this term by using only negative gains?
No. Daisy says that the sum of the gains must be
equal to +1. This is hard to do, if all the gains are negative. You
will need some positive gain. It could be ground gain.
ZP+ and ZP- contain the stray capacitance
present at the op-amp inputs. The capacitance will create a low impedance
value at high frequencies. No problem for ZP+, but a problem for ZP-.
Small values in the denominator create large gain values. If the
low values occur at frequencies where the op-amp gain is less than one,
the circuit is probably stable. If the op-amp still has gain, the circuit
may oscillate.
Low values in the denominator of the gain
equation are a sign of danger. Plato is trying to tell you that if you
need to probe the circuit (add stray capacitance), don't probe the (-) op-amp
terminal. It's ok to probe the (+) terminal.
Zpi , Zni
The input impedance may be a just resistor or a resistor plus capacitor for AC coupling. Just use the series impedance value when calculating gain.
You need at least one
positive input impedance, Zpi. This will
allow the circuit to have positive gain. You need at least one Zpi with a DC component to create DC gain. You can’t
violate Daisy’s theorem; the circuit won’t work.
Zpi can be zero (short). This will force ZP+ to be zero. When calculating the gain, you need to cancel the ZP+ and Zpi terms to avoid zero / zero in the gain equation.
You don’t need any negative input impedances, Zni. Zf connects to the (-) terminal and keeps the value of ZP-
finite.
Zni may not be zero. Zero Zni will force ZP- to zero. In this case there is no cancellation, just inifinate gain.
Gain Interaction
The negative gain formula contains only two terms, Zf and Zni. If you have other components, they won't alter the negative gain value. The positive gain contains the ZP+/ZP- term. This term contains all components. Changing any single component will change all positive gains.
Since negative gains are independent, many Legacy circuits prefer negative gains. For an audio mixer, a negative summer circuit is often used. Alternate circuits are a positive gain summer and a mixed gain summer. The best choice is the mixed gain summer. It will have less noise and better bandwidth, (see Noise) With Legacy procedures, a mixed gain circuit is very difficult to design, hence rarely used. Not a problem for K9, just use Rf/Ri.
Design
Plato's gain formula suggests a simple circuit design procedure: Assume p =1. Pick an Rf value (100k). Use Rf/|gain| to calculate Ri values. Check p and add a resistor to ground to balance the design. Easy ? Shadow's procedure is even easier.
Summary
Plato’s Gain
Formula combines the legacy single op-amp formulas into a single
formula. You obtain the output equation by simply adding the terms
for each input. There is just one simple gain formula to remember.
The distinction
between inverting amplifiers and non-inverting amplifiers that is emphasized in
legacy analysis is partially hidden.
Balanced circuits, that have p=1, are preferred by op-amps. Plato says “Use Rf/Ri for all cases and apply the p factor to positive gains”.
Note:
If you have learned Legacy op-amp analysis, you may be skeptical. How can the non-inverting gain be Rf/Ri? For a non-inverting amplifier, if you vary the input resistance at the + terminal, the gain will not change. This is true. For a single positive input, ZP+ = Ri. The terms cancel and the remaining terms don't include Ri.
In Legacy texts, non-inverting circuits are complicated. The non-inverting formulas bear little resemblence to the inverting gain formula. Plato provides a single gain formula that includes the circuit variations. The ZP terms allow a simple equation.
Legacy analysis
emphasizes the difference between non-inverting and inverting circuit
configurations. Plato prefers to hide this. He likes to keep things
simple.
Plato is no longer with us. I
ask that you honor him by keeping the gain formula name. Whenever you see
Rf/Ri, think of Plato. His formula applies to
all gains, not just negative gains. Thank him for making Single Op-amp
circuits Dog-Gone Simple.
If you have comments or questions, I can be reached at k9analysis@netzero.net.