Middlebrook’s input/output impedance theorem

A “low-entropy” equation that depicts the circuit behavior can help analog designers work backwards, finding values of the circuit elements in order to meet the specification.

Let’s start from the output impedance of a simple linear circuit.

Fig. 1

A common procedure of determining the output impedance is by: shorting input to ground, and look into the output port, to “see” the output impedance. In this simple case (Fig. 1), we can “see” R₁ ∥ R₂

When the circuit becomes more complicated, it is difficult to “see” the output impedance directly. One way is to inject a current and calculate the resulting voltage. But it is often difficult to inject the signal “backwards” into an output stage of the circuit.

One technique is to calculate the output impedance as:
open circuit voltage / short circuit current

In general, according to Middlebrook’s input/output impedance theorem, knowing the gain A, the output impedance can be calculated as

Note that A term didn’t cancel out on the numerator and denominator, because it might contain the term ZL , which takes the limit to zero.

Use the previous example

It’s the same result.

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