Feedback and Impedances

-June 03, 2014

Negative feedback has a profound impact on the input and output impedances of a circuit.  For an intuitive feel, consider the popular op amp circuit of Figure 1, which we assume to be a well-designed circuit with ri >> R1//R2 and ro << R1 + R2.  To find the closed-loop input resistance Ri, apply a test voltage vI, find the ensuing current iI, and then let Ri = vI/iI.  So long as the gain av is suitably high, vD will be very small, in turn making iI quite small and therefore Ri quite large.  Indeed,

where A = av/(1 + T) is the closed-loop gain, and Tav/(1 + R2/R1) is the loop gain [1].  Consequently,

To find the closed-loop output resistance Ro, subject the output port to a test current iO, find the ensuing voltage vO, and then let Ro = vO/iO.  Again, vD is bound to be very small, implying a very small current through R1 and, hence, through R2.  So, virtually all of iO will flow right into ro, giving

so

Figure 1.  Circuits to find the closed-loop resistances Ri and Ro of the series-shunt configuration.

The above transformations reveal typical negative-feedback features, namely, the tendency to raise the impedance of a series-type port and to lower the impedance of a shunt-type port.  In fact, two-port analysis (TPA) predicts the impedance transformations

where zpa is the open-loop impedance presented by the port under scrutiny, TTP is the loop gain, Z is the closed-loop impedance, and we use the exponent +1 in the series case, and –1 in the shunt case.  As discussed previously [2], TPA requires that we identify which of the four feedback topologies is in use so we can suitably manipulate the basic amplifier to find TTP as well as its open-loop impedances zia and zoa

Return-ratio analysis (RRA) expedites the process by expressing the impedance Z between any two nodes (not just the nodes of the input port or the output port) via Blackman’s Impedance Formula

where z0 is the impedance between the given node pair with the amplifier’s dependent source set to zero (av → 0 for op amps, gm → 0 for transistors); moreover, TRR(sc) and TRR(oc) are the return ratios with the node pair short circuited and open circuited, respectively.  In the case of a purely series-type port we have TRR(oc) = 0, and in the case of a purely shunt-type port we have TRR(sc) = 0.  As we know, RRA is generally quicker than TPA because it does not require any topology-dependent circuit manipulations [2].

Impedance transformations can be quite dramatic at low frequencies, where the loop gain is usually very high.  However, as the open-loop gain rolls off with frequency, so does the loop gain, indicating that what’s high is bound to drop with frequency, and what’s low is bound to rise with frequency, at least over a certain frequency range.  Put another way, series-type impedances exhibit capacitive behavior, and shunt-type impedances exhibit inductive behavior.

The drop of a series-type impedance may not necessarily be an issue so long as the impedance remains suitably high over the frequency range of interest.  Shunt-type impedances, however, may pose problems, because when terminated on a capacitance, whether parasitic or intentional, the inductive component tends to resonate with the external capacitance, possibly leading to ringing and peaking, or even destabilizing the circuit [3].

It is precisely this destabilizing tendency that I want to address in the current blog.  I will illustrate via the inverting amplifier of Figure 2, where CL is the output load capacitance, and Cn is the total capacitance of the inverting input pin (Cn is the sum of the stray capacitances associated with the op amp’s input stage, the stray capacitances of R1 and R2, and the stray capacitances of the wires [3]).  For simplicity I will discuss the two cases separately, leaving their simultaneous treatment to my next blog.

Figure 2.  Circuit to investigate the destabilizing effect of capacitively-terminated shunt ports.