For feedback amplifiers with single-pole loop gains, it is
where τbw is the time-constant of the bandwidth, fbw = 1/2·π·τbw. G0 and H0 are the quasi-static (frequency-independent) values of G and H that would apply at 0 Hz or sufficiently low frequencies without the effect of reactances. Then the frequency-dependent feedback factor is
The general feedback port impedance for ii and vo is
This low-Z formula can be rewritten by distributing the quasi-static feedback factor in the denominator as
The equivalent circuit is drawn from the continued-fraction form of Z(cl), as shown below. The multiplication of Z by s gyrates it as an impedance by +90° so that a resistive Z becomes inductive.
A trans-impedance amplifier (Zm = vo/ii) has input and output impedance of the same form. For resistive open-loop input and output ports, Z = R, and the closed-loop equivalent circuit for both is R shunting a series RL, as shown below.
For an op-amp trans-impedance amplifier, the open-loop input resistance is the feedback resistance Rf = R. The combined quasi-static resistance of the equivalent circuit is
Input resistance is reduced from Rf to Rf/(1 + G0·H0) by the feedback factor. The series inductance is
where the unity-gain frequency,
The inductance forms a time constant in the series RL branch with R/(G0·H0) of τbw.
The dual case of vi and io has a closed-loop port impedance in the general form of
Some more algebra results in a topologically-explicit form by multiplying Z·(1 + G0·H0) through the numerator and dividing;
Z(cl) in this high-Z formula has Z in series with a parallel combination of two elements in series. The equivalent circuit is shown below and is the dual circuit to that of the previous Z(cl). The gyrated Z is divided by s this time, which gyrates Z by –90° so that a resistance becomes capacitive.
For Z = R, the quasi-static port impedance has two resistances in series having a combined resistance of R·(1 + G0·H0). The gyrated R is a capacitive reactance with capacitance τbw/(R·G0·H0). It forms a parallel RC with R·(G0·H0) having a time constant of τbw.
Feedback Amplifier Example
A dominant single-pole voltage amplifier with feedback having a loop gain of
G0·H0 = 10k
and a unity-gain frequency of fT = 1 MHz has the following open-loop port impedances:
Zi = 1 kΩ ; Zo = 10 Ω || (1/s·(22 pF))
What are the closed-loop input and output impedances and what, if any, are the hf resonances?
First, the loop bandwidth of the amplifier is
fbw = fT/(G0·H0) = (1 MHz)/(10 k) = 100 Hz
with time constant τbw = 1/2·π·fbw = 1.590 ms.
Second, the input port impedance of a voltage amplifier with feedback-loop input variable vi has an equivalent circuit of the high-Z form, Z·(1 + G·H). Substituting Zi = Ri = 1 kΩ, the result is 1 kΩ in series with 1 kΩ·(10k) = 10 MΩ. It shunts a capacitance of
Ci = τT/Ri = (1/2·π·(1 MHz))/1 kΩ » 159 ns/1 kΩ » 159 pF
Ci shunts 10 MΩ and the RC time constant is τbw. The capacitance is rather considerable, and above bandwidth its reactance becomes small relative to the 10 MΩ shunting it. This results in an approximate hf impedance of 159 pF in series with 1 kΩ. This is quite different than the quasi-static input resistance of 10.001 MΩ, and if the input source is inductive (such as a connecting cable terminated at the source end in a voltage source), then a resonance is formed.
Finally, and most interestingly, the output port quantity, vo, is also a voltage and the low-Z closed-loop impedance formula applies: Zo/(1 + G·H). The circuit is assumed linear and superposition can be used to combine the equivalent circuits for Ro = 10 Ω and Co = 22 pF as shown below.
The contribution to Zo(cl) of Ro is 10 Ω of resistance shunting 1 mΩ in series with an inductance of 1.59 μH. Co is in parallel with Ro; consequently, their equivalent circuits are also in parallel. Co of 22 pF shunts a series RC with capacitance of 0.22 μF in series with the gyrated capacitance, which becomes a resistance of 7.23 kΩ.
In a hf approximation (valid only in the hf region) of Zo(cl), the series RL collapses to L and series RC to R. The resulting parallel resistance is 10 Ω || 7.23 kΩ which is about 10 Ω. This resistance damps a parallel LC resonant circuit with resonant characteristics,
This amplifier will not oscillate at its output because fn is beyond fT (where there is insufficient gain to oscillate) and also because it is highly over-damped. Critical damping resistance for parallel resonance is Zn/2 = 134 Ω >> 10 Ω. However, were Co to increase significantly, Zn and fn would decrease and the possibility of oscillations arises. An open cable connecting a high-resistance load or CRT deflection plates can be instances of such large capacitance.