In Seemingly Simple Circuits: Transresistance Amplifier, Part 1-- Approximating Op-Amps, the op-amp was approximated from a single-pole, finite gain amplifier to an infinite-gain, single-pole op-amp, and the gain of the transresistance amplifier circuit, shown repeated below, was derived. In this Part 2 we look at the consequences.

From Part 1, the derived gain - a transimpedance – is

with poles at

The amplifier gain gives us an opportunity to apply control theory to the circuit. This example will illustrate how useful and important control theory is in understanding circuit dynamics. Taken a logical step at a time, it is not all that overwhelming and hopefully will offer some insights into control techniques and how they apply.

A pole-pair (quadratic) polynomial is expressed in general as

The *resonant time constant*, *τ*_{n} = 1/*ω*_{n} = 1/(2 x *π* x *f*_{n}) and *damping*, *ζ* for the amplifier are

The poles become a complex pole-pair whenever *π* < 1, and the pole angle,

For real poles, *π* > 1 and *φ* = 0.

For a constant group (or envelope) delay (Maximally Flat Envelope Delay/MFED or Bessel) response, phase decreases linearly with frequency, and occurs at a pole angle of *φ* = 30^{o}. The delay is the same in time for all frequencies, preserving the waveshape. Then

For the transimpedance amplifier MFED response,

For critical damping (fastest step response without overshoot), *π* = 1 and *τ*_{T} = 4 x *τ*_{i} or *f*_{T} = *f*_{i}/4. The two poles are borderline real and are both at *f*_{i}/2.

As *R*_{R} is made larger and *f*_{i} decreases, the amplifier shows greater overshoot in *v*_{ix}. To some extent, this is advantageous for a Z-meter in that for a pole angle of *φ* = 45^{o}, then damping, *π* = cos(*φ*) = cos(45^{o}) ≈ 0.707, and the frequency magnitude (or amplitude) response is constant, or *flat*, to near bandwidth frequency. This is a *maximally-flat amplitude* (MFA) frequency response. The MFA response is optimal for steady-state (frequency-domain) applications and the MFED response for transient (time-domain) applications where the ideal is a step response. (Oscilloscope vertical-amplifier design is caught between the conflicting criteria of optimizing both responses.)

**Op-Amp Speed and Amplifier Stability**

A slow op-amp has low *f*_{T} and *τ*_{T} >> *τ*_{i}, resulting in wide pole separation of the two real poles. In the limit,

These are poles at the origin and *f*_{i}. *f*_{T} must be kept small enough to keep *f*_{T} << *f*_{i}. As *f*_{T} is decreased, however, there is less loop gain and it might be insufficient to maintain acceptable op-amp gain error. Precision in this case requires some speed.

As op-amp *f*_{T} is increased, *Z*_{m} becomes less damped and less stable. For a given *ς* and *f*_{i},

For *f*_{T} = 1 MHz and *G*_{0} = 10^{5}, then *f*_{G} = 10 Hz, and a critically-damped loop (*π* = 1) has *f*_{i} = 40 Hz. Given that *C*_{i} = 10 pF, then *R*_{R} = 398 MΩ, sufficiently large that any lesser value would keep *f*_{i} > 40 Hz.

The *migration* of the closed-loop poles (in bold) with increasing *f*_{T} (faster op-amp) is shown on the *s*-domain *root-contour* plot below. The separated poles, at the origin and *f*_{i} (–1/*τ*_{i}), come together at *f*_{i} /2 (where *π* = 1) and then become a complex pole-pair and leave the real (*σ*) axis vertically. As *f*_{T} increases, the pole angle increases and *π* decreases. The amplifier becomes less stable and more oscillatory in response.

The location, or *locus*, of pole migration shown in the plot occurs whenever the varied parameter (*f*_{T} or *τ*_{T} in this plot) is in both the *s*^{2} and *s* terms of the polynomial. The amplifier becomes least damped at infinite *f*_{T} and the pole locations in the limit as *τ*_{T} → 0 *s* are

There are two values on the *j* x *ω*-axis where response is stable (and not oscillatory): at the origin and at +/-*j* x ∞. Both are non-finite. (Zero (0) is infinitesimal, or “infinitely small”.) As *τ*_{T} → 0 s, both terms in *s* of the pole polynomial approach zero, leaving the constant 1 term and no frequency-dependent effect. In the limit, the poles are on the *j* x *ω*-axis and *ζ* = 0 - the conditions of an oscillator - but at finite values of *s*, their magnitudes are zero. The poles are so high in frequency that their damping no longer matters. They are too far removed from *f*_{i} to affect loop dynamics. This is the condition of the ideal op-amp. Thus, we can conclude that for either a very slow or very fast op-amp, the poles are sufficiently separated that the response is stable. It is only in a range of *f*_{T} where the op-amp and *C*_{i} poles are too close together that damping is reduced excessively at a low enough pole frequency, *f*_{n}, that oscillatory behavior occurs in the amplifier at a significant magnitude.

Returning to the transimpedance amplifier, if the op-amp is nearly ideal - that is, fast enough to where *τ*_{T} ≈ 0 s - the pole polynomial collapses to about one. For a sufficiently fast op-amp, *f*_{T} >> *f*_{i}, and the poles are separated, stabilizing the loop. To provide additional damping so that the op-amp *f*_{T} (and loop gain) need not be impractically low, capacitor *C*_{f} is shunted across *R*_{R}. Then cranking through the circuit algebra with *C*_{f} included,

The effect of *C*_{f} is to add *τ*_{f} to *τ*_{i} in the quadratic coefficient, and more importantly, to *τ*_{T} in the linear term, which increases damping. For *τ*_{i} = *τ*_{T}, then

;

For critical damping, set *π* = 1; then *τ*_{T} = (3 + 2 x √2) x *τ*_{i} ≈ 3.414 x *τ*_{i} and *τ*_{n} ≈ 1.848 x *τ*_{i}. Without *C*_{f} (*C*_{f} = 0 pF), as calculated previously, *τ*_{T} = 4 x *τ*_{i}. With *C*_{f}, the op-amp can be faster - that is, have higher *G*_{0} and achieve higher precision - with the same dynamic response.

The frequency-response magnitude and phase are

An ideally-fast op-amp (*τ*_{T} = 0 s) and *C*_{f} = *C*_{i} (*τ*_{f} = *τ*_{i}) have a response at frequency *f*_{g} (or *ω*_{g}) of

If *f*_{i} = 10 x *f*_{g}, then amplitude error ≈ 0.5 %. For phase,

For *f*_{i} = 10 x *f*_{g}, phase error ≈ 6^{o}. Phase error is more sensitive to frequency effects than is magnitude error. This is important in impedance-meter circuit design and sometimes in photodetector amplifiers where the photodetected waveform is intended to be synchronous with some other waveform.