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 , τn = 1/ωn = 1/(2 x π x fn ) and damping , ζ for the amplifier are
The poles become a complex pole-pair whenever ζ < 1, and the pole angle,
For real poles, ζ > 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 = 30o . 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), ζ = 1 and τT = 4 x τi or fT = fi /4. The two poles are borderline real and are both at fi /2.
As RR is made larger and fi decreases, the amplifier shows greater overshoot in vix . To some extent, this is advantageous for a Z-meter in that for a pole angle of φ = 45o , then damping, ζ = cos(φ ) = cos(45o ) ≈ 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 fT and τT >> τi , resulting in wide pole separation of the two real poles. In the limit,
These are poles at the origin and fi . fT must be kept small enough to keep fT << fi . As fT 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 fT is increased, Zm becomes less damped and less stable. For a given ς and fi ,
For fT = 1 MHz and G 0 = 105 , then fG = 10 Hz, and a critically-damped loop (ζ = 1) has fi = 40 Hz. Given that Ci = 10 pF, then RR = 398 MΩ, sufficiently large that any lesser value would keep fi > 40 Hz.
The migration of the closed-loop poles (in bold) with increasing fT (faster op-amp) is shown on the s -domain root-contour plot below. The separated poles, at the origin and fi (–1/τi ), come together at fi /2 (where π = 1) and then become a complex pole-pair and leave the real (σ ) axis vertically. As fT increases, the pole angle increases and ζ 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 (fT or τT in this plot) is in both the s2 and s terms of the polynomial. The amplifier becomes least damped at infinite fT and the pole locations in the limit as τT → 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 τT → 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 fi 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 fT where the op-amp and Ci poles are too close together that damping is reduced excessively at a low enough pole frequency, fn , 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 τT ≈ 0 s – the pole polynomial collapses to about one. For a sufficiently fast op-amp, fT >> fi , and the poles are separated, stabilizing the loop. To provide additional damping so that the op-amp fT (and loop gain) need not be impractically low, capacitor Cf is shunted across RR . Then cranking through the circuit algebra with Cf included,
The effect of Cf is to add τf to τi in the quadratic coefficient, and more importantly, to τT in the linear term, which increases damping. For τi = τT , then
For critical damping, set π = 1; then τT = (3 + 2 x √2) x τi ≈ 3.414 x τi and τn ≈ 1.848 x τi . Without Cf (Cf = 0 pF), as calculated previously, τT = 4 x τi . With Cf , 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 (τT = 0 s) and Cf = Ci (τf = τi ) have a response at frequency fg (or ωg ) of
If fi = 10 x fg , then amplitude error ≈ 0.5 %. For phase,
For fi = 10 x fg , phase error ≈ 6o . 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.
Circuit that Avoids Large Feedback Resistor
RR is made large – 10 MΩ or greater – for some Z-meter (ZM) designs with transimpedance amplifiers. A ZM must amplify a wide range of currents, often down to the nA or less range. Photodetector signals can also be very small. As RR becomes very large, the Cf shunting it must be made impractically small for the desired damping, and resistor shunt parasitic capacitance can be excessive. To avert this problem, the following circuit can be used instead.
Let the op-amp be a high-gain, single-pole op-amp, G ≈ –1/s x τT . (See Seemingly Simple Circuits: Transresistance Amplifier, Part 1– Approximating Op-Amps for derivation of G .) The feedback divider transfer function is
and τf = RR x Cf . When the circuit is solved with Rp = R 1 ||R 2 ,
An ideal op-amp (τT = 0 s) Zm reduces to
For Rp = 0 Ω, the transimpedance reduces further to
The R 1 , R 2 divider output resistance need not be made small (Rp << RR ) if a fast × 1 buffer amplifier is inserted between its output and RR and Cf . Then for Rp = 0 Ω and with an op-amp having τT ,
This circuit is of the same form as without the output divider but with two differences: RR and τT are both effectively increased by 1/Hdiv .
This two-part article has shown that even a two-component circuit can involve significant dynamics derivations. The derivations are sometimes avoided by designers to reduce math angst, but having the equations offers greater insight into not only how a given circuit behaves but how it will behave under a wide variety of conditions. The transimpedance amplifier analysis, now that it has been carried out, sets forth a design template for such a circuit generally, and it also offers a guiding example of how to go about analyzing the dynamics of amplifiers.
Do not be discouraged from using s -domain algebra to solve circuit dynamics because of cubic or higher polynomials. We encountered a cubic in this exercise but did not have to solve it because some realistic simplifying assumptions reduced the degree of the polynomial to the quite workable quadratic we subsequently analyzed. This is typically the case because circuits tend to be modularized in design into stages that are either isolated from each other or have controlled interactions through controlled port impedances. Then template solutions can be applied to the stages, and are most usually limited to quadratic equations in the s -domain.