A seemingly simple circuit with only two components, an op-amp and a feedback resistor, is commonly used to convert a current to a voltage. This circuit is familiar as a photodetector or impedance-meter current-sense amplifier. The circuit is shown below, where the input capacitance of the op-amp, Ci is shown externally for analysis. Op-amps have typically around Ci = 10 pF across their inputs.
This kind of circuit is found in impedance meters, to convert the unknown current, ix, through the device-under-test (DUT) to a voltage, vix. The feedback resistance, RR in such an instrument is range-switched to cover a wide range of currents over a wide range of DUT impedances.
To keep the circuit seemingly simple, let the op-amp be very ordinary - a voltage-feedback op-amp with a single dominant pole at fG. G is the forward path gain of the feedback loop and H is the feedback path, in accordance with control theory nomenclature; no A for G and
β for H as is commonly found in active-circuits textbooks because BJTs have an over-riding claim to the use of β. Gain magnitude decreases with frequency to a gain of one at fT from a high quasistatic (0+ Hz) gain of G0. The op-amp is the forward path of the feedback loop with a voltage gain of
where G0 is the open-loop op-amp quasistatic gain magnitude and fG = ˝ x π x τG is the open-loop bandwidth. In the dynamic or frequency-dependent factor of G that depends on s, it is sometimes easier to work with the math by using pole or zero time constants such as τG instead of pole or zero frequencies.
Two simplifications of this op-amp transfer function or gain can be made. The first is often closely approximated in practice and is what makes op-amps “operational”: high gain. By approximating the quasistatic gain, G0, as infinite, or G0 → ∞, the first op-amp simplification occurs. Divide numerator and denominator of G by 1/G0, then let G0 approach infinity;
where τT = 1/2 x π x fT = τG/G0. This is the response of an infinite-G0, finite-fT op-amp. The graph below is a frequency response plot of an op-amp having an open-loop bandwidth of fbw(ol) extended to zero as a pole at the origin by infinite open-loop quasistatic gain. The open-loop gain is G and closed-loop gain includes the feedback.
The second op-amp simplification is to let its frequency response be unlimited by letting the bandwidth of G become infinite - an “infinitely fast” op-amp. As τT decreases (or fT increases), the gain plot shifts to the right (or upward), which increases gain. A faster amplifier has more quasistatic gain, higher quasistatic loop gain, GH0, higher precision, and can be less stable in a circuit with slow poles. An op-amp with infinite G0 and infinite fT is an ideal op-amp and is the op-amp assumed in the commonplace inverting and noninverting gain formulas for op-amps.
This second approximation of infinite bandwidth is not as practical as infinite gain and if assumed can sometimes result in an oscillating amplifier. Consequently, to understand seemingly simple op-amp circuits, it is sometimes - nay, usually - necessary to include the single pole in the gain derivation, then assess it for amplifier dynamic behavior. Unless the op-amp is being used in a slow, high-precision application, infinite G0 is usually valid in practice - but not infinite fT.
The closed-loop amplifier is a transimpedance (current in, voltage out) amplifier with transfer function,
The rational factor is the closed-loop feedback formula, with voltage gain vix /v–. Ti is the ix-to-v– transfer function before the input to the feedback loop. It is not in the loop but is entangled with it. Ci forms an input impedance with RR;
The feedback part of the amplifier is a voltage amplifier that has input, v– = ix x Zin.
Because a voltage divider in one direction can be H and is Ti in the other direction, it is a good practice to always include Ti (and for similar reasons, To) in the general feedback formula, so that it
This feedback formula corresponds to the general block diagram shown below where x can be voltage or current.
For the noninverting op-amp configuration, Ti = 1 as is To, but there are circuits (such as this transresistance amplifier) for which it is important to recognize that circuit elements involved in the loop also have a pre-loop or post-loop effect on gain.
H is the feedback-path transfer function with the same Ci, RR as in Ti but as a voltage divider from output to error voltage, v–:
The feedback error-summing (the circle with Σ in the block diagram) is accomplished in the circuit by the superposition of the divider and input voltages: the input quantity, ix through Ti adds to the feedback quantity, the output of H. The feedback error-summer of the closed-loop formula subtracts the input voltage from the fed-back voltage, but because the voltage divider adds (does not invert), the negative sign in H corrects for the non-inversion so that the formula remains correct and consistent with the block diagram.
GH = G x H is the loop gain and is of interest because it determines loop stability. G, H, and Ti, when substituted into the feedback formula for a single-pole, infinite-G0 op-amp result in the closed-loop transimpedance of
Although RR is a resistance, the frequency-dependent Ci and op-amp make the closed-loop response an impedance which simplifies to
At s = 0+ Hz, the gain is an inverting transresistance, –RR, as expected. Frequency effects appear with the poles in the denominator. The amplifier has a two-pole (quadratic) response, where the poles are
In Part 2, we will examine the consequences for circuit behavior of Zm(s).