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, C_i is shown externally for analysis. Op-amps have typically around C_i = 10 pF across their inputs.

This kind of circuit is found in impedance meters, to convert the unknown current, i_x , through the device-under-test (DUT) to a voltage, v_ix . The feedback resistance, R_R 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 f_G . 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 f_T 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 f_G = ½ x π x &#964_G 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 τ_G 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 τ_T = 1/2 x π x f_T = τ_G /G₀ . This is the response of an infinite-G₀ , finite-f_T op-amp. The graph below is a frequency response plot of an op-amp having an open-loop bandwidth of f_bw (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 τ_T decreases (or f_T 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 f_T 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 f_T .

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 v_ix /v _– . T_i is the i_x -to-v _– transfer function before the input to the feedback loop. It is not in the loop but is entangled with it. C_i forms an input impedance with R_R ;

The feedback part of the amplifier is a voltage amplifier that has input, v _– = i_x x Z_in .

Because a voltage divider in one direction can be H and is T_i in the other direction, it is a good practice to always include T_i (and for similar reasons, T_o ) in the general feedback formula , so that it becomes

This feedback formula corresponds to the general block diagram shown below where x can be voltage or current.

For the noninverting op-amp configuration, T_i = 1 as is T_o , 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 C_i , R_R as in T_i 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, i_x through T_i 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 T_i , when substituted into the feedback formula for a single-pole, infinite-G₀ op-amp result in the closed-loop transimpedance of

Although R_R is a resistance, the frequency-dependent C_i and op-amp make the closed-loop response an impedance which simplifies to

At s = 0+ Hz, the gain is an inverting transresistance, –R_R , 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 Z_m (s ).