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 *G*_{0}. The op-amp is the forward path of the feedback loop with a voltage gain of

where *G*_{0} is the open-loop op-amp quasistatic gain magnitude and *f*_{G} = ˝ x π x *τ*_{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, *G*_{0}, as infinite, or *G*_{0} → ∞, the first op-amp simplification occurs. Divide numerator and denominator of *G* by 1/*G*_{0}, then let *G*_{0} approach infinity;

where *τ*_{T} = 1/2 x π x *f*_{T} = *τ*_{G}/*G*_{0}. This is the response of an infinite-*G*_{0}, 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, *GH*_{0}, higher precision, and can be less stable in a circuit with slow poles. An op-amp with infinite *G*_{0} 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 *G*_{0} 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-*G*_{0} 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*).