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*= 10 pF across their inputs.

_{i}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*. The feedback resistance,

_{ix}*R*in such an instrument is range-switched to cover a wide range of currents over a wide range of DUT impedances.

_{R}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*from a high quasistatic (0+ Hz) gain of

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

_{0}where *G _{0} * is the open-loop op-amp quasistatic gain magnitude and

*f*= ½ x π x

_{G}*τ*is the open-loop bandwidth. In the dynamic or frequency-dependent factor of

_{G}*G*that depends on

*s*, it is sometimes easier to work with the math by using pole or zero time constants such as

*τ*instead of pole or zero frequencies.

_{G}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*→ ∞, the first op-amp simplification occurs. Divide numerator and denominator of

_{0}*G*by 1/

*G*, then let

_{0}*G*approach infinity;

_{0}where *τ _{T} * = 1/2 x π x

*f*=

_{T}*τ*/

_{G}*G*. This is the response of an infinite-

_{0}*G*, finite-

_{0}*f*op-amp. The graph below is a frequency response plot of an op-amp having an open-loop bandwidth of

_{T}*f*(ol) extended to zero as a pole at the origin by infinite open-loop quasistatic gain. The open-loop gain is

_{bw}*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*increases), the gain plot shifts to the right (or upward), which increases gain. A faster amplifier has more quasistatic gain, higher quasistatic loop gain,

_{T}*GH*, higher precision, and can be less stable in a circuit with slow poles. An op-amp with infinite

_{0}*G*and infinite

_{0}*f*is an

_{T}*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*is the

_{i}*i*-to-

_{x}*v*

_{–}transfer function before the input to the feedback loop. It is not in the loop but is entangled with it.

*C*forms an input impedance with

_{i}*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*(and for similar reasons,

_{i}*T*) in the

_{o}*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*, 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.

_{o}*H* is the feedback-path transfer function with the same *C _{i} * ,

*R*as in

_{R}*T*but as a voltage divider from output to error voltage,

_{i}*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*adds to the feedback quantity, the output of

_{i}*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*op-amp result in the closed-loop transimpedance of

_{0}Although *R _{R} * is a resistance, the frequency-dependent

*C*and op-amp make the closed-loop response an impedance which simplifies to

_{i}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*).

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