# What circuits textbooks haven’t told you about feedback amplifier and transistor port impedances

The recent EDN article of Bruce Trump (thesignal@list.ti.com) on the input impedance of an op-amp circuit took into account the single-pole roll-off of its loop gain. He showed with a simple derivation how the equivalent circuit for the input impedance of a trans-impedance amplifier (with current as the input variable) has inductance. As we know, inductance can resonate with capacitance, and this offers an important clue to some of the unexplained oscillations not uncommonly encountered in circuits.

# Background of an Enigma

Trump’s article is a good lead-in to an important aspect of circuit design involving high-frequency (hf) theory, the theory of what happens in circuits above bandwidth. It applies not only to feedback amplifiers but also to transistor circuits above fβ, the β bandwidth of BJTs. (FETs have a corresponding bandwidth.)

This topic is not developed in any active-circuits textbook that I have found and is one of the neglected topics of electronics. It has long been known by oscilloscope vertical-amplifier designers at Tektronix and was taught for years in a course, “Amplifier Frequency and Transient Response” by Carl Battjes and others within Tektronix. Over the last half century, it has not diffused very extensively into the electronics industry. It is tangentially covered in filter theory about negative-impedance converters but is not applied more generally to circuits.

This hf circuit theory can be used to identify spurious resonances or oscillations in feedback amplifier and discrete transistor circuits, such as photodiode-input trans-resistance op-amps or emitter followers with capacitive loads. Keep in mind that “high frequency” means above bandwidth, not “microwave”. For a typical op-amp, the hf region starts at the open-loop bandwidth, typically 10 Hz. The hf region upper end is the gain-bandwidth product, or f_{T}, of the amplifier or transistor.

What I intend to do in this article is present a basic tutorial on a theory that explains a variety of circuit behaviors that are often not understood and are consequently fixed with experimental circuit hacks (whether on the bench or simulator). Ironically, despite its relative obscurity, hf theory is not hard to understand. The only math needed is basic algebra.

# Closed-Loop Port Impedances

Trans-resistance amplifier input impedance is a special case of a more general principle that can be applied to both feedback amplifier input and output ports. From basic feedback theory, amplifiers can have either voltage or current inputs or outputs, resulting in four basic kinds of amplifiers and four port impedances, for current or voltage variables at input or output.

A feedback amplifier with voltage output has a closed-loop output port impedance of

where G is the forward-path gain, H is the feedback-path gain, G·H is the loop gain and 1 + G·H is what I call the feedback factor. (It is also less descriptively called the return ratio in control literature.) It is the improvement factor attributable to feedback and reduces output impedance so that with feedback the output is closer to that of an ideal voltage source.

Many circuits textbooks use A for G and β for H, but the BJT β parameter appears in feedback circuit analysis, causing ambiguity. I am using the well-established control-theory notation instead.

A feedback amplifier with current input has an input port impedance that is of the same form as the above feedback amplifier with voltage output;

where Z_{i} is the open-loop input port impedance. The ideal current-input amplifier has 0 Ω input, and feedback also causes the open-loop impedance to be reduced toward the ideal.

The other two cases, those of closed-loop input voltage and output current, have the form

These are the two basic feedback equations for port impedances.

**Next: High-Frequency Theory**

System level design and integration challenges with multiple ADCs on single chip

Understanding the basics of setup and hold time

Product How-to: Digital isolators offer easy-to-use isolated USB option

Managing noise in the signal chain, Part 2: Noise and distortion in data converters

War of currents: Tesla vs Edison

Simple reverse-polarity-protection circuit has no voltage drop

Control an LM317T with a PWM signal

Start with the right op amp when driving SAR ADCs

Vintage electrical measuring instruments from the 1950s

Emergency strobe flasher generates 250V

NASA Space Station’s top 10 benefits to humanity

Feedback circuit enhances phototransistor's linear operation

Simple circuits enable oscilloscope art

Iterated-map circuit creates chaos