Return ratio

Key takeaways

Return ratio belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Return ratio to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Return ratio from memory before moving on to harder problems.

Reference excerpt

The return ratio of a dependent source in a linear electrical circuit is the negative of the ratio of the current (voltage) returned to the site of the dependent source to the current (voltage) of a replacement independent source. The terms loop gain and return ratio are often used interchangeably; however, they are necessarily equivalent only in the case of a single feedback loop system with unilateral blocks.

Calculating the return ratio

The steps for calculating the return ratio of a source are as follows:

Set all independent sources to zero. Select the dependent source for which the return ratio is sought. Place an independent source of the same type (voltage or current) and polarity in parallel with the selected dependent source. Move the dependent source to the side of the inserted source and cut the two leads joining the dependent source to the independent source. For a voltage source the return ratio is minus the ratio of the voltage across the dependent source divided by the voltage of the independent replacement source. For a current source, short-circuit the broken leads of the dependent source. The return ratio is minus the ratio of the resulting short-circuit current to the current of the independent replacement source.

Other Methods These steps may not be feasible when the dependent sources inside the devices are not directly accessible, for example when using built-in "black box" SPICE models or when measuring the return ratio experimentally. For SPICE simulations, one potential workaround is to manually replace non-linear devices by their small-signal equivalent model, with exposed dependent sources. However this will have to be redone if the bias point changes. A result by Rosenstark shows that return ratio can be calculated by breaking the loop at any unilateral point in the circuit. The problem is now finding how to break the loop without affecting the bias point and altering the results. Middlebrook and Rosenstark have proposed several methods for experimental evaluation of return ratio (loosely referred to by these authors as simply loop gain), and similar methods have been adapted for use in SPICE by Hurst. See Spectrum user note or Roberts, or Sedra, and especially Tuinenga.

Example: Collector-to-base biased bipolar amplifier

Figure 1 (top right) shows a bipolar amplifier with feedback bias resistor Rf driven by a Norton signal source. Figure 2 (left panel) shows the corresponding small-signal circuit obtained by replacing the transistor with its hybrid-pi model. The objective is to find the return ratio of the dependent current source in this amplifier. To reach the objective, the steps outlined above are followed. Figure 2 (center panel) shows the application of these steps up to Step 4, with the dependent source moved to the left of the inserted source of value it, and the leads targeted for cutting marked with an x. Figure 2 (right panel) shows the circuit set up for calculation of the return ratio T, which is

T = − i r i t . {\displaystyle T=-{\frac {i_{r}}{i_{t}}}\ .}

The return current is

i r = g m v π . {\displaystyle i_{r}=g_{m}v_{\pi }\ .}

The feedback current in Rf is found by current division to be:

i f = R D / / r O R D / / r O + R F + r π / / R S i t . {\displaystyle i_{f}={\frac {R_{D}//r_{O}}{R_{D}//r_{O}+R_{F}+r_{\pi }//R_{S}}}\ i_{t}\ .}

The base-emitter voltage vπ is then, from Ohm's law:

v π = − i f ( r π / / R S ) . {\displaystyle v_{\pi }=-i_{f}\ (r_{\pi }//R_{S})\ .}

Consequently,

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Return ratio

Start with the simplest possible case. Write down what Return ratio claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Return ratio before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Return ratio ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Return ratio

In research: Return ratio appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Return ratio in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Return ratio is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Electronic feedback, Signal processing, so understanding it makes those chapters shorter.
In everyday life: Look for Return ratio outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Return ratio in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Return ratio means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Return ratio out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Return ratio in simple terms?

The return ratio of a dependent source in a linear electrical circuit is the negative of the ratio of the current (voltage) returned to the site of the dependent source to the current (voltage) of a replacement independent source. The terms loop gain and return ratio are often used interchangeably…

Why does Return ratio matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Return ratio?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Return ratio.