Return ratio

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. cite book
author=Richard R Spencer & Ghausi MS
title=Introduction to electronic circuit design
page=p. 723
year= 2003
publisher=Prentice Hall/Pearson Education
location=Upper Saddle River NJ
isbn=0-201-36183-3
url=http://worldcat.org/isbn/0-201-36183-3]

Calculating the return ratio

The steps for calculating the return ratio of a source are as follows:cite book
author=Paul R. Gray, Hurst P J Lewis S H & Meyer RG
title=Analysis and design of analog integrated circuits
page=§8.8 pp. 599-613
year= 2001
edition=Fourth Edition
publisher=Wiley
location=New York
isbn=0-471-32168-0
url=http://worldcat.org/isbn/0-471-32168-0]
# 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 [ [http://www.informaworld.com/smpp/content~content=a771365730~db=all Middlebrook, RD:"Loop gain in feedback systems 1"; Int. J. of Electronics, vol. 38, no. 4, (1975) pp. 485-512 ] ] and Rosenstark [ [http://www.informaworld.com/smpp/content~content=a777774065~db=all Rosenstark, Sol: "Loop gain measurement in feedback amplifiers"; Int. J. of Electronics, vol. 57, No. 3 (1984) pp.415-421] ] 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. [ [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=99170 Hurst, PJ: "Exact simulation of feedback circuit parameters"; IEEE Trans. on Circuits and Systems, vol. 38, No. 11 (1991) pp.1382-1389] ] See [http://www.spectrum-soft.com/news/spring97/loopgain.shtm Spectrum user note] or Roberts, or Sedra, and especially Tuinenga.cite book
author=Gordon W. Roberts & Sedra AS
title=SPICE
edition=Second Edition
year= 1997
pages=Chapter 8; pp. 256-262
publisher=Oxford University Press
location=New York
isbn=0-19-510842-6
url=http://worldcat.org/isbn/0-19-510842-6] cite book
author=Adel S Sedra & Smith KC
title=Microelectronic circuits
edition=Fifth Edition
year= 2004
pages=Example 8.7; pp. 855--859
publisher=Oxford University Press
location=New York
isbn=0-19-514251-9
url=http://worldcat.org/isbn/0-19-514251-9] cite book
author=Paul W Tuinenga
title=SPICE: a guide to circuit simulation and analysis using PSpice
edition=Third Edition
year= 1995
pages=Chapter 8: "Loop gain analysis"
publisher=Prentice-Hall
location=Englewood Cliffs NJ
isbn=0134360494
url=http://worldcat.org/isbn/0134360494]

Example: Collector-to-base biased bipolar amplifier

Figure 1 (top right) shows a bipolar amplifier with feedback bias resistor "R_f" 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.cite book
author=Richard R Spencer & Ghausi MS
title=Example 10.7 pp. 723-724
isbn=0-201-36183-3
url=http://worldcat.org/isbn/0-201-36183-3] 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 "i_t", 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

::$T\; =\; -\; frac\; \{i\_r\}\; \{i\_t\}\; .$

The return current is

::$i\_r\; =\; g\_m\; v\_\{pi\}\; .$

The feedback current in "R_f" is found by current division to be:::$i\_f\; =\; frac\; \{R\_D//r\_O\}\; \{R\_D//r\_O\; +R\_F\; +r\_\{pi\}//\; R\_S\}\; i\_t\; .$

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

::$v\_\{pi\}\; =\; -i\_f\; (\; r\_\{pi\}//\; R\_S\; )\; .$

Consequently,::$T\; =\; g\_m\; (r\_\{pi\}//\; R\_S\; )\; frac\; \{R\_D//r\_O\}\; \{R\_D//r\_O\; +R\_F\; +r\_\{pi\}//\; R\_S\}\; .$

Application in asymptotic gain model

The overall transresistance gain of this amplifier can be shown to be:

::$G\; =\; frac\; \{v\_\{out\; \{i\_\{in\; =\; frac\; \{(1-g\_m\; R\_F)R\_1\; R\_2\}\; \{R\_F+R\_1+R\_2+g\_m\; R\_1R\_2\}\; ,$

with "R₁ = R_S || r_π" and "R₂ = R_D || r_O".

This expression can be rewritten in the form used by the asymptotic gain model, which expresses the overall gain of a feedback amplifier in terms of several independent factors that are often more easily derived separately than the overall gain itself, and that often provide insight into the circuit. This form is:

::$G\; =\; G\_\{\; infty\; \}\; frac\; \{T\}\; \{1+T\}\; +\; G\_0\; frac\; \{1\}\; \{\; 1+T\}\; ,$

where the so-called asymptotic gain "G_∞" is the gain at infinite "g_m", namely:

::$G\_\{infty\}\; =\; -\; R\_F\; ,$

and the so-called feed forward or direct feedthrough "G₀" is the gain for zero "g_m", namely:

::$G\_\{0\}\; =\; frac\; \{\; R\_1\; R\_2\; \}\; \{R\_F\; +R\_1\; +R\_2\}\; .$

For additional applications of this method, see asymptotic gain model.

References

ee also