# Calculating gain of compound feedback network

Discussion in 'Homework Help' started by bitrex, Mar 31, 2010.

1. ### bitrex Thread Starter Active Member

Dec 13, 2009
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I'm trying to figure out how to calculate the gain and input impedance of a certain amplifier circuit that has two stages of feedback. Basically it's a two-stage amplifier that has voltage-sampling series feedback from the output into the emitter of the first stage, and then has a current-sensing resistor that does shunt mixing into the base of the first stage. $\beta$ for the voltage sampling series feedback stage is going to be the ratio of the feedback voltage to the output voltage, and $\beta$ for the current-sampling shunt mixing stage is going to be the ratio of the feedback current to the output current. Those ratios are easy enough to find, but what I'm not sure about is how to apply the standard feedback equation $\frac{A}{1+A\beta}$. Would the gain just be derived using the sum of the two betas? In this case the series mixing stage is trying to raise the input impedance, while the shunt mixing stage is trying to lower it. Does one calculate the rise and the fall independently, and then sum the result? Thanks for any advice.

2. ### Jony130 AAC Fanatic!

Feb 17, 2009
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Can you post the schematics ?

3. ### bitrex Thread Starter Active Member

Dec 13, 2009
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4
Sure. Attached is an example of the compound feedback structure I'm talking about. It's a slightly different setup than my previous post described: there's voltage sampling series mixing from the output to the first stage (R5 and R4) and there's current sampling shunt mixing from the output to the first stage (R3 and R8). The input would be capacitively coupled to R6 and R7, with the output taken at R2. The feedback fraction for the first type of feedback, $\beta_1$, is going to be the ratio of Vf to Vo, with the gain $A$ defined as $\frac{]V_o}{V_i}$. The feedback fraction for the second type, is going to be the ratio of If to Io, with the gain $A$ defined as $\frac{I_o}{I_i}$. I think I can find these quantities easily enough using small signal analysis, but what I am not sure about is how to combine the values to find the total gain of the amplifier with the two types of feedback applied.

Edit: there needs to be a DC blocking capacitor between R5 and the emitter of Q1. The DC path between R3 and R8 is probably all right with proper selection of components...

• ###### Feedback.jpg
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4. ### t_n_k AAC Fanatic!

Mar 6, 2009
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I would take a "dumb" approach. I think I would have to draw up the small signal model and solve the problem using standard circuit solution methods such as nodal analysis. I would probably make useful assumptions along the way to make things easier - such as equal current gains (betas) for the two transistors and beta >>1.

This isn't a trivial exercise - is it an actual homework problem you have been set as an assignment? If so the teacher is a hard taskmaster.

Some actual values would be useful so that you could check the worked solution by running a simulation.

The Electrician has some good insights on such matters and might be able to suggest a more elegant approach.

5. ### The Electrician AAC Fanatic!

Oct 9, 2007
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I'll say. It's amazing how complicated the exact result for a circuit with just a few components can get. Steveb and I exchanged a few comments about this "high entropy expression" problem and what to do about it.

Shekel's method, which I've demonstrated in another thread, comes to the rescue.

I assumed separate β and re for the two transistors--β1, re1, β2, re2.

The voltage gain from input to collector of Q2 is calculated and gives a very complicated expression.

I then allowed the 4 transistor parameters to become idealized one at a time like this:

β1 -> ∞
re1 -> 0
β2 -> ∞
re2 -> 0

You can see the gain expression become simpler and simpler, finally ending in an expression that might just be considered manageable.

All this is what I get from Mathematica. It's likely that if a human were to work on the expression it could be simplified further.

I fully expect Jony130 to give the same analysis!

Edit: I forgot the input impedance. See the second image.

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• ###### Bitrex2.png
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6. ### The Electrician AAC Fanatic!

Oct 9, 2007
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When you've got mixed feedback like this, I've never been able to get the classical feedback formulas to work. I think you just have to bite the bullet and do a full analysis.

It would be interesting to see if what you get when you combine the two feedback expressions is anything like what I got from the full analysis.

7. ### bitrex Thread Starter Active Member

Dec 13, 2009
79
4
Thanks for the replies, everyone! To t_n_k - no, I'm the only taskmaster here - I'm trying to get a feeling for how these multiple feedback circuits work in practice. I can think of some situations where they would be quite useful...
I'm attempting to analyze it using the above feedback equations and a cascade hybrid-pi model of the transistors. It's slow going because I have to be very careful not to make any errors in setting up the models and equations, and even so I've had to make some simplifications to make it tractable (ro = infinity for example). Eventually I will have a set of equations for the current gain, voltage gain, voltage feedback ratio, and current feedback ratio, and at that point I'll be able to use Maxmia (CAS) to help simplify things and do the feedback equations.

An old circuit analysis book I have here briefly mentions compound feedback, and states that as far as gain is concerned the total feedback fraction is the sum of the two feedback fractions. This could lead to a beta greater than one, but I don't think that's a problem since there are two independent paths. In the case of the input and output impedances the two betas are used separately since they push those values in opposite directions. In the end I doubt I'll get a set of equations that looks like the result The Electrician got with Shekel's method, so I'll probably try setting up the circuit in LTSpice and plugging in some values and see if my equations produce anything similar.

My reason for doing this is that while the full nodal analysis method produces the correct answer, it's hard to get a sense from the equations of what effect altering the component values in one feedback path or the other has on the gain and input impedance. I've been studying the design of some high-performance discrete microphone preamps, and they often use a compound feedback technology where a little variable voltage feedback is used to set the gain, and fixed current feedback is used to drop the input impedance. Being able to get a handle on what effect each feedback loop has in isolation could make design an easier task.

8. ### The Electrician AAC Fanatic!

Oct 9, 2007
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For reference, here's the current gain. The output current is taken as the current in R2.

The output current into a short would be somewhat different, but that would eliminate the feedback through R5.

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