# Feedback Amplifier

#### KCHARROIS

Joined Jun 29, 2012
302
Hello,

I've been looking at feedback amplifiers such as the one below and the overall gain of the amplifier is equal to R11/R3. Knowing this is great but I'm trying to understand the gain of each of the two stages. Figuring this out is easy without the feedback path, you calculate Zin, Zbase, gain of each stage, etc. But with feedback path its not so easy, so how should I go about calculating the gain of each stage?

Thanks

#### kubeek

Joined Sep 20, 2005
5,625
I think that Q2 will have the same gain as without feedback, and Q1 will have the rest of what is needed to maintain the overall gain set by R11/R3.

#### KCHARROIS

Joined Jun 29, 2012
302
I can't figure what the gain of the first stage will be but for the second stage I think that:

R10||R9||(R11+R3)

Can someone confirm this for the second stage? Any ideas for the first stage?

Thanks

#### KCHARROIS

Joined Jun 29, 2012
302
After playing around with the feedback resister R11 the gain of both stages is changing there for its safe to say that R11 has a direct effect on both amplifiers but still don't understand how. Not sure if this helps anyone?

Thanks

#### LvW

Joined Jun 13, 2013
875
After playing around with the feedback resister R11 the gain of both stages is changing there for its safe to say that R11 has a direct effect on both amplifiers but still don't understand how. Not sure if this helps anyone?
Thanks
I am afraid, there is no other way than to apply Black´s classical feedback formula for the closed-loop gain Acl:

Acl=Aol/(1-T)

with Aol=gain without feedback and T=Loop gain (negative sign).

Loop gain is the gain around the complete feedback loop, which for the purpose of calculation must be opened at a suitable node. Suitable means: A node where a low output impedance is connected to a large input impedance. By doing this, loading errors are minimized. Otherwise, a kind of load mirror has to be connected.

EDIT 1: After looking into your circuit, I would propose to open the loop at the collector of Q1 and inject a test signal (for calculation of T) into the second amplifier circuit. However, dont forget to load the collector of Q1 with a good approximation for the input resistance of the second stage. This node provides the output voltage for loop gain calculation.
A a result, you will see that the overall closed-loop gain will not be equal to R11/R3.

EDIT 2: For calculation of Aol you must not forget to load Q2 with R11.

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#### LvW

Joined Jun 13, 2013
875
With the aim to justify the above formula I have simulated the whole circuit (according to the mentioned procedure).
Results (for f>300Hz):
Acl=10.5
Aol=220
T=-20

Thus: Acl=Aol(1+20)=10.47 (good agreement with the measured value of 10.5).

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#### LvW

Joined Jun 13, 2013
875
.... for the second stage I think that:
R10||R9||(R11+R3)
Can someone confirm this for the second stage? Any ideas for the first stage?
R11 is not only connected to R3. This node is also connected to the very low impedance emitter node. For this reason, you cannot simply isolate the voltage divider R11-R3.
More than that, I dont think it makes sense to ask separately for the gain of the 1st stage because you have TWO signal inputs (base and emitter).
Therefore my former answer (suggestion) to use the loop gain concept.

#### KCHARROIS

Joined Jun 29, 2012
302
I will try your loop method and see if I get the same results that you are getting.

Thanks

#### PRS

Joined Aug 24, 2008
989
I can't figure what the gain of the first stage will be but for the second stage I think that:

R10||R9||(R11+R3)

Can someone confirm this for the second stage? Any ideas for the first stage?

Thanks
The gain of the first stage is (Rin2//R5)/(R3+re1)
The gain of the second: (R10//R9)/(R8+re2)

These are the gains without feedback, of course. The overall gain without feedback is the gain of stage one times the gain of stage two.

#### Jony130

Joined Feb 17, 2009
5,022
To find open loop gain you need to use this small-signal diagram. So if we assume Ic1 = Ic2 ≈ 1mA and Hfe = 150.
The first stage voltage gain is equal to

$$Av1 = \frac{R5||R6||R7||(Hfe+1)*re2}{re1 +R3||R11} = \frac{918}{117} = 7.8V/V$$

And Second stage voltage gain

$$Av2 = \frac{R10||R9||(R3 + R11)}{re2} = \frac{777}{26} = 29.8V/V$$

So the Aol = Av1 * Av2 = 232V/V

And the close loop gain (Acl)

Acl = Aol/(1 + Aol * B) =
232/(1 + 232*(0.1/1.1)) = 10.5V/V

where
Aol - open-loop voltage gain. The gain without the feedback.

B - feedback factor (feedback gain) in our example

B = Ve1/Vc2 = R3/(R3 + R11) = 100/(100Ω + 1KΩ) = 0.1/1.1 = 0.090

And is Aol>>1/B we can say that Acl ≈ 1 + R11/R3 ≈ 11V/V

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#### LvW

Joined Jun 13, 2013
875
Hi Jony, good calculation.
Just one comment: I think at the output we should have R11 in series with R3||(1/gm1).
This would reduce the gain Aol to about 198. However, the gain Acl remains nearly the same.

#### The Electrician

Joined Oct 9, 2007
2,750
Hello,

I've been looking at feedback amplifiers such as the one below and the overall gain of the amplifier is equal to R11/R3. Knowing this is great but I'm trying to understand the gain of each of the two stages. Figuring this out is easy without the feedback path, you calculate Zin, Zbase, gain of each stage, etc. But with feedback path its not so easy, so how should I go about calculating the gain of each stage?

Thanks
I am afraid, there is no other way than to apply Black´s classical feedback formula for the closed-loop gain Acl:

Acl=Aol/(1-T)

with Aol=gain without feedback and T=Loop gain (negative sign).
Of course there's another way; analysis of the entire circuit. I explained how to do this in an older thread: http://forum.allaboutcircuits.com/showthread.php?t=26710

More than that, I dont think it makes sense to ask separately for the gain of the 1st stage because you have TWO signal inputs (base and emitter).
It makes sense if the question is asked precisely. V1 applies a signal to the base of Q1, which leads to signal voltages of various amplitudes at the several nodes.

One can ask, "What is the ratio of the signal at the collector of Q1 to the signal at the base of Q1?", and call this the base to collector gain of Q1.

Then ask "What is the ratio of the signal at the collector of Q1 to the signal at the emitter of Q1?", and call this the emitter to collector gain of Q1.

Using the method described in the older thread, we can set up the admittance matrix. I left R8 in the matrix even though it is bypassed, so that a non-zero emitter resistor for the Q2 stage could be added if desired later; it's set to a very low value to account for the effect of the bypass capacitor.

Node 1 is the base of Q1; node 2 is the emitter of Q1; node 3 is the collector of Q1 and the base of Q2; node 4 is the emitter of Q2; node 5 is the collector of Q2: First let's analyze the circuit without feedback. We do this by setting R11 to ∞. Here's the overall voltage gain and the impedances at the nodes. Notice that without feedback, the impedance at the emitter of Q1 is 30.4852 ohms (assuming the source driving the base of Q1 has zero internal impedance). Here are the individual stage gains as defined above (without feedback): #### The Electrician

Joined Oct 9, 2007
2,750
Now repeat the analyses with R11 set to 1000 ohms. First the overall amplifier voltage gain, and the nodal impedances; the impedance at the emitter is now 1.719 ohms with feedback, a very low value: And the individual stage gains as defined in the previous post: #### Jony130

Joined Feb 17, 2009
5,022
Just one comment: I think at the output we should have R11 in series with R3||(1/gm1).
I'm not so sure about that. Simulation show that R11 + R3 gives closer result for Acl.

First let's analyze the circuit without feedback. We do this by setting R11 to ∞.
Why you don't include R11 loading effect on the open loop gain?

#### The Electrician

Joined Oct 9, 2007
2,750
Why you don't include R11 loading effect on the open loop gain?
I could, but I used the phrase "no feedback" to mean that R11 is not there.

When preparing to analyze the way you and LvW did, the presence of R11 as a load on the output is necessary to preserve the impedances at various nodes when the loop is open to be the same as when the loop is closed.

It's easy enough to analyze the circuit with R11 present as a load, but not connected as feedback:

Edit: I added the effect of R11 to the emitter of Q1, and added 100 ohms in series with R11 at the collector of Q2 to make the circuit exactly the same as yours:  Last edited:

#### KCHARROIS

Joined Jun 29, 2012
302
To find open loop gain you need to use this small-signal diagram. So if we assume Ic1 = Ic2 ≈ 1mA and Hfe = 150.
The first stage voltage gain is equal to

$$Av1 = \frac{R5||R6||R7||(Hfe+1)*re2}{re1 +R3||R11} = \frac{918}{117} = 7.8V/V$$

And Second stage voltage gain

$$Av2 = \frac{R10||R9||(R3 + R11)}{re2} = \frac{777}{26} = 29.8V/V$$

So the Aol = Av1 * Av2 = 232V/V

And the close loop gain (Acl)

Acl = Aol/(1 + Aol * B) =
232/(1 + 232*(0.1/1.1)) = 10.5V/V

where
Aol - open-loop voltage gain. The gain without the feedback.

B - feedback factor (feedback gain) in our example

B = Ve1/Vc2 = R3/(R3 + R11) = 100/(100Ω + 1KΩ) = 0.1/1.1 = 0.090

And is Aol>>1/B we can say that Acl ≈ 1 + R11/R3 ≈ 11V/V

Jony130, thanks this does make sense but here is what really gets to me. When I measure the signal voltage across R5 I measure 300uV and when I measure across R3 to ground I get ≈ 1mV there for the gain is about 1/3, how is that possible I can't see or calculate a gain of 1/3 for the first stage. (This is with feedback included)

Thanks to all other op's as well

#### The Electrician

Joined Oct 9, 2007
2,750
When I measure the signal voltage across R5 I measure 300uV and when I measure across R3 to ground I get ≈ 1mV there for the gain is about 1/3, how is that possible I can't see or calculate a gain of 1/3 for the first stage. (This is with feedback included)

Thanks to all other op's as well
Look at my analysis in post #13. I get a gain of about .32, emitter to collector.

#### KCHARROIS

Joined Jun 29, 2012
302
Great seems like you got it but I can't figure out how you calculated that vale I also looked at post #12.

#### Jony130

Joined Feb 17, 2009
5,022
Jony130, thanks this does make sense but here is what really gets to me. When I measure the signal voltage across R5 I measure 300uV and when I measure across R3 to ground I get ≈ 1mV there for the gain is about 1/3, how is that possible I can't see or calculate a gain of 1/3 for the first stage. (This is with feedback included)
My analysis that I show at post 10 describes how to find Open Loop Gain.

But if you want the individual gains with the feedback loop closed you need to analysis the whole circuit at once.
See this small-signal representation, so all you need to do the small-signal analysis for the whole circuit.

#### The Electrician

Joined Oct 9, 2007
2,750
Great seems like you got it but I can't figure out how you calculated that vale I also looked at post #12.
As Jony130 says, you have to analyze the whole circuit at once, with feedback included.

To fully understand what I'm doing in post #12 and #13, you really must read and understand the old thread I referenced: