
#1
Oct2805, 04:39 PM

P: 1

Hi there.
i need some help from you guys can any of you tell me how to prove the effects of negative feedback mathematically? i need to prove that negative feedback causes reduction in gain reduction of distortion increase of gain stability increase in bandwidth increase in input resistance thanks in advance Jon 



#2
Oct3105, 11:41 AM

Mentor
P: 39,628

Reduction in gain is easy, just compare the open loop gain and closed loop gain for an opamp circuit like a follower. Distortion  depends on the circuit. Gain Stability  not sure what that means, unless you mean like emitter degeneration. Just use that as the example, if you mean bias point stability instead of Gain Stability. Where did you get the question from? They must have had something more specific in mind. What textbook are you using? 



#3
Nov105, 06:34 PM

P: 2,451

Berkeman, I would say all of those are true. You question input impedance. Take a simple common emmiter configured transistor amplifier. Suppose it an emitter resistor of 1000 ohms with a capacitor in parallel that presents a low (let's call it zero) impedance to the signal being amplified on the emitter. So, knowing this the input impedance will be R'e + (beta * the emitter resistor). This gives us a Zin of R'e. Remove the capacitor and you are introducing negative feedback because of the emitter resistor. Do the math again and you will find that the Zin has gone up.
Increasing bandwidth? You bet. It is at the expense of lowering the gain, but it does. Suppose you have an amplifier running open loop and due to parasitic capacitance and etc. of the circuit it has a funny frequency response curve. If you introduce some of the output signal back onto the input (inverted of course) you will reduce the gain and assuming that the feedback path has a flat frequency response the output will try to follow input closer than before. Think of the output as being a scaled version of what is on the inverting input. Basic opamp theory tells us that inverting and noninverting inputs always try to track each other. By definition you cannot flatten the frequency response without also reducing distortion. Not necessarily ALL types of distortion. I think gain stability means stability of the gain over a temperature range. 



#4
Nov205, 01:07 AM

P: 70

Effects of negative feedback
http://users.ece.gatech.edu/~mleach/...k/fdbkamps.pdf
I only read the first ten pages but this paper should cover everything you want except maybe bandwidth calculation. But even if it doesn't the technique is the same but you do it in the frequency domain. (If you're not familiar with Laplace transform now is the perfect time to start) I think jonathansd actually is referring to the "effective bandwidth" of a system. As I recall the effective bandwidth is defined as the position of the 3db point on the frequency axes. 



#5
Nov205, 09:52 AM

Mentor
P: 39,628




#6
Nov205, 10:43 AM

P: n/a

Comparing in series increases the input resistence, while comparing in parallel decreases it. Sampling the ouput voltage decreases the output resistence, while sampling the output current increases it. The example of the common emmiter amplifier, suggested by ASN is one of sampling voltage ang comparing in series, so we should expect an increasing of input resistence and a decrease in output resistence. 



#7
Nov205, 04:17 PM

P: 2,451

SGT, I don't think the output Z of a common emitter is determined by the emitter resistor. The emitter resistor does have a play in voltage gain though. 



#8
Nov305, 06:17 PM

Mentor
P: 39,628

I checked my basic transistor text (Electronic Circuits by Holt), and I couldn't find any reference to negative feedback in the discussions about Re. I did check the feedback chapter, and found the statements that you were referring to, about how voltage sum feedback can increase Zin, and current sum feedback can decrease Zin. But the polarity of the voltage feedback in the examples is positive, so of course that can increase Zin. That's the bootstrap effect that I was referring to. Taking your example of an inverting amplifier  the input impedance would be Ri+Rf with no negative feedback (assuming a low input impedance for the stage that the output of the opamp is driving), and Ri with negative feedback. How does the negative feedback increase the input resistance? It sure looks like it lowers it using that example. 



#9
Nov305, 10:27 PM

P: 2,451

Berkeman I was just thumbing through Malvino's Electronic Principles Third edition. Page 507 (chapter 16) describes what I was referring to about the emitter resistor. They have a common emitter NPN transistor amplifier. It is set up with voltage divider bias on the base, a collector resistor and 2 emitter resistors in series. The lower emitter resistor is bypassed with a capacitor. The upper one is the one considered the feedback resistor. Also known as a swamping resistor. They call this configuration noninverting current feedback. BUT, the effect is negative feedback. They also show some other configurations of discreet feedback schemes. They show a resistor going from the collector back to the base and it is described as inverting voltage feedback. It would lower the input impedance. They show 2 other schemes but they both involve a second transistor stage. They are: Inverting current feedback (lower input impedance) and Noninverting voltage feedback which a second inverting stage feeds the signal on its collector back to the first stage emitter. This configuration raises input impedance.
They also describe in another area of the book how negative feedback increases the bandwidth of a circuit. I thought I covered that well enough earlier. 


#10
Nov405, 04:46 AM

P: n/a

If you sample the collector signal and feed it back to the base through a resistor, you have parallel comparison: the input current and the feedback current sum up at the input node, so you must have opposite phases in order to have negative feedback. The opposite is true for the bootstrat configuration you mentioned. Since you have parallel comparison and in phase currents, you have positive feedback. 



#11
Nov1505, 10:50 AM

P: 69

1. Reduction in Gain Closedloop gain = Openloop gain / (1 + Openloop gain*B) where B = feedback factor > 0 Can be derived from Transfer Function analysis as you can se fro the relation, the denominator > 1, so Closedloop gain is effectively reduced by negative feedback! 2. Reduction in Linear Distortion Consider the voltage transfer characteristics of both openloop and closedloop amplifier(with B=0.01). Assume openloop voltage transfer characteristics to be piecewise linear. Openloop gain varies from 1000 to 100 to 0. Plot the openloop voltage transfer characteristics vo/vi on graph paper. Then apply negative feedback B = 0.01. Closedloop gain 1 = 1000 / (1000 + 1000*0.01) = 90.9 Closed loop gain 2 = 100 / (100 + 100*0.01) = 50 Closedloop gain varies from 90.9 to 50 to 0. Plot the closedloop voltage transfer characteristics vo/vi on graph paper. From the plot, you can see that the slope=gain change is gradual for closedloop voltage TF compared to openloop TF. So, linear distortion is effectively reduced by negative feedback! 



#12
Nov1505, 11:16 AM

P: 69

3. Increase in gain stability / Gain Desensitivity Consider closedloop gain = openloop gain / (1 + openloop gain*B) Af = A / (1 + AB) Applying calculus approximation method, d(Af) = dA/(1+AB)^2 Divide by Af, d(Af)/Af = (dA/A) * (1/(1+AB) only accurate for small dA/A As can be seen from above, a significantly large change in openloop gain (dA/A) will cause little change in closedloop gain (d(Af)/Af), due to the division of negative feedback densensitivity factor (1+AB) >> 1. Thereby, the gain is stabilized by applying negative feedback! 4. Increase in BW Consider highfrequency amplifier TF with dominant pole Wh. A = Ao / (1+(s/Wh)) Applying the formula of negative feedback, Af = A/(1+AB) Af = (Ao/(1+AoB)) * (1/(1+(s/(Wh*(1+AoB))))) From the above, we can see that Aof = 1st term = Ao/(1+AoB) Whf = last term = Wh*(1+AoB) (1) From equation (1), we see that the high frequency 3dB point is increased by a factor of (1+AoB) with negative feedback applied! Uising the same transfer function analysis for lowfrequency point, we arrive Wlf = Wl / (1+AoB) effectively reduced by factor (1+AoB) with negative feedback applied! So overall, BW = Whf  Wlf is effectively increased due to negative feedback! 



#13
Nov1505, 11:26 AM

P: 69

5. Increase in input resistance This is not necessary true! It depends on the actual feedback topology you employed in the amplifier circuit. Basically, you need to apply voltage feedback to input signal (Series mixing) so that the input resistance of the feedback amplifier is effectively increased by a factor of (1+AB)! If you apply current feedback to input (Shunt mixing), the input resistance is effectively reduced by factor (1+AB)! Hope you have a hapy time reading! 


Register to reply 
Related Discussions  
passive sign convention (negative watts, and negative current confusion)  Advanced Physics Homework  7  
Negative Feedback  Biology  1  
Negative feedback in the dopaminergic system  Medical Sciences  6  
Negative Feedback Control  Biology  1  
Negative Feedback Mechanisms  Biology  2 