Calculating Voltage Gain for Electrical Circuits

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Discussion Overview

The discussion focuses on calculating the voltage gain for specific electrical circuits, particularly using Kirchhoff's laws and analyzing the role of various resistances in the circuits. Participants are attempting to derive expressions for voltage gain based on their circuit diagrams and previous knowledge.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants express difficulty in deriving the voltage gain despite knowing the answers, indicating potential misunderstandings in their calculations.
  • One participant questions the significance of the amplifier input resistance (r_i) in the context of the voltage divider rule and its effect on the input voltage of the amplifier.
  • Another participant suggests that r_i does matter and confirms it is part of a voltage divider, indicating that it influences the calculations for the amplifier's input voltage.
  • There is a discussion about the relationship between input voltage (V_in) and output voltage (V_out), with references to controlled current sources and output impedances.
  • One participant derives a formula for voltage gain (A_v) but realizes it differs from the textbook solution, prompting a discussion about the conditions under which the textbook solution applies.
  • Another participant emphasizes the importance of keeping the input current as a variable, noting that it cancels out when calculating voltage gain.
  • Multiple participants share their calculations and reasoning, with some expressing uncertainty about specific steps or results, particularly regarding the derivation of A_v for different parts of the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach to calculating voltage gain, with multiple competing views and methods presented. There is ongoing debate about the significance of certain parameters and the correctness of derived formulas.

Contextual Notes

Participants mention various assumptions and conditions related to the circuit configurations, such as the arrangement of resistances and the application of Kirchhoff's laws. Some calculations appear to depend on specific interpretations of the circuit diagrams, which may not be clear to all participants.

krotan
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Homework Statement



I've been reviewing my electrical theory and I'm having trouble trying to calculate an expression for the voltage gain for these two specific circuits. (Images c and d of the question.png attachment). I already know the answers, however I am unable to derive the same solution. Answers are given with my solution attempt.

Homework Equations


Kirchoff's loop law and current junction law.

Voltage gain = Output voltage/Input Voltage

R_s is source resistance, R_L is load resistance.
r_o is amplifier output resistance and r_i is amplifier input resistance.

The Attempt at a Solution


I've attached my solution for b if it's unclear what I am asking. I'm pretty sure I've misunderstood something while working it out but haven't been able to figure out the correct steps. Any help is appreciated.

Edit: Attachments are now there.
 

Attachments

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  • question.png
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krotan said:

Homework Statement



I've been reviewing my electrical theory and I'm having trouble trying to calculate an expression for the voltage gain for these two specific circuits. (Images c and d of the question.png attachment). I already know the answers, however I am unable to derive the same solution.



Homework Equations


Kirchoff's loop law and current junction law.

Voltage gain = Output voltage/Input Voltage

R_s is source resistance, R_L is load resistance.
r_o is amplifier output resistance and r_i is amplifier input resistance.


The Attempt at a Solution


I've attached my solution for b if it's unclear what I am asking. I'm pretty sure I've misunderstood something while working it out but haven't been able to figure out the correct steps. Any help is appreciated.

I'm not seeing any attachments. Try again?
 


Part (c): well, the input voltage is Vin and the output voltage is what if the controlled current source is μVin feeding an output impedance of ro?

What function, if any, does ri have?
 


r_i is a constant but I'll see where I get with your suggestions on C for now. It's just late here right now but I'm going back to it in the morning. Thank you very much.
 
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krotan said:
r_i is a constant but I'll see where I get with your suggestions on C for now. It's just late here right now but I'm going back to it in the morning. Thank you very much.

Yeah, ask yourself if ri has any significance at all ...
 


I thought r_i would have mattered as it is used in the voltage divider rule to figure out the input voltage of the amplifier and from there mu*v_in would have been the output voltage of the amplifier.

If I considered the first loop in C then we have:

Vs = VSource Resistance + VAmp input resistance.
Vs = I(Rs + ri)
I = Vs/(Rs + ri)

then multiply by r_i to get
VAmp input resistance = riVs/(Rs + ri)

The diagrams also aren't too clear on the symbols but I believe by vin and Iin is referring to the voltage/current across/through the amp. input resistor. I'll post the full picture for B that is given as a text example.
 

Attachments

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Now that you've included the entire circuit - yes, it matters. It's a voltage divider as you said.

You're on the right track.
 


For C:

Then if I know the vin voltage, the amp. output voltage would just be μvin. But then by Kirchoff's loop law, since the load is in parallel with the amplifier output voltage, this would imply the load voltage, amplifier output resistor voltage and the amplifier voltage are equal.

vL = μvin = μriVs/(Rs + ri)

I just realized I forgot to put in Av for part C so:

And so gain is Av = vL/vs = ri/(Rs + ri)

However my text has the solution as Av = μrirsRL/[(ri + Rs)(ro + RL)] so I'm off by a factor of RL/(ro + RL).

The textbook seems correct only if the load ,amp. output resistance and amp. were in series since I'd be able to use voltage divider again to multiply in the extra factor I'm missing mentioned above or unless I've misunderstood something for the second part of the solution.
 
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The input voltage is given by the input current times the input resistance. Just like Ohms Law.

The output voltage is given by the output current (hfe* iin) times the parallel combined resistance of the load and the output z of the transistor.



If you keep the input current as your only variable, it will cancel when you calculate the voltage gain, which is just the ratio of the two calculations above.

Voltage gain = output voltage / input voltage
 
  • #10


Gotcha, so referring to C:

Av = vL/vs

where:

By Ohm's Law:
vs = IRtotal = I(Rs + ri)

So far then I have:

Av = vL/I(Rs + ri)

Then vL = μvin = μIinReq

Where Req = RLro/(ro + RL)

Then vL = μIin*RLro/(ro + RL)

And now Av = μIin*RLro/vin[(ro + RL)(Rs + ri)]

But vin/Iin = ri by Ohm's Law thus:

Av = μri*RLro/[(ro + RL)(Rs + ri)]

Thanks so much for your help. Let me know if I had followed any of the above steps incorrectly. I'll try to work through D with what I now know.
 
  • #11


Bah ignore this...I thought I had D right for a moment...


For D:

For Av = vL/vs

vs = iin(ri + Rs)

μiin = μvin/Req

where Req = RL + Ro

μiin = μvin/(RL + Ro)

Since the right loop of D is a series circuit, multiplying the output current on both sides by RL will yield vL on the left.

vL = RLμiin = μvinRL/(RL + Ro)

Av =
 

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