# Combine cascaded amplifier into single model

1. Aug 22, 2015

### TheRedDevil18

1. The problem statement, all variables and given/known data
1) vs = 1 mV. Find v1, v2, v3 and vL

2) Model the three-stage cascaded amplifier by a single voltage amplifier model. What are the values of Ri , Avo, and Ro?

Diagram:

2. Relevant equations

3. The attempt at a solution

1) v1 = 0.909 mV
v2 = 9 mV
v3 = 818 mV
vl = 744 mV

These are correct, same as the book

2) I got 900.42 to be the gain. But why is Ri and Ro the same values of the input and output of the 3 stage amplifier ?, I am thinking because Ri depends on the source voltage which is the same and Ro depends on the voltage given to the load which is also the same in both cases ?

Last edited by a moderator: Aug 23, 2015
2. Aug 22, 2015

### Zondrina

Assuming the values of $v_1, v_2, v_3$, and $v_L$ are correct, I have a few questions.

What do you mean by $A_{vo}$? Do you mean the gain across all cascading stages, or the overall gain from source to load? $A_{vo}$ is usually used to represent open-loop gain, and probably shouldn't be used in this context. I would suggest using something such as $A_v$ for the gain across all cascading stages, and $A_{vs}$ for the gain from source to load.

The gain across the cascading stages is $818.48 \frac{V}{V}$. The overall gain would be $744 \frac{V}{V}$. How did you get the result of $900.42 \frac{V}{V}$?

More importantly, why don't you show us how you calculated $R_i$ and $R_o$?

3. Aug 23, 2015

### TheRedDevil18

I am talking about the open circuit voltage gain. I am referring to question two where they say you have to model the three amplifiers as a single voltage amplifier and find the values of Ri, Ro and Av. I am not sure how to calculate Ri and Ro but from the answers in the book they are 1 mega ohm and 10 ohms

4. Aug 23, 2015

### Zondrina

If they want you to model all three stages as a single voltage amplifier, why would the cascade gain or overall gain be any different?

To find $R_i$, you must find $\frac{v_i}{i_i}$. What about $R_o$?

5. Aug 23, 2015

### LvW

I think the term Avo applies to the gain without any load and without consideration of Rs because Rs is part of the source only. Hence, it is the voltage ratio V3/V1.

What are the values of Ri , Avo, and Ro?
Simply look into the input node and the output node of the amplifier without connected source and load. Simple solution by visual inspection.

6. Aug 23, 2015

### TheRedDevil18

I think we are actually considering Rs because then v1 would be exactly 1mV which is the source voltage ?

Can you explain how to find Ri and Ro by disconnecting the source and load ?

7. Aug 23, 2015

### LvW

I think your task is to model the amplifier - in my view that means: Without any source resistance and without any load. Otherwise, it is not the amplifier itself what you are modelling.
Can you explain how to find Ri and Ro by disconnecting the source and load ?
Visual inspection: Look into the amplifier and identify the resistor you see. And the same from the other side. Disconnect the load and look into the output. Not very tricky!

Last edited: Aug 23, 2015
8. Aug 23, 2015

### TheRedDevil18

Do you mean find the no load and full load voltage ?

9. Aug 23, 2015

### Zondrina

When you look into the amplifier input, what is the input resistance, i.e what resistance is initially seen by the signal at the input?

Now pretend the load resistor isn't there. How much output resistance does the signal experience before it gets to the load resistor?

10. Aug 23, 2015

### TheRedDevil18

Okay, so the first resistance the signal sees is the 1MOhm resistor, so the input impedance of the single voltage amplifier circuit is 1Mohm ?

In order to get to the output it passes the remaining resistors which are all in parallel so the equivalent resistance is 1/(1*10^6) + 1/(1*10^3) + 1/(100*10^3) + 1/(1*10^3) + 1/(10*10^3) + 1/10 = 1/10

So the equivalent resistance is 10 ohms which is equal to the output resistance of the single voltage amplifier ?

11. Aug 23, 2015

### LvW

Looking into the output node of the amplifier model you must know that the source resistance of the last voltage source is ZERO !

12. Aug 24, 2015

### donpacino

that is correct for the input resistance.
Ro is output resistance, which means the resistance seen from the output of the amplifier. This has already been stated, but I will say it again.

V=IR

so input resistance is R=Vin/Iin

how would you calculate output resistance?

13. Aug 24, 2015

### TheRedDevil18

v3-vl = 74mV = Voltage across Ro
Io = Vl/Rl = 0.744/100 = 0.00744 A

Ro = Vo/Io = 0.074/0.00744 = 9.95 ohms

Is there any other way to calculate it ?

14. Aug 24, 2015

### donpacino

so you forgot something important!!!!
you must zero out all independent sources (which includes the voltage source at the input). If that voltage source is zero, what is V3?

15. Aug 24, 2015

### TheRedDevil18

V3 depends on Vs so V3 is zero. If their is no voltage source then there is no voltage in the circuit

16. Aug 24, 2015

### donpacino

correct, which means in your solving for Ro Vs is zero...
soo Ro is............

17. Aug 24, 2015

### TheRedDevil18

Zero because R = V/I ?, but this is wrong

18. Aug 24, 2015

### donpacino

So I made an assumption that you knew how to find equivalent resistances. When I say V/I, i mean zero out all independent sources, remove the load, then put a voltage source on the output (which is vL). Then find what the current is, and that will tell you what the output resistance is. does that make sense, tell me if it doesn't

But for a problem like this, there is a shortcut, that was hinted at before. since v3 is zero, there is just a single resistor, so you can tell what the output resistance is by inspection.

Last edited: Aug 24, 2015
19. Aug 24, 2015

### LvW

The RedDevil - here is what i wrote in my post#7, Didn`t you read it?

"Visual inspection: Look into the amplifier and identify the resistor you see. And the same from the other side. Disconnect the load and look into the output. Not very tricky!"

You must not calculate anything - which resistor do you identify looking into the output? (The voltage source has a zero source resistance!)

20. Aug 24, 2015

### TheRedDevil18

I know it's the 10 ohm resistor but what about the output resistances in stage 1 and 2 ?, doesn't that play a role in finding the equivalent output resistance for the single voltage amplifier ?