# Combine cascaded amplifier into single model

## Homework Statement

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:

## 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 ?

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STEMucator
Homework Helper
What are the values of Ri , Avo, and Ro?
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##?

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##?
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

STEMucator
Homework Helper
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
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##?

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.

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.
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 ?

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!

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

STEMucator
Homework Helper
Do you mean find the no load and full load voltage ?
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?

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?
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 ?

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

STEMucator
donpacino
Gold Member
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 ?
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?

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?
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 ?

donpacino
Gold Member
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 ?
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?

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

donpacino
Gold Member
V3 depends on Vs so V3 is zero. If their is no voltage source then there is no voltage in the circuit
correct, which means in your solving for Ro Vs is zero...
soo Ro is............

correct, which means in your solving for Ro Vs is zero...
soo Ro is............
Zero because R = V/I ?, but this is wrong

donpacino
Gold Member
Zero because R = V/I ?, but this is wrong
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.

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LvW
The RedDevil - here is what i wrote in my post#7, Didnt 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!)

The RedDevil - here is what i wrote in my post#7, Didnt 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!)
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 ?

donpacino
Gold Member
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 ?
no. thats the beauty of electronics. You can isolate all those other stages to get the desired output resistance!

no. thats the beauty of electronics. You can isolate all those other stages to get the desired output resistance!
So if I had a voltage amplifier connected to a current amplifier which is connected to the load and if I wanted to model it as a single transconductance amplifier then the input and output resistance would be the same as the resistor that you see when looking into the input and output terminals of the voltage/current amplifier ?

donpacino
Gold Member
So if I had a voltage amplifier connected to a current amplifier which is connected to the load and if I wanted to model it as a single transconductance amplifier then the input and output resistance would be the same as the resistor that you see when looking into the input and output terminals of the voltage/current amplifier ?
Yes!!!! please keep in mind that sometimes it is not just one resistor, and you will have to use the voltage/current method (there are other methods too).

The Electrician
Gold Member
Yes!!!! please keep in mind that sometimes it is not just one resistor, and you will have to use the voltage/current method (there are other methods too).
Not necessarily. He postulates "a voltage amplifier connected to a current amplifier which is connected to the load". Just saying "connected" is ambiguous. If there's any internal feedback, for example, things get more complicated.

donpacino
Gold Member
Not necessarily. He postulates "a voltage amplifier connected to a current amplifier which is connected to the load". Just saying "connected" is ambiguous. If there's any internal feedback, for example, things get more complicated.
very true, which is why I specified that you would have to use the voltage/current method and that the systems would have to be isolated for it to work perfectly and simply

edit: i did not state that they would have to be isolated. I must have been thinking it and just never typed it :(