Calculating gain of an overall system. Analogue electronics

Question

Our professor has given us a review for a upcoming exam. But he has never done anything like this question and it has stumped a quite few of us. We are hoping someone can walk us through this so we can understand, as he will not assist.

A modular two amplifier system consists of a first stage amplifier with an open loop gain of 100, an input impedance of 25kΩ and an output impedance of 2kΩ, and a second stage with an input impedance of 2kΩ and an open loop gain of 50 and an output impedance of 500Ω.

Calculate the gain of the overall system.

Using a single op-amp, specify a third modular amplifier, using the original two in the order or otherwise, such that the input impedance of the overall system is 20kΩ, the output impedance is 2kΩ, and the magnitude of the overall voltage gain is 2000.

I thought you could just multiply the gains (gain = - Z2/Z1) of each op amp together, but some classmates disagree with me.

Should I approach it with op amps or signal circuit equivalents? Does a voltage divider have anything to do with it?

I can attach screenshots of circuit diagrams we have tried.

Answer 1

As Dan Mills says, stop thinking of the amplifier system as inverting operational amplifier as the question does not explicitly mention so. Just imagine them to be gain blocks, they could be inverting operational amplifiers but that does not matter. The following is the schematic I came up with the information given.

^{simulate this circuit – Schematic created using CircuitLab}

Note that there is a voltage divider that halves the output voltage of the first stage to the input of the second stage. Hence the total voltage gain of the system would be 100*0.5*50 = 2500.

For the second part, the question insists that the third stage should be built with a single opamp. So let it be an inverting opamp. We can use an inverting opamp with Rin of 20k as the first stage and the amplifier with output impedence of 2k as the final stage as shown below:

^{simulate this circuit}

To calculate the feedback resistance R2 of the first stage inverting amplifier, we need to use the fact that the overall gain of the system is 2000.

Overall gain = (R/20)*50*(25/25.5)*100=2000. (Note that voltage divider between the second and third stage)

Therefore, the feedback resistance R2 of the inverting amplifier will be 8.16k.

Answer 2

Stop thinking opamps, thing opaque gain blocks having the specified voltage gains and input/output impedance, they might be opamps, but nothing says they have to be.

There are two tricks to this question, firstly that the output impedance of the first stage is 2k and the input impedance of the second stage is also 2k, so you have a 2:1 voltage divider between the stages formed by the input and output impedances.

The second trick, is to remember when calculating gain in dB, that dB is a POWER ratio, so impedances matter.

For the first part:

Av(first stage) = 100 Av(second stage) = 50 Av(interstage) = 2k/(2k+2k) = 0.5 Total voltage gain = 2,500

Power gain = 20 * Log (2,500) + 10 log (25k/500r) = 68dB + 17dB = 85dB.

A hint for the second part, reverse the stage ordering to get the 2k output impedance, recalculating the interstage attenuation, see what total Av you get, then calculate an opamp (Probably inverting as you are likely to need a |gain| smaller then 1) to put the gain and input impedance where you need it.