Circuit Analysis – Solving a Two Op-Amp Circuit

circuit analysisoperational-amplifier

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We are supposed to find \$R\$ such that \$I_S=0\$ and then find the maximum \$V_{SRC}\$ for operation in the linear region. I tried to use KCL at the nodes but it didn't work out eventually since I got a value of R that is negative. And for the second part of the question how does one determine the maximum possible value in the linear range: I am not sure how to go about this. I would appreciate any help.

Best Answer

Step by step:

The current, from left to right, through \$R\$ is

$$I_R = \frac{V_{SRC} - V_{O2}}{R} $$

The current, from left to right, through the left-most 10k resistor is

$$I_{10k} = \frac{V_{SRC}}{10k\Omega} $$

KCL at the input node yields

$$I_S = I_R + I_{10k}$$

Using the well-known inverting op-amp gain formula, the two op-amp cascade has a gain of

$$\frac{V_{O2}}{V_{SRC}} = (-\frac{40k}{10k}) \cdot (-\frac{20k}{10k}) = 8 $$

Now, set \$I_S = 0\$ and solve.

A rewarding exercise is to solve for the input resistance seen by the input voltage source:

$$R_{IN} = \frac{V_{SRC}}{I_S} = \frac{V_{SRC}}{I_R + I_{10k}} = \frac{1}{\frac{1}{10k\Omega} - \frac{7}{R}}$$

Note that the input resistance is positive for \$R > 70k\Omega\$, is negative for \$R < 70k\Omega\$ (the circuit supplies power to the voltage source), and is 'infinite' (open circuit) for \$R = 70k\Omega\$

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