Let's solve the circuit **step by step** using nodal analysis and op-amp properties. --- ## **Step 1: Label Node Voltages** - Let the non-inverting input of the first op-amp be at ground ( V). - Let the voltage at the output of the first op-amp be \( v_{o1} \). - The voltage at the inverting input of the second op-amp is at the node between \( R_3 \) and \( R_4 \) (call it \( v_x \)). - The output of the second op-amp is \( v_o \). --- ## **Step 2: First Op-Amp (Ideal Op-Amp, Negative Feedback)** The non-inverting input is grounded, so by virtual short, the inverting input is also at ** V**. **Apply KCL at the inverting input:** \[ \frac{v_s - }{R_1} = \frac{ - v_{o1}}{R_2} \] \[ \frac{v_s}{R_1} = -\frac{v_{o1}}{R_2} \] \[ v_{o1} = -\frac{R_2}{R_1} v_s \] --- ## **Step 3: Second Op-Amp (Negative Feedback)** The non-inverting input is at ground ( V), so by virtual short, the inverting input is also at ** V**. - The current through \( R_3 \): \(\frac{v_{o1} - }{R_3}\) - The current through \( R_4 \): \(\frac{ - v_o}{R_4}\) - The current through \( R_5 \): \(\frac{ - v_o}{R_5}\) **Apply KCL at the inverting input of the second op-amp:** \[ \frac{v_{o1} - }{R_3} = \frac{ - v_o}{R_4} + \frac{ - v_o}{R_5} \] \[ \frac{v_{o1}}{R_3} = -v_o \left( \frac{1}{R_4} + \frac{1}{R_5} \right) \] \[ v_o = -\frac{v_{o1}}{R_3 \left( \frac{1}{R_4} + \frac{1}{R_5} \right)} \] \[ v_o = -\frac{v_{o1}}{R_3} \cdot \left( \frac{R_4 R_5}{R_4 + R_5} \right) \] Now substitute \(v_{o1}\): \[ v_{o} = -\frac{1}{R_3} \left( \frac{R_4 R_5}{R_4 + R_5} \right) \left( -\frac{R_2}{R_1} v_s \right) \] \[ v_o = \frac{R_2}{R_1 R_3} \left( \frac{R_4 R_5}{R_4 + R_5} \right) v_s \] --- ## **Step 4: Output Current \( i_o \) through \( R_5 \)** \[ i_o = \frac{ - v_o}{R_5} = -\frac{v_o}{R_5} \] Plug in the expression for \( v_o \): \[ i_o = -\frac{1}{R_5} \cdot \frac{R_2}{R_1 R_3} \left( \frac{R_4 R_5}{R_4 + R_5} \right) v_s \] \[ i_o = -\frac{R_2 R_4}{R_1 R_3 (R_4 + R_5)} v_s \] --- ## **Step 5: Boxed Answers** \[ \boxed{ v_o = \frac{R_2}{R_1 R_3} \left( \frac{R_4 R_5}{R_4 + R_5} \right) v_s } \] \[ \boxed{ i_o = -\frac{R_2 R_4}{R_1 R_3 (R_4 + R_5)} v_s } \] --- ### **Summary Table** | Quantity | Expression (in terms of \( v_s, R_1, R_2, R_3, R_4, R_5 \)) | |:--------:|:----------------------------------------------------------:| | \( v_o \) | \( \dfrac{R_2}{R_1 R_3} \left( \dfrac{R_4 R_5}{R_4 + R_5} \right) v_s \) | | \( i_o \) | \( -\dfrac{R_2 R_4}{R_1 R_3 (R_4 + R_5)} v_s \) |