Questions: Figure 9: Instrumentation amplifier. Consider the circuit in Figure 9, which represents an instrumentation amplifier (A1, A2, A3) with additional functionality (A4, A5). All operational amplifiers are assumed to be ideal. (a) [5 points] Derive the transfer function H1(s) = (V2-V1)/(Vint-Vin). Hint: Set up independent equations that involve Ig, the current through Rg and eliminate Ig. Use your knowledge about ideal operational amplifiers. (b) [5 points] Derive an expression for (V01)/(Vout) of the sub-circuit with A 5 and for (V02)/(Vout) of the sub-circuit with A4. Calculate the transfer function H2(s) = (V02-V01)/(Vout). (c) [4 points] Set up the circuit equations for operational amplifier A3: Find a relationship between VN and VP, then set up circuit equations for V1, Vol, V2, and Vo 2 to derive the transfer function H3(s) = (V02-V0)/(V2-V1).

Figure 9: Instrumentation amplifier. Consider the circuit in Figure 9, which represents an instrumentation amplifier (A1, A2, A3) with additional functionality (A4, A5). All operational amplifiers are assumed to be ideal. (a) [5 points] Derive the transfer function H1(s) = (V2-V1)/(Vint-Vin). Hint: Set up independent equations that involve Ig, the current through Rg and eliminate Ig. Use your knowledge about ideal operational amplifiers. (b) [5 points] Derive an expression for (V01)/(Vout) of the sub-circuit with A 5 and for (V02)/(Vout) of the sub-circuit with A4. Calculate the transfer function H2(s) = (V02-V01)/(Vout). (c) [4 points] Set up the circuit equations for operational amplifier A3: Find a relationship between VN and VP, then set up circuit equations for V1, Vol, V2, and Vo 2 to derive the transfer function H3(s) = (V02-V0)/(V2-V1).
Transcript text: Figure 9: Instrumentation amplifier. Consider the circuit in Figure 9, which represents an instrumentation amplifier (A1, A2, A3) with additional functionality (A4, A5). All operational amplifiers are assumed to be ideal. (a) [5 points] Derive the transfer function $H_{1}(s)=\frac{V_{2}-V_{1}}{V_{\text {int }}-V_{\text {in }}}$. Hint: Set up independent equations that involve $I_{\mathrm{g}}$, the current through $R_{\mathrm{g}}$ and eliminate $I_{\mathrm{g}}$. Use your knowledge about ideal operational amplifiers. (b) [5 points] Derive an expression for $\frac{V_{01}}{V_{\text {out }}}$ of the sub-circuit with A 5 and for $\frac{V_{02}}{V_{\text {out }}}$ of the sub-circuit with A4. Calculate the transfer function $H_{2}(s)=\frac{V_{02}-V_{01}}{V_{\text {out }}}$. (c) [4 points] Set up the circuit equations for operational amplifier A3: Find a relationship between $V_{\mathrm{N}}$ and $V_{\mathrm{P}}$, then set up circuit equations for $V_{1}, V_{\mathrm{ol}}, V_{2}$, and $V_{\mathrm{o} 2}$ to derive the transfer function $H_{3}(s)=\frac{V_{02}-V_{0}}{V_{2}-V_{1}}$.
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Solution

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Step 1: Derive the transfer function \( H_1(s) = \frac{V_2 - V_1}{V_{in+} - V_{in-}} \)

  1. Identify the voltages at the non-inverting inputs of A1 and A2:

    • \( V_{in+} \) is at the non-inverting input of A1.
    • \( V_{in-} \) is at the non-inverting input of A2.

  2. Assume ideal op-amps:

    • The voltage at the inverting input of A1 is \( V_2 \).
    • The voltage at the inverting input of A2 is \( V_1 \).

  3. Write the current equations:

    • The current through \( R_g \) is \( I_g = \frac{V_2 - V_1}{R_g} \).

  4. Apply Kirchhoff's Current Law (KCL) at the inverting inputs of A1 and A2:

    • For A1: \( \frac{V_{in+} - V_2}{R_f} = I_g \)
    • For A2: \( \frac{V_{in-} - V_1}{R_f} = I_g \)

  5. Combine the equations:

    • \( \frac{V_{in+} - V_2}{R_f} = \frac{V_2 - V_1}{R_g} \)
    • \( \frac{V_{in-} - V_1}{R_f} = \frac{V_2 - V_1}{R_g} \)

  6. Solve for \( V_2 - V_1 \):

    • \( V_2 - V_1 = \left( \frac{R_g}{R_f} \right) (V_{in+} - V_{in-}) \)

  7. Derive the transfer function \( H_1(s) \):

    • \( H_1(s) = \frac{V_2 - V_1}{V_{in+} - V_{in-}} = \frac{R_g}{R_f} \)
Step 2: Derive an expression for \( \frac{V_{O1}}{V_{out}} \) and \( \frac{V_{O2}}{V_{out}} \)

  1. Identify the sub-circuits:

    • \( V_{O1} \) is the output of A5.
    • \( V_{O2} \) is the output of A4.

  2. Assume ideal op-amps:

    • The voltage at the inverting input of A4 is \( V_{O2} \).
    • The voltage at the inverting input of A5 is \( V_{O1} \).

  3. Write the voltage divider equations:

    • For A4: \( V_{out} = V_{O2} \left( 1 + \frac{R4}{R3} \right) \)
    • For A5: \( V_{out} = V_{O1} \left( 1 + \frac{R5}{R2} \right) \)

  4. Solve for \( \frac{V_{O1}}{V_{out}} \) and \( \frac{V_{O2}}{V_{out}} \):

    • \( \frac{V_{O1}}{V_{out}} = \frac{1}{1 + \frac{R5}{R2}} \)
    • \( \frac{V_{O2}}{V_{out}} = \frac{1}{1 + \frac{R4}{R3}} \)

  5. Derive the transfer function \( H_2(s) \):

    • \( H_2(s) = \frac{V_{O2} - V_{O1}}{V_{out}} \)
Step 3: Set up the circuit equations for operational amplifier A3

  1. Identify the voltages at the inputs of A3:

    • \( V_P \) is at the non-inverting input of A3.
    • \( V_N \) is at the inverting input of A3.

  2. Assume ideal op-amps:

    • The voltage at the output of A3 is \( V_{out} \).

  3. Write the voltage equations:

    • \( V_P = V_2 \)
    • \( V_N = V_1 \)

  4. Combine the equations:

    • \( V_{out} = A3(V_P - V
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