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).

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.

Assume ideal op-amps:

• The voltage at the inverting input of A1 is $V_2$.
• The voltage at the inverting input of A2 is $V_1$.

Write the current equations:

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

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: $\frac{V_{in-} - V_1}{R_f} = I_g$

Combine the equations:

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

Solve for $V_2 - V_1$:

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

Derive the transfer function $H_1(s)$:

• $H_1(s) = \frac{V_2 - V_1}{V_{in+} - V_{in-}} = \frac{R_g}{R_f}$

Identify the sub-circuits:

• $V_{O1}$ is the output of A5.
• $V_{O2}$ is the output of A4.

Assume ideal op-amps:

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

Write the voltage divider equations:

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

Solve for $\frac{V_{O1}}{V_{out}}$ and $\frac{V_{O2}}{V_{out}}$:

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

Derive the transfer function $H_2(s)$:

• $H_2(s) = \frac{V_{O2} - V_{O1}}{V_{out}}$