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 $H_{3}(s)=\frac{V_{02}-V_{0}}{V_{2}-V_{1}}$.
failed

Solution

failed
failed
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
Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
All of the following describe perfectly competitive firms, except each produces a negligible share of industry output they produced standardized products they make no attempt to differentiate their products they face large barriers to entry
failedAnswered
What is the main difference between balloon and platform framing?
failedAnswered
Which can be a cause of pallor? Cold temperatures Liver malfunctions Bacterial infection Increased blood supply
failedAnswered
reaction highlighted atom is being... ------------ oxidized reduced neither oxidized nor reduced CH₄(g)+H₂O(g) → CO(g)+3H₂(g) 0 0 H₂S(aq)+2NaOH(aq) → Na₂S(aq)+2H₂O(l) FeO(s)+CO(g) → Fe(s)+CO₂(g) 0 N₂(g)+3H₂(g) → 2NH₃(g)
failedAnswered
A rede hidrográfica da Ásia é de fundamental importância para determinadas regiões. em qual das regiões asiáticas encontramos os seguintes rios: A) Tigre e Eufrates. B) Ganges. C) Yang-Tsé.