To determine the transfer function of this operational amplifier circuit, let’s analyze it step-by-step. We’ll make the following assumptions based on the ideal op-amp characteristics:

1. The input impedance of the op-amp is infinite, meaning that no current flows into the inverting or non-inverting inputs.
2. The output impedance of the op-amp is zero.
3. The op-amp maintains a virtual short, so $V_+ = V_-$.

Let's denote:

• $V_{\text{in}}$: Input voltage at the non-inverting terminal.
• $V_{\text{out}}$: Output voltage of the circuit.
• $R$: Resistance value of each resistor.
• $\Delta R$: Small change in resistance of one of the resistors in the feedback path.

Step 1: Set up the circuit voltages

1. Non-Inverting Input ($V_+$):

• There are two resistors, both of value $R$.
• One resistor connects $V_{\text{in}}$ to $V_+$, and the other resistor connects the non-inverting input $V_+$ to a constant 5V.
• Using the voltage divider rule, we can write the voltage at the non-inverting terminal ($V_+$) as: $V_+ = \frac{V_{\text{in}} + 5}{2}$

2. Inverting Input ($V_-$):

• There are two resistors: one of value $R$ connected to ground and another of value $R + \Delta R$ connected between the inverting terminal and the output $V_{\text{out}}$.
• Since $V_+ = V_-$ due to the virtual short, $V_-$ is also equal to $\frac{V_{\text{in}} + 5}{2}$.

Using Kirchhoff's Current Law (KCL) at the inverting terminal:

$\frac{V_-}{R} = \frac{V_{\text{out}} - V_-}{R + \Delta R}$

Step 2: Solve the Equation for the Transfer Function

Substitute $V_- = \frac{V_{\text{in}} + 5}{2}$ into the equation:

$\frac{\frac{V_{\text{in}} + 5}{2}}{R} = \frac{V_{\text{out}} - \frac{V_{\text{in}} + 5}{2}}{R + \Delta R}$

Multiply through by $R(R + \Delta R)$ to eliminate the denominators:

$\frac{(V_{\text{in}} + 5)(R + \Delta R)}{2} = R \left(V_{\text{out}} - \frac{V_{\text{in}} + 5}{2}\right)$

Expanding and rearranging terms will yield the expression for $V_{\text{out}}$ in terms of $V_{\text{in}}$, which represents the transfer function.

Would you like to see the detailed solution to solve for $V_{\text{out}}$ explicitly?

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Related Questions:

1. What is a virtual short in operational amplifiers?
2. How do we apply Kirchhoff's Current Law in operational amplifier circuits?
3. What assumptions make an op-amp "ideal"?
4. How does changing resistor values affect the transfer function?
5. What are the effects of $\Delta R$ on the output voltage?

Tip: In analyzing op-amp circuits, always remember to apply the virtual short principle if dealing with an ideal op-amp. This can simplify finding voltages at the inverting and non-inverting terminals.