Ideal Op-Amp Circuit Problem

Find the value of R if the AC Gain (Vout / Vin) = 8.

📘 Question

Given an ideal operational amplifier, determine the resistor value R that produces a voltage gain of 8.

Step 1 — Ideal Op-Amp Rule

For an ideal op-amp operating with negative feedback:

No current flows into the input terminals
The voltages at the inputs are equal

👉 V⁺ = V⁻

This condition is often called the virtual short. Even though the terminals are not physically connected, their voltages are identical.

Step 2 — Find V⁺ (Non-Inverting Input)

The positive input is connected to a voltage divider formed by:

90kΩ from Vout
10kΩ to ground

V⁺ = Vout × (10k / (90k + 10k))
V⁺ = 0.1 Vout

Since V⁺ = V⁻:

V⁻ = 0.1 Vout

Step 3 — Analyze the Inverting Node

The negative input is another voltage divider:

10kΩ from Vin
R to ground

V⁻ = Vin × (R / (R + 10k))

Step 4 — Substitute the Gain

Given:

Gain = Vout / Vin = 8
Vout = 8Vin

Substitute into V⁻:

V⁻ = 0.1(8Vin)
V⁻ = 0.8Vin

Now set both expressions for V⁻ equal:

Vin × (R / (R + 10k)) = 0.8Vin

Cancel Vin:

R / (R + 10k) = 0.8

Step 5 — Solve for R

R = 0.8(R + 10k)

R = 0.8R + 8k

0.2R = 8k

R = 40kΩ

✅ Final Answer: R = 40 kΩ

⚡ Engineering Insight

Many students notice the 10kΩ resistor at the output and wonder if it affects the gain. It does NOT. An ideal op-amp has zero output resistance, meaning it can drive loads without changing the output voltage.