Open-Loop vs Closed-Loop Op-Amp Gain

 

Mathematical Difference Between Open-Loop and Closed-Loop Op-Amp Gain

The mathematical difference between open-loop gain and closed-loop gain in an operational amplifier (op-amp) is mainly based on the use of feedback.

---

1. Open-Loop Gain

An op-amp without feedback is called an open-loop amplifier. Its gain is extremely large.

Formula

A_OL = V_out / (V⁺ - V^-)

Where:

• A_OL = Open-loop gain
• V⁺ = Non-inverting input voltage
• V^- = Inverting input voltage

Typical open-loop gain:

A_OL = 10⁵ to 10⁶

Example of Open-Loop Gain

Suppose:

• A_OL = 100000
• Input difference V_d = 1mV = 0.001V

V_out = A_OL × V_d

V_out = 100000 × 0.001 = 100V

Since the op-amp supply voltage is limited, the output saturates near the supply voltage (for example +15V).

Applications:

• Comparator
• Zero-crossing detector

---

2. Closed-Loop Gain

When negative feedback is added, the op-amp operates in closed-loop mode.

General Closed-Loop Formula

A_CL = A_OL / (1 + A_OLβ)

Where:

• A_CL = Closed-loop gain
• A_OL = Open-loop gain
• β = Feedback factor

Since A_OL is very large:

A_CL ≈ 1 / β

Therefore, gain depends mainly on external resistors.

Example: Inverting Amplifier

Given:

• R₁ = 1kΩ
• R_f = 10kΩ

A_CL = -R_f / R₁

A_CL = -10k / 1k = -10

If input voltage:

V_in = 0.2V

Output voltage:

V_out = -10 × 0.2 = -2V

This amplification is stable and predictable because of feedback.

---

Mathematical Difference

Feature	Open-Loop	Closed-Loop
Feedback	No	Yes
Gain Formula	AOL = Vout / Vd	ACL = AOL / (1 + AOLβ)
Gain Value	Very High	Controlled
Stability	Poor	Good
Bandwidth	Small	Larger
Applications	Comparator	Amplifier

---

Important Insight

V⁺ ≈ V^-

This is called the virtual short concept. Due to the huge open-loop gain, only a tiny input difference is needed.

---

Summary

Open-Loop

V_out = A_OL(V⁺ - V^-)

• No feedback
• Very high gain
• Output saturates easily

Closed-Loop

A_CL = A_OL / (1 + A_OLβ)

• Uses feedback
• Stable gain
• Linear amplification