The direct answer for most bench applications is this: for an ideal non-inverting operational amplifier, the closed loop gain formula is ACL = 1 + (Rf / Rin). For an inverting configuration, it is ACL = -(Rf / Rin). However, when dealing with real-world silicon where open-loop gain is finite, the universal feedback equation ACL = AOL / (1 + AOLβ) dictates the actual output behavior. Below, we break down the exact derivations, rearrange the algebra for component selection, and solve bench-realistic problems with strict unit tracking.
The Core Closed Loop Gain Formula and Symbol Definitions
Before applying the simplified resistor ratios, you must understand the foundational control theory equation derived by Harold Black. The general closed loop gain formula applies to any negative feedback system, including discrete transistor amplifiers and integrated op-amps.
General Feedback Equation:
ACL = AOL / (1 + AOLβ)
| Symbol | Parameter | Definition | Typical Real-World Value |
|---|---|---|---|
| ACL | Closed-Loop Gain | The actual voltage gain of the circuit with feedback applied (V/V). | 1 to 1,000 V/V |
| AOL | Open-Loop Gain | The intrinsic gain of the amplifier without feedback (V/V). | 100,000 to 10,000,000 V/V |
| β | Feedback Factor | The fraction of the output voltage fed back to the inverting input (dimensionless). | 0.001 to 1.0 |
| Rf | Feedback Resistor | The resistor connecting the output pin to the inverting input pin. | 1 kΩ to 100 kΩ |
| Rin | Input/Ground Resistor | The resistor connecting the inverting input to ground (non-inverting) or the signal source (inverting). | 1 kΩ to 100 kΩ |
When AOL approaches infinity (the ideal op-amp assumption), the term (1 + AOLβ) simplifies such that ACL ≈ 1/β. For a standard non-inverting voltage divider feedback network, β = Rin / (Rin + Rf). Substituting this into 1/β yields the familiar ideal formula:
Ideal Non-Inverting: ACL = 1 + (Rf / Rin)
Ideal Inverting: ACL = -(Rf / Rin)
For deeper reading on how internal op-amp compensation affects these limits, refer to the Analog Devices MT-033 Tutorial on Op-Amp Feedback.
Rearranged Forms and Solving for Hidden Variables
On the bench, you rarely solve for ACL directly; you usually have a target gain and need to find the missing component value or verify the feedback factor. Here are the algebraically rearranged forms of the core equations.
- Solving for Feedback Factor (β): β = (AOL - ACL) / (AOL × ACL)
- Solving for Open-Loop Gain (AOL): AOL = ACL / (1 - ACLβ)
- Solving for Feedback Resistor (Rf) in Non-Inverting: Rf = Rin × (ACL - 1)
- Solving for Input Resistor (Rin) in Non-Inverting: Rin = Rf / (ACL - 1)
- Solving for Feedback Resistor (Rf) in Inverting: Rf = |ACL| × Rin
Worked Examples with Unit Tracking
Abstract math hides real-world errors. The following two problems track units explicitly to demonstrate how resistance cancels out and where finite open-loop gain introduces measurable error.
Problem 1: Ideal Non-Inverting Amplifier Design
Scenario: You need to amplify a 0.5V peak-to-peak sensor signal to exactly 5.0V peak-to-peak using an ideal op-amp in a non-inverting configuration. You have a 10 kΩ precision resistor available for Rin. Find the required Rf.
- Calculate Target ACL:
ACL = Vout / Vin
ACL = 5.0 V / 0.5 V = 10 V/V - Select the Rearranged Formula:
Rf = Rin × (ACL - 1) - Substitute and Track Units:
Rf = 10 kΩ × (10 V/V - 1 V/V)
Note: The (V/V) term is dimensionless. We are multiplying resistance by a scalar.
Rf = 10 kΩ × 9
Rf = 90 kΩ
Verification: ACL = 1 + (90 kΩ / 10 kΩ) = 1 + 9 = 10 V/V. The units of kΩ cancel perfectly.
Problem 2: Real-World Finite AOL Error Calculation
Scenario: You build the circuit from Problem 1 using a vintage LM741 op-amp. The LM741 datasheet specifies a typical open-loop gain (AOL) of 200,000 V/V. What is the actual closed loop gain, and what is the percentage error compared to the ideal 10 V/V?
- Determine β from the Resistor Network:
β = Rin / (Rin + Rf)
β = 10 kΩ / (10 kΩ + 90 kΩ) = 10 / 100 = 0.1 (dimensionless) - Apply the General Feedback Equation:
ACL(actual) = AOL / (1 + AOLβ)
ACL(actual) = 200,000 V/V / (1 + (200,000 V/V × 0.1))
ACL(actual) = 200,000 / (1 + 20,000)
ACL(actual) = 200,000 / 20,001 ≈ 9.9995 V/V - Calculate Percentage Error:
Error = ((Ideal - Actual) / Ideal) × 100
Error = ((10 - 9.9995) / 10) × 100 = 0.005%
Takeaway: The finite AOL of the LM741 introduces a mere 0.005% gain error. In practice, the 1% tolerance of your physical resistors will cause vastly more gain deviation than the op-amp's finite open-loop gain. For a comprehensive breakdown of op-amp DC error sources, consult the Electronics Tutorials Op-Amp Guide.
Assumptions, Unit Traps, and Realistic Magnitudes
The closed loop gain formula is robust, but it breaks down if you violate its underlying assumptions or mishandle the units.
When the Formula Applies (and When It Doesn't)
The formulas ACL = 1 + (Rf/Rin) and the general Black equation strictly apply only when the op-amp is operating in its linear region with negative feedback. If the output hits the supply rails (clipping/saturation), the gain formula becomes invalid because Vout can no longer scale linearly with Vin. Furthermore, if you accidentally wire positive feedback, the circuit becomes a comparator with hysteresis (Schmitt trigger), and ACL approaches infinity, latching to the rails.
Unit Mistakes That Break the Math
- The dB vs. V/V Trap: The formula requires linear scalar ratios (V/V or A/A). If a specification sheet lists open-loop gain as 120 dB, you cannot plug '120' into the AOL variable. You must convert it first: AOL = 10(120/20) = 1,000,000 V/V.
- Treating β as Resistance: β is a voltage division ratio, not a resistance value. It is strictly dimensionless. If you measure β in ohms, your network analysis is fundamentally flawed.
- Ignoring the Inverting Sign: In the inverting topology, the negative sign indicates a 180-degree phase shift. If you are cascading stages and drop the negative sign, your total system gain calculation will be inverted, potentially turning negative feedback into positive feedback in your math model.
What a Realistic Answer Magnitude Looks Like
If your calculated ACL falls outside typical engineering bounds, you likely have a resistor typo. Here are realistic closed loop gain magnitudes by application:
| Application | Typical ACL (V/V) | Typical ACL (dB) |
|---|---|---|
| DAC Buffer / Line Driver | 1 to 2 V/V | 0 to 6 dB |
| Audio Preamplifier | 10 to 100 V/V | 20 to 40 dB |
| Current Shunt Amplifier | 50 to 500 V/V | 34 to 54 dB |
| Photodiode Transimpedance (V/A) | 10k to 1M V/A | N/A (Transimpedance) |
If you calculate a required closed loop gain of 50,000 V/V for an audio circuit using a single op-amp stage, stop. You will destroy your bandwidth. Cascade two stages of ~224 V/V instead.
Frequently Asked Questions
How does closed loop gain affect bandwidth in op-amps?
Closed loop gain and bandwidth are inversely proportional in voltage-feedback op-amps due to the Gain-Bandwidth Product (GBW) constant. The formula is Bandwidth = GBW / ACL. If your op-amp has a GBW of 1 MHz and you set your closed loop gain formula to yield 100 V/V, your -3dB bandwidth drops to exactly 10 kHz. If you need high gain and high bandwidth, you must select an op-amp with a higher GBW or distribute the gain across multiple cascaded stages.
What happens to the closed loop gain formula if the feedback is positive?
The general equation ACL = AOL / (1 + AOLβ) assumes negative feedback (where the feedback signal opposes the input). If you apply positive feedback, the denominator becomes (1 - AOLβ). Because AOL is massive, the denominator quickly crosses zero and becomes negative, driving the mathematical gain toward infinity. In physical reality, the op-amp output instantly slams into the positive or negative supply rail and stays there. The linear gain formula no longer applies.
Why is my measured closed loop gain lower than the formula predicts at high frequencies?
If your DC gain matches the formula perfectly but drops off at higher frequencies, you are hitting either the GBW limit or the Slew Rate limit. Slew rate (measured in V/μs) dictates how fast the output pin's physical voltage can change. If your closed loop gain demands a 10V peak output at 100 kHz, the required slew rate is 2π × f × Vpeak = 6.28 V/μs. If your op-amp (like the LM741 at 0.5 V/μs) cannot charge its internal compensation capacitor that fast, the output waveform distorts into a triangle wave, and the effective measured gain collapses.
Does the closed loop gain formula apply to current feedback amplifiers (CFAs)?
No. The standard closed loop gain formula and the concept of a constant Gain-Bandwidth Product apply strictly to Voltage Feedback Amplifiers (VFAs). Current Feedback Amplifiers (CFAs) have a different internal topology where the bandwidth is primarily set by the value of the feedback resistor (Rf), largely independent of the closed loop gain. For CFAs, you must consult the manufacturer's specific Rf vs. Bandwidth graphs rather than relying on the standard VFA gain equations.
