Closed-Loop Gain Calculator
Solve the classic negative-feedback relation Gcl = G / (1 + G·H) for any of its three unknowns — closed-loop gain, forward gain, or feedback factor — with step-by-step math, a live block-diagram schematic, and companion sensitivity and desensitivity metrics.
Feedback trades raw gain for precision
Without feedback, the overall gain is simply G, which drifts with temperature, supply voltage, device tolerances and ageing. Closing the loop reduces the gain from G to G/(1+GH) but makes the result depend primarily on H — a passive resistive or capacitive network that can be made extremely stable. For large loop gain, Gcl ≈ 1/H, so a ×10 amplifier designed with H = 0.1 keeps its ×10 gain even if the open-loop gain varies by ±50 %. This is the single most important idea in analog and control-system design.
The desensitivity factor (1 + GH) is the key metric
The quantity D = 1 + GH is called the desensitivity factor, return difference, or amount of feedback. It tells you by how much the closed-loop gain is reduced relative to the open-loop gain, and by the same factor it tells you how much the sensitivity to changes in G is reduced. With G = 100 and H = 0.1 (loop gain 10), D = 11: the closed-loop gain is 1/11 of the open-loop gain, and a 1 % change in G produces only a 1/11 % ≈ 0.09 % change in Gcl. Higher loop gain means stronger desensitivity and better precision.
As G → ∞, the closed-loop gain converges to 1/H
The limit of G/(1+GH) as G → ∞ is 1/H, the "ideal" closed-loop gain. The difference between the actual Gcl and this ideal is (Gcl,ideal − Gcl) = 1/(H(1+GH)), which shrinks as GH grows. In practical terms, a designer first chooses H to set the desired closed-loop gain (H = 1/Gcl), then chooses G large enough that the ideal approximation is accurate to the required tolerance. A typical op-amp circuit runs at GH ≈ 10³–10⁶, so the closed-loop gain tracks 1/H to within 0.001 %.
Sign matters — positive feedback causes instability
The formula G/(1+GH) assumes negative feedback, which is the case when the loop gain GH is positive and the summing node subtracts. If the sign of H is reversed (positive feedback), the denominator becomes 1 − GH and the closed-loop gain grows without bound as GH approaches 1. This is the oscillation condition that makes oscillators oscillate but destroys amplifiers. Even with negative feedback, GH becomes a complex function of frequency, and the loop can become unstable if |GH| > 1 when the phase of GH reaches −180°. This is why stability analysis (Nyquist, Bode, root-locus) is the central discipline of feedback design.
How to Use This Calculator
The closed-loop relation links three quantities: Gcl, G and H. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick Gcl, G or H from the toggle. The calculator rearranges Gcl = G/(1+G·H) automatically for whichever variable you choose.
2. Use realistic presets
Presets cover typical values: closed-loop gains from 0.5 to 100, forward gains from 1 to 10⁶, and feedback factors from 0.001 to 2. The default G = 100, H = 0.1 gives a ×10 amplifier with loop gain 10.
3. Watch the sign convention
The formula assumes negative feedback. Signs of G and H matter: if GH is negative, the denominator becomes 1 − GH and the loop is positive — the calculator flags this and shows a warning. For stability analysis in real systems, GH must be evaluated as a frequency-dependent complex quantity.
4. Read the result and verdict
The result card shows the solved value plus loop gain, desensitivity, ideal gain (1/H), and the ratio of the actual to ideal. A colour-coded verdict classifies the loop from very weak through weak, moderate, strong and very strong feedback.
5. Check the schematic and charts
The schematic shows a live block diagram with the input, summing junction, G block, output, and feedback H block — all labelled with current values. The main chart plots Gcl against G for three feedback factors.
6. Switch to Advanced for depth
Advanced mode adds a Gcl vs. H chart, a sensitivity analysis, and companion metrics including loop gain, desensitivity in dB, ideal gain, gain error, and the sensitivity dGcl/Gcl per unit dG/G.
What This Tool Offers
Built around the classic negative-feedback relation Gcl = G/(1+G·H), the fundamental link between forward gain, feedback factor and closed-loop behaviour.
Three-way solving
Solve for closed-loop gain, forward gain, or feedback factor. The calculator rearranges the formula for whichever variable you choose.
Live block diagram
A classic negative-feedback schematic with input, summing junction, forward block G, output node, and feedback block H — all labelled with current numerical values.
Feedback-strength verdict
Each result is classified from very weak (little feedback) through weak, moderate, strong and very strong, with a coloured badge and comparison against the ideal gain 1/H.
Companion metrics
Loop gain L = GH, desensitivity D = 1+L in linear and dB, ideal gain 1/H, gain error, and sensitivity dGcl/Gcl per unit dG/G — all live.
Dual comparison charts
Gcl vs. G for three feedback factors, plus an advanced Gcl vs. H chart for three forward gains.
Reference tables
A Gcl lookup grid, typical values by application (op-amp circuits, servo loops, instrumentation), and a full variable reference.
Reference Tables & Conversions
A quick three-field solver, a Gcl lookup grid, and typical feedback values by application.
Quick feedback solver — enter any two, read the third
Gcl Lookup — Forward Gain vs. Feedback Factor
Computed from Gcl = G / (1 + G·H). Values are the closed-loop gain.
| G | H = 0.01 | H = 0.05 | H = 0.1 | H = 0.25 | H = 0.5 | H = 1.0 |
|---|
Typical Values by Application
Representative forward gain, feedback factor, and resulting closed-loop gain. Actual values depend on design constraints and device tolerances.
| Application | Forward gain G | Feedback H | Gcl | Notes |
|---|---|---|---|---|
| Op-amp voltage follower (×1 buffer) | 10⁵ – 10⁷ | 1.0 | ≈ 1.000 | Output wired directly back to inverting input |
| Op-amp non-inverting ×2 | 10⁵ – 10⁷ | 0.5 | ≈ 2.000 | Equal feedback resistors |
| Op-amp non-inverting ×10 | 10⁵ – 10⁷ | 0.1 | ≈ 10.00 | Feedback divider 9R and 1R |
| Op-amp non-inverting ×100 | 10⁵ – 10⁷ | 0.01 | ≈ 100.0 | Feedback divider 99R and 1R |
| Precision instrumentation amp | 10⁴ – 10⁶ | 0.001 – 0.1 | 10 – 1000 | Three-op-amp topology with trimmed resistors |
| Motor speed loop (PI control) | 10 – 1000 | 0.5 – 1.0 | 1 – 2 | Tachometer or encoder as feedback |
| Position servo (with encoder) | 50 – 5000 | 1.0 | ≈ 1.0 | Unity-gain position feedback |
| Temperature control loop | 5 – 200 | 0.1 – 1.0 | 1 – 10 | Thermocouple or RTD feedback |
| Voltage regulator (linear) | 100 – 10⁵ | 0.3 – 0.9 | 1.1 – 3.3 | Resistive feedback divider |
| Switching buck regulator | 10 – 100 | 0.2 – 0.9 | 1.1 – 5.0 | Compensator plus modulator |
| PLL frequency synthesiser | 10³ – 10⁶ | 0.001 – 0.1 | 10 – 1000 | Divider ratio N in feedback path |
| Audio power amplifier (×20) | 10⁴ – 10⁶ | 0.05 | ≈ 20.00 | Feedback factor set by divider ratio |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| G | Forward (open-loop) gain | 1 – 10⁷ | — |
| H | Feedback factor (β) | 0.001 – 1 | — |
| Gcl | Closed-loop gain | 0.5 – 10⁴ | — |
| L | Loop gain = G·H | 0.1 – 10⁶ | — |
| D | Desensitivity = 1 + G·H | 1.1 – 10⁶ | — |
| S | Sensitivity = 1 / (1 + G·H) | 10⁻⁶ – 0.9 | — |
| Aol | Alternative symbol for G (open-loop gain) | — | — |
| β | Alternative symbol for H (feedback factor) | — | — |
What Does Gcl = G / (1 + G·H) Measure?
The closed-loop gain is the overall input-to-output gain of a feedback system once the loop is closed. It replaces the open-loop gain G of the forward path with a smaller but far more precise value Gcl, and the reduction is exactly what buys the improvement in stability, linearity, bandwidth and noise immunity that feedback provides. The relation is the single most important formula in analog electronics and classical control theory: it applies to op-amp circuits, motor drives, power supplies, servomechanisms, oscillators, PLLs and virtually every other feedback system ever designed.
The formula reads as a simple ratio. The numerator is the forward gain G — the gain the signal would see if the loop were open. The denominator 1 + G·H is the "desensitivity factor", so named because it also describes how much the closed-loop gain is desensitised to changes in G. When the loop gain L = G·H is large, the denominator is dominated by GH and the closed-loop gain simplifies to Gcl ≈ 1/H, a value set almost entirely by the passive feedback network rather than the active forward path.
Closed-Loop Gain Formula
The defining relationship for negative feedback is:
Rearranged to solve for the other two unknowns:
H = ( G − Gcl ) / ( G · Gcl ) = 1/Gcl − 1/G
Useful derived quantities:
Desensitivity: D = 1 + L (sometimes called return difference)
Sensitivity: S = 1 / D = dGcl/Gcl ÷ dG/G
Ideal closed-loop gain: Gcl,ideal = 1 / H (as G → ∞)
Gain error: Δ = Gcl,ideal − Gcl = 1 / ( H · (1 + G·H) )
For positive feedback the sign in the denominator flips:
Feedback Variables
| Symbol | Meaning | Notes |
|---|---|---|
| G | Forward gain | Open-loop gain of the forward path before feedback |
| H | Feedback factor (β) | Fraction of output returned to the summing junction |
| Gcl | Closed-loop gain | Overall input-to-output gain with the loop closed |
| L | Loop gain | L = G·H; the gain around the closed loop if it were broken |
| D | Desensitivity | D = 1 + L; factor by which Gcl is smaller than G |
| S | Sensitivity | S = 1/D; fractional change in Gcl per fractional change in G |
| E | Error signal | E = R − H·C; the input to the forward path |
How Each Variable Affects the Result
Increasing G increases Gcl but with diminishing returns — as G → ∞ the closed-loop gain approaches 1/H and stops growing. Increasing H decreases Gcl and simultaneously increases the loop gain for a fixed G, so the closed-loop gain becomes more precise. The two effects are coupled: H sets the target gain (1/H) and the strength of feedback (L = GH) at the same time, so designers usually choose H first from the gain requirement and then choose G large enough to make the ideal approximation accurate.
Loop gain is the master variable. It determines the desensitivity D = 1+L, the sensitivity S = 1/D, the bandwidth extension (closed-loop bandwidth ≈ open-loop bandwidth × D for a single-pole system), the reduction in nonlinear distortion (also by D), and the reduction in output impedance (again by D). All of the classic benefits of negative feedback — precision, linearity, wide bandwidth, low output impedance, high input impedance — come from the same factor, and all of them scale with L. This is why control engineers speak of "how much feedback" as the primary design parameter and why the loop gain, not the closed-loop gain, is the number that matters.
The sign convention is critical. If G and H have the same sign, GH > 0 and the loop is negative — the closed-loop gain is G/(1+GH), which is smaller than G. If G and H have opposite signs, GH < 0 and the loop is positive — the closed-loop gain is G/(1−GH), which grows with GH and becomes infinite at GH = 1. The oscillator exploits this pole; the amplifier avoids it. In practice, GH is complex and frequency-dependent, so even a nominally negative feedback loop can become effectively positive at some frequency, which is why stability analysis is needed on top of the simple algebraic formula.
Closed-Loop Gain Calculation Examples
Loop gain L = 100 × 0.1 = 10
Desensitivity D = 1 + 10 = 11
Gcl = 100 / 11 = 9.0909
Ideal value 1/H = 10.00 (only 10 % above the actual)
L = 100, D = 101
Gcl = 1000 / 101 = 9.901
Ideal = 10.00 — now within 1 %
G = 10 / (1 − 10 × 0.1) = 10 / 0 = ∞
(The required G is infinite because Gcl = 1/H exactly — the ideal limit.)
L = 1000, D = 1001
Gcl = 10⁵ / 1001 = 99.90
Ideal 1/H = 100.00 — matches to 0.1 %
Sensitivity S = 1/D = 1/1001 ≈ 0.001
ΔGcl/Gcl ≈ 0.05 % — a 50 % open-loop variation shrinks to 0.05 % closed-loop
Frequently Asked Questions
Because feedback subtracts a scaled copy of the output from the input, so the signal entering the forward path is smaller than the external input. The forward path must then work harder to produce the same output, and the overall input-to-output ratio falls. In mathematical terms, if the forward path produces C = G·E and the summing junction produces E = R − H·C, then substituting gives C = G(R − H·C), so C(1 + GH) = G·R, and C/R = G/(1+GH). The gain is reduced by exactly the desensitivity factor. This is not a penalty but a deliberate trade: the designer gives up raw gain in exchange for precision, bandwidth, linearity and stability.
G is the open-loop gain of the forward path alone — the gain you would measure with the feedback path disconnected. Loop gain L = G·H is the product of the forward and feedback gains, and it represents the total gain around the closed loop if you were to break the loop at any point and inject a test signal. Loop gain is the more fundamental quantity for feedback design: it determines the desensitivity D = 1+L, the sensitivity S = 1/D, the bandwidth extension factor, the distortion reduction factor, and the stability margin. The closed-loop gain Gcl is only one of the outputs of the design; the loop gain is the parameter you actually optimise.
Because the denominator 1 + GH is dominated by GH for large G, and the gain becomes G/(GH) = 1/H. Physically, this means the forward path's imperfections no longer matter: as long as G is large, the feedback network H sets the gain almost entirely. This is the "golden rule" of feedback design — the closed-loop gain is set by the passive feedback network and is insensitive to the active forward path. In an op-amp circuit, the 1/H rule means the gain is set by two resistors and is limited only by the tolerance of those resistors (typically ±0.1 % with precision components), regardless of the op-amp's own open-loop gain drifting with temperature or supply voltage.
The desensitivity factor D = 1 + GH is the ratio of the open-loop gain to the closed-loop gain (D = G/Gcl), and simultaneously it is the factor by which the sensitivity of the closed-loop gain to changes in the open-loop gain is reduced. The two roles are the same thing stated two ways. If D = 100, then the closed-loop gain is 100 times smaller than the open-loop gain, AND a 1 % change in G causes only 0.01 % change in Gcl. This is why strong feedback (large D) is the goal in precision applications: an amplifier with D = 10⁴ has a closed-loop gain that is essentially independent of the active device's characteristics. The same D also extends the closed-loop bandwidth, reduces harmonic distortion, lowers output impedance, and improves the linearity of the system.
The static formula Gcl = G/(1+GH) is exact for a memoryless feedback system with a linear forward path and a linear feedback path. It breaks down in three situations. First, when the forward path saturates or clips — the loop cannot produce the output demanded by the input, so the effective G collapses and the closed-loop gain no longer tracks 1/H. Second, when the loop gain at some frequency exceeds unity with −180° phase shift — the system oscillates or becomes unstable, and the simple algebraic model no longer applies. Third, when the forward path is nonlinear (motor dead zones, diode drops, quantisers), the effective G depends on the operating point and the closed-loop gain becomes signal-dependent. In all three cases, the static formula is still useful as a starting point, but a proper analysis requires simulation or frequency-domain tools.
Directly. For a non-inverting op-amp, G is the open-loop gain of the op-amp (typically 10⁵–10⁷), and H = R₁/(R₁+R₂) is set by the feedback resistor divider. The closed-loop gain is then Gcl = 1 + R₂/R₁, which is exactly the 1/H limit of the general formula for large G. The same logic applies to the inverting configuration, transimpedance amplifiers, active filters, and virtually every other op-amp circuit. Designers choose the resistors to set H, then rely on the op-amp's large open-loop gain to make the closed-loop gain track 1/H. The bandwidth of the closed-loop amplifier is the gain-bandwidth product of the op-amp divided by the closed-loop gain, which follows directly from the desensitivity factor D = G/Gcl.
The sign in the denominator flips: Gcl = G/(1 − GH). The closed-loop gain grows without bound as GH approaches 1 from below, and the system self-oscillates when GH ≥ 1. Positive feedback is not "wrong" — it is the basis of every oscillator, Schmitt trigger, latch and flip-flop ever designed. But for an amplifier, it is catastrophic: the output latches to one supply rail or oscillates, and the input-to-output transfer function is no longer linear. The difference between negative and positive feedback is often a single sign in the feedback network — a resistor connected to the wrong input of an op-amp, or a Hall sensor wired with reversed polarity — which is why manufacturers of critical systems put great effort into ensuring the sign is correct during assembly and testing.
For a forward path with a single dominant pole at ωp and DC gain G0, the open-loop gain falls as G0/(1 + jω/ωp). Closing the loop with a feedback factor H gives a closed-loop gain that is Gcl ≈ G0/(1+G0H) at DC and rolls off at a new corner frequency ωp(1+G0H) — the bandwidth is extended by exactly the desensitivity factor D = 1 + G0H. The product of DC gain and bandwidth is conserved, which is the famous gain-bandwidth product or GBP of an op-amp. Choosing a lower closed-loop gain (larger H) gives a proportionally wider bandwidth, and vice versa. This is why a ×1 buffer has a much wider bandwidth than a ×1000 amplifier built from the same op-amp.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| G_cl | Closed-loop system gain | |
| G | Forward-path (open-loop) gain | |
| H | Feedback-path gain |
What it means
This is the same as id=86. It gives the closed‑loop gain. It is used to compute the system’s output for a given input in the frequency domain. Understanding this formula is fundamental for control system design and analysis.
Worked example
Closed‑Loop Gain – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| G | 10 |
| H | 0.1 |
| Parameter | Value |
|---|---|
| G | 100 |
| H | 0.01 |
Common mistakes
- Closed‑loop gain (negative feedback): G_cl = G / (1 + G·H) – same as ID 86.
- Frequency response: For sinusoidal inputs, use s = jω – magnitude and phase.
- Stability margin: Gain margin and phase margin indicate how close the system is to instability.
- Loop gain: The open‑loop transfer function is L(s) = G(s)H(s) – used in Nyquist/Bode.
- Steady‑state tracking: For a unity feedback system, the steady‑state error depends on the type of input.
Applications
Closed‑loop gain (negative feedback), G_cl = G/(1 + G·H), gives the overall transfer function of a system with feedback. This is used to analyse the effect of feedback on system behaviour, such as reducing sensitivity to disturbances and improving linearity. Engineers use it to design feedback controllers and to predict system response. By increasing the loop gain, they can achieve better tracking and disturbance rejection, but at the risk of instability. This formula is essential for understanding feedback control and is the basis for many design techniques.
- Feedback controller design and stability analysis
- Performance improvement in servo systems
- Disturbance rejection and noise reduction
- Analysis of control system sensitivity
- Educational foundation for automatic control
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