Formula & Calculator
Closed-Loop Gain (Negative Feedback)
Calculates the overall gain of a negative-feedback control system from the forward-path gain and feedback-path gain.
Interpretation
G_cl = G/(1+GH). Overall gain of a negative feedback system. Used to analyse steady‑state and frequency response.
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
Frequently Asked Questions
For a system with forward gain G and feedback gain H, the closed‑loop gain is T = G / (1 + G·H). The negative feedback sign is implicit in the denominator with a plus sign (since the loop gain is G·H).
Using the positive‑feedback formula (T = G/(1 – G·H)) for a system that actually uses negative feedback, which produces an incorrect result.
It is the loop gain (open‑loop transfer function). The closed‑loop gain is reduced by feedback if the loop gain is large, which improves robustness and reduces sensitivity.
Negative feedback reduces the overall gain (compared to open‑loop) but makes the gain less sensitive to variations in G. It also improves linearity and bandwidth.
When |G·H| >> 1, T ≈ 1/H. This is the benefit of high loop gain: the closed‑loop gain is determined mainly by the feedback network, which can be made very stable.
Amplifier design, control systems, and any feedback system analysis.
A higher loop gain reduces the steady‑state error for a given input. The error is proportional to 1/(1+G·H).
Negative feedback increases the bandwidth of the system, as long as the system is stable. This is a key benefit of feedback.
Stability is determined by the denominator 1 + G·H. The roots of 1 + G·H = 0 give the closed‑loop poles. All poles must be in the left half‑plane for stability.
Unity feedback has H = 1. Non‑unity feedback has H ≠ 1, and the closed‑loop gain is as given. The analysis is similar but with different equations.