Engineering · Electrical engineering/Control

Closed-loop transfer function

With negative feedback, the closed-loop transfer function combines the forward path and loop gain.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

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Formula

T (s) = \frac{G ( s )}{1 + G ( s ) H ( s )}

G, H and T dimensionless for compatible input/output quantities; s in s⁻¹

Variables & units

Variables & units
Symbol	Meaning	Unit
$T$	Closed-loop setpoint-to-output transfer function	dimension-dependent
$G$	Forward-path transfer function	dimension-dependent
$H$	Feedback-path transfer function	dimension-dependent

LaTeX

T(s)=\frac{G(s)}{1+G(s)H(s)}

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Derivation & background

Assumes linear time-invariant blocks and negative feedback. Positive feedback changes the denominator sign; stability is not implied by the algebraic form and must be checked from closed-loop poles.

Exam blueprint

Validity range

Assumes linear time-invariant blocks and negative feedback. Positive feedback changes the denominator sign; stability is not implied by the algebraic form and must be checked from closed-loop poles.

Derivation steps

With negative feedback, the closed-loop transfer function combines the forward path and loop gain.

1State the model assumptions and list quantities with units.
2Substitute into T = G/(1+GH); convert quantities to SI first.
3At one frequency, G=10 and H=1 give T=10/11≈0.909.

Rearrangements

Base equation for the target quantity

T(s)=\frac{G(s)}{1+G(s)H(s)}

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Closed-loop transfer function” and check its unit and magnitude.

At one frequency, G=10 and H=1 give T=10/11≈0.909.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: G, H and T dimensionless for compatible input/output quantities; s in s⁻¹.

Exam context

Find setpoint-to-output behavior of a feedback system.

Formula cluster

Electrical engineering/Control

With negative feedback, the closed-loop transfer function combines the forward path and loop gain.

C(s) = KP + KI/s + KDs

Ideal PID controller transfer function

G(s) = K/(τs+1)

Transfer function of a first-order system

S = 1/(1+GH)

Feedback sensitivity function

Worked example

At one frequency, G=10 and H=1 give T=10/11≈0.909.

Applications

Find setpoint-to-output behavior of a feedback system.

How do you calculate with Closed-loop transfer function?

Here is how to work through a typical Closed-loop transfer function (T = G/(1+GH)) task step by step:

1
Task
Work through the numerical example for “Closed-loop transfer function” and check its unit and magnitude.
Solution path
At one frequency, G=10 and H=1 give T=10/11≈0.909.

Frequently asked questions about Closed-loop transfer function

What does denominator 1+GH mean?+

It follows from combining output and feedback algebraically. The product GH is loop gain. Positive rather than negative feedback changes the sign, so the feedback convention must be clear. Assumes linear blocks and negative feedback; check closed-loop poles for stability. Application: Find setpoint-to-output behavior of a feedback system. The numerical example At one frequency, G=10 and H=1 give T=10/11≈0.909 anchors the idea in a concrete measurement case. The corresponding units are G, H and T dimensionless for compatible input/output quantities; s in s⁻¹. At one frequency, G=10 and H=1 give T=10/11≈0.909. This checks that the reference quantities are compatible.

What does T(s) say about the setpoint?+

T(s) maps reference input to output with disturbances considered separately. Its frequency response shows which setpoint changes are tracked well. It alone does not describe disturbance rejection or stability margins. Assumes linear blocks and negative feedback; check closed-loop poles for stability. The corresponding units are G, H and T dimensionless for compatible input/output quantities; s in s⁻¹. At one frequency, G=10 and H=1 give T=10/11≈0.909. This checks that the reference quantities are compatible. Scope: Assumes linear time-invariant blocks and negative feedback. Positive feedback changes the denominator sign; stability is not implied by the algebraic form and must be checked from closed-loop poles. A practical application is Find setpoint-to-output behavior of a feedback system.

Can a small denominator indicate instability?+

A small magnitude of 1+GH at a frequency creates a large closed-loop response and may indicate resonance or low robustness. Stability is determined by poles of the full closed-loop transfer function, not one frequency value. Assumes linear blocks and negative feedback; check closed-loop poles for stability. Scope: Assumes linear time-invariant blocks and negative feedback. Positive feedback changes the denominator sign; stability is not implied by the algebraic form and must be checked from closed-loop poles. A practical application is Find setpoint-to-output behavior of a feedback system.

How do I check the result and units for Closed-loop transfer function?+

Write the quantity and unit from the variable list next to every value: G, H and T dimensionless for compatible input/output quantities; s in s⁻¹. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example At one frequency, G=10 and H=1 give T=10/11≈0.909. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Assumes linear blocks and negative feedback; check closed-loop poles for stability.

Which model limit matters for Closed-loop transfer function?+

The background states this starting condition: Assumes linear time-invariant blocks and negative feedback. Positive feedback changes the denominator sign; stability is not implied by. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Assumes linear blocks and negative feedback; check closed-loop poles for stability.

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Which formula describes Closed-loop transfer function?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]MIT OpenCourseWare, Feedback Control Systems, closed-loop transfer functions.

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U_C(t)=U_0(1-e^{-t/(RC)}),\qquad \tau=RC

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Symbol

Meaning

Unit

T

Closed-loop setpoint-to-output transfer function

dimension-dependent

G

Forward-path transfer function

dimension-dependent

H

Feedback-path transfer function

dimension-dependent

Exam blueprint

Validity range

Assumes linear time-invariant blocks and negative feedback. Positive feedback changes the denominator sign; stability is not implied by the algebraic form and must be checked from closed-loop poles.

Derivation steps

With negative feedback, the closed-loop transfer function combines the forward path and loop gain.

1State the model assumptions and list quantities with units.
2Substitute into T = G/(1+GH); convert quantities to SI first.
3At one frequency, G=10 and H=1 give T=10/11≈0.909.

Frequently asked questions about Closed-loop transfer function

What does denominator 1+GH mean?+

What does T(s) say about the setpoint?+

Can a small denominator indicate instability?+

How do I check the result and units for Closed-loop transfer function?+

Which model limit matters for Closed-loop transfer function?+