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What is Transfer function?

The input–output relation of a linear time-invariant system expressed as the ratio G(s) = Y(s)/X(s) of the Laplace transforms of output and input with all initial values zero. It equals the Laplace transform of the impulse response. Values of s that make the denominator zero are poles and those that make the numerator zero are zeros; the pole positions show stability and speed of response. Setting s = jω gives the frequency response. For discrete-time systems it is written H(z) using the z-transform.

Japanese name: 伝達関数

Transfer function appears in 25 practice questions across 3 topics, including Control Engineering Basics, Digital Signal Processing Basics, Circuit Analysis: AC, Linear Circuits and Transients.

RC low-pass filter and RC high-pass filter (a) vin vout R C (b) vin vout C R H(s) = 1/(1 + sRC) H(s) = sRC/(1 + sRC)
RC low-pass filter and RC high-pass filter · View in diagrams

Field: Control Engineering Basics

Calculating with block diagrams

With transfer functions, elements in series multiply and elements in parallel add. In a negative-feedback loop with forward transfer function G(s) and feedback transfer function H(s), the closed-loop transfer function is G(s)/(1 + G(s)H(s)). The product G(s)H(s) is called the loop transfer function (open-loop transfer function) and is the starting point for examining stability and stability margins.

Typical forms

An RC low-pass filter is the first-order system 1/(1 + sRC), while RLC circuits and motor position loops are often represented as second-order systems. An integrator is 1/s, a differentiator s and a time delay L is e^(−Ls). A transfer function whose denominator degree is at least the numerator degree is called proper, and real physical systems normally take this form.

What poles and zeros tell you

A pole p produces a component e^(pt) in the response, so the system is stable if all poles have negative real parts. Poles closer to the imaginary axis decay more slowly, and larger imaginary parts oscillate faster. Zeros change the shape of the response; a system with a zero in the right half of the s-plane can show an inverse response, in which the step response first moves the wrong way, and fast control becomes difficult.

Try it: questions about Transfer function

In the RC low-pass filter in figure (a), R = 1 kΩ and C = 1 µF. Which of the following is the pole of the transfer function H(s) = Vout(s)/Vin(s)?

  1. As = −1,000 s⁻¹
  2. Bs = −159 s⁻¹
  3. Cs = −0.001 s⁻¹
Show answer and explanation

Answer: A. s = −1,000 s⁻¹

Method 1 (from the transfer function): the voltage divider gives H(s) = (1/(sC))/(R + 1/(sC)) = 1/(1 + sRC), and the pole is s = −1/(RC), where the denominator becomes 0. Since RC = 1×10³ × 1×10⁻⁶ = 1×10⁻³ s, the pole is at s = −1,000 s⁻¹. Method 2 (link to the time response): a pole at s = −a corresponds to a term e^(−at) in the step response and other responses, so from the RC circuit response e^(−t/RC) the pole reads −1/τ = −1/(1 ms). The magnitude of the pole, 1,000 rad/s, is the cutoff angular frequency, about 159 Hz as a frequency.

When the frequency response of a discrete-time LTI system is obtained from its transfer function H(z), along which of the following is z moved?

  1. AAlong the unit circle of the z-plane (z = e^(jω), |z| = 1)
  2. BAlong the positive real axis of the z-plane (z = r, where r is a real number greater than 0)
  3. CAlong the imaginary axis of the z-plane (z = jω)
Show answer and explanation

Answer: A. Along the unit circle of the z-plane (z = e^(jω), |z| = 1)

The frequency response is H(z) with z = e^(jω) substituted; as ω goes from −π to π, z travels once around the unit circle. z = 1 (ω = 0) corresponds to DC and z = −1 (ω = π) to the Nyquist frequency. For the continuous-time Laplace transform the frequency response is found on the imaginary axis s = jω, and in discrete time the unit circle plays that role.

FAQ

What is the difference between a transfer function and a frequency response?

A transfer function G(s) is a function of the complex variable s, and restricting s to jω on the imaginary axis gives the frequency response G(jω). Plotting the magnitude and phase of the frequency response against frequency gives a Bode plot.

Do nonlinear systems have transfer functions?

Not as they are. Linearizing around an operating point allows a transfer function to be used for small changes near that point. Strong nonlinearities such as saturation and dead zones are handled by other methods.

Transfer function in the explanations

Questions that mention Transfer function (25)

  1. In the RC low-pass filter in figure (a), R = 1 kΩ and C = 1 µF. Which of the following is the pole of the transfer function H(s) = Vout(s)/Vin(s)?
  2. When the frequency response of a discrete-time LTI system is obtained from its transfer function H(z), along which of the following is z moved?
  3. Which of the following is the transfer function H(z) = Y(z)/X(z) of the system described by the difference equation y[n] = x[n] + 0.5x[n − 1] + 0.25y[n − 1]?
  4. Which of the following correctly gives the poles and the stability of the causal system with transfer function H(z) = 1/(1 − 1.3z⁻¹ + 0.4z⁻²)?
  5. An FIR filter with transfer function H(z) = 1 + z⁻² (y[n] = x[n] + x[n − 2]) runs at a sampling frequency of 8 kHz. Which of the following is the frequency of a sine-wave input for which the output is 0?
  6. Which of the following correctly describes the bilinear transform used to design a digital filter from the transfer function H(s) of an analog filter?
  7. In a negative-feedback block diagram, the forward path has the transfer function G(s) and the feedback path has the transfer function H(s). Which of the following is the closed-loop transfer function Y(s)/R(s) from the setpoint R(s) to the output Y(s)?
  8. A step of size 1 is applied to a system with the transfer function G(s) = 5/(s + 2). Which of the following is the final value of the output (its value as t → ∞)?
  9. Which of the following correctly describes the transfer function of a linear time-invariant system?
  10. A step of size 1 is applied at time 0 to a first-order system with the transfer function G(s) = 2/(1 + 0.5s). Which of the following is closest to the output at t = 1 s?
  11. When a step of size 2 was applied to a first-order system, the output finally settled at 6 and reached about 63% of its final value 0.4 s after the input. Which of the following is the transfer function of this system?
  12. Which of the following is the correct pair of natural angular frequency ωn and damping ratio ζ for the second-order system with the transfer function G(s) = 25/(s² + 4s + 25)?
  13. In a unity-feedback system the forward transfer function is G(s) = 9/(1 + 2s). Which of the following is the steady-state error for a step setpoint of size 1?
  14. In a unity-feedback system with a stable closed loop, the forward transfer function G(s) contains one integrator 1/s (a type 1 system). Which of the following correctly describes the steady-state error?
  15. A unity-feedback system has the loop transfer function K/(s(s + 1)(s + 2)). Which of the following is the range of K for which the closed-loop system is stable?

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Comparisons that include this term

Semiconductor, Electronics and Computing Calculators

Bode plot calculatorDraw magnitude and phase plots from a transfer function and find phase and gain margins

Related terms

Nyquist frequency Phase margin Laplace transform First-order system Time constant Gain margin z-transform Impulse response Second-order system Steady-state error

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