Control Engineering Basics
Which of the following correctly describes the transfer function of a linear time-invariant system?
- AThe ratio y(t)/x(t) of the output and input waveforms, calculated at each instant
- BThe ratio of output to input amplitude for a sine-wave input, which is set by the system alone and so is constant regardless of the frequency of the applied sine wave
- CThe ratio Y(s)/X(s) of the Laplace transform of the output to that of the input, with all initial values zero
Show answer and explanation
Answer: C. The ratio Y(s)/X(s) of the Laplace transform of the output to that of the input, with all initial values zero
The transfer function is the ratio of the Laplace transforms of the output and the input with all initial values zero. It equals the Laplace transform of the impulse response of the system and expresses the system as a ratio of polynomials in s. Values of s that make the denominator zero are poles and those that make the numerator zero are zeros; the positions of the poles show stability and speed of response. Setting s = jω gives the frequency response, and for discrete-time systems H(z), expressed with the z-transform, plays the same role.
- A. The ratio of waveforms changes from instant to instant and does not describe the system.
- B. The amplitude ratio is the gain of the frequency response, which generally depends on frequency.
- C. Correct. Setting the initial values to zero lets it express only the relation between input and output.