Function specifying the behavior of a component in an electronic or control system
Inengineering, a transfer function (also known as system function[1]ornetwork function) of a system, sub-system, or component is a mathematical function that models the system's output for each possible input.[2][3][4] It is widely used in electronic engineering tools like circuit simulators and control systems. In simple cases, this function can be represented as a two-dimensional graph of an independent scalar input versus the dependent scalar output (known as a transfer curveorcharacteristic curve). Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.
Dimensions and units of the transfer function model the output response of the device for a range of possible inputs. The transfer function of a two-port electronic circuit, such as an amplifier, might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical actuator might be the mechanical displacement of the movable arm as a function of electric current applied to the device; the transfer function of a photodetector might be the output voltage as a function of the luminous intensity of incident light of a given wavelength.
Descriptions are given in terms of a complex variable, . In many applications it is sufficient to set (thus ), which reduces the Laplace transforms with complex arguments to Fourier transforms with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case in signal processing and communication theory), not the fleeting turn-on and turn-off transient response or stability issues.
For continuous-time input signal and output , dividing the Laplace transform of the output, , by the Laplace transform of the input, , yields the system's transfer function :
Discrete-time signals may be notated as arrays indexed by an integer (e.g. for input and for output). Instead of using the Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the z-transform (notated with a corresponding capital letter, like and ), so a discrete-time system's transfer function can be written as:
Direct derivation from differential equations[edit]
where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space transforms u into r. That kind of equation can be used to constrain the output function u in terms of the forcing function r. The transfer function can be used to define an operator that serves as a right inverse of L, meaning that .
A general sinusoidal input to a system of frequency may be written . The response of a system to a sinusoidal input beginning at time will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady-state response; it corresponds to the homogeneous solution of the differential equation. The transfer function for an LTI system may be written as the product:
where sPi are the N roots of the characteristic polynomial and will be the poles of the transfer function. In a transfer function with a single pole where , the Laplace transform of a general sinusoid of unit amplitude will be . The Laplace transform of the output will be , and the temporal output will be the inverse Laplace transform of that function:
The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σP is positive. For a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be:
The frequency response (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude:
which is the absolute value of the transfer function evaluated at . This result is valid for any number of transfer-function poles.
is input to a linear time-invariant system, the corresponding component in the output is:
In a linear time-invariant system, the input frequency has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. The frequency response describes this change for every frequency in terms of gain
and phase shift
The phase delay (the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is
The group delay (the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency ,
Incontrol engineering and control theory, the transfer function is derived with the Laplace transform. The transfer function was the primary tool used in classical control engineering. A transfer matrix can be obtained for any linear system to analyze its dynamics and other properties; each element of a transfer matrix is a transfer function relating a particular input variable to an output variable. A representation bridging state space and transfer function methods was proposed by Howard H. Rosenbrock, and is known as the Rosenbrock system matrix.
^Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN0-471-98800-6 p. 50
^M. A. Laughton; D.F. Warne (27 September 2002). Electrical Engineer's Reference Book (16 ed.). Newnes. pp. 14/9–14/10. ISBN978-0-08-052354-5.
^E. A. Parr (1993). Logic Designer's Handbook: Circuits and Systems (2nd ed.). Newness. pp. 65–66. ISBN978-1-4832-9280-9.
^Ian Sinclair; John Dunton (2007). Electronic and Electrical Servicing: Consumer and Commercial Electronics. Routledge. p. 172. ISBN978-0-7506-6988-7.
^Birkhoff, Garrett; Rota, Gian-Carlo (1978). Ordinary differential equations. New York: John Wiley & Sons. ISBN978-0-471-05224-1.[page needed]
^Valentijn De Smedt, Georges Gielen and Wim Dehaene (2015). Temperature- and Supply Voltage-Independent Time References for Wireless Sensor Networks. Springer. p. 47. ISBN978-3-319-09003-0.