Tables: LaPlace Transform to Difference
Equation
The following tables were derived from
LaPlace transfer functions using backward rectangular
approximation for the integration. This method assures mapping of
poles and zeros within the unit circle. In real applications, I
have found that for most simple feedback control applications, no
additional benefits are obtained by using higher order
approximations, including Tustin's method. For precision digital
filtering, or simulation higher order methods and compensation
for warping may be required.
The following symbols are used throughout
the tables:
- T is the sampling time in seconds.
- a,b, and c represent real poles in
radians per second.
- omega and zeta represent the bandwidth
and damping for a complex pole (or zero) pair
respectively. Omega is in radians per second.
- K represents the transfer function
(high frequency) gain.
- x is the input and y is the output
- s is the LaPlace operator (complex
frequency) and n is the difference equation sample index;
n=[0,1,2,3, ....] sampled T seconds apart.
- Initial conditions are given as zero
for difference equation states. Of course these may be
selected as non-zero where required.
Table 1 1st Order Transfer Functions
Table 2 2nd Order Transfer Functions
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