Rectangular Pulse Response of RC Circuit via Difference Equation.
Cuthbert Nyack
The RC Circuit response to an input sequence can be obtained
by deriving a discrete transfer function for the RC circuit starting
from the continuous transfer function G(s) = 1/(Ts + 1) (T = RC). If this
discrete transfer function G(z) = Y(z)/X(z) is converted to a difference equation (y(n) = f(y(n - m) + x(n - p)))
then the output can be obtained. In this page, the input is assumed
to be a sampled rectangular pulse and the output obtained using
transfer functions obtained from different approximations are compared.
The approximations used are, Bilinear, Back Difference, Forward
Difference, Impulse Invariant, Step Invariant and the Matched Z.
The Bilinear Approximation replaces s with (2/T)(z - 1)/(z + 1) and the
discrete transfer function obtained is shown below.
The Back difference approximation replaces s with (1 - z-1)/T
and the resulting discrete transfer function is shown below.
The Forward approximation replaces s by (z - 1)/T and the
resulting discrete transfer function is shown below.
The Impulse Invariant replaces a 1/(s + a) factor with
aT/(1 - e-aTz-1) resulting in the
following transfer function.
The Step Invariant derives a Discrete t/f which has the same
step response as the continuous transfer function.
The Matched Z approximation assumes that a continuous pole at a
produces a discrete pole at eaT. The transfer function arising is
shown below.
The applet below shows the responses for the different approximations. The DTFT of the input and output as well as the location of the discrete pole
are also shown.
Fn = 0 , Bilinear Approximation.
Fn = 1 , Back Difference Approximation.
Fn = 2 , Forward Differencer Approximation.
Fn = 3 , Impulse Invariant Approximation.
Fn = 4 , Step Invariant Approximation.
Fn = 5 , Matched Z Approximation.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 1.0, 0, na, 0.15, 0.05) show the Bilinear approximation giving very close agreement with the
continuous result.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 0.1, 0, na, 0.2, 0.05) show the Bilinear approximation having a pole at the origin Tc = Ts/2 and the output reaches the input after one sample time. Furthur reduction
in Tc to 0.01 brings the pole to -0.818 and the output
displays the oscillatory behaviour characteristic of poles on the
negative real axis.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 1.0, 1, na, 0.15, 0.05) show the Back Difference approximation giving very close agreement with the
continuous result with the pole remaining on the +ve z axis as Tc and Ts are varied.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 1.0, 2, na, 0.15, 0.05) show the Forward Difference approximation already showing noticeable differences from the
continuous result. Reducing Tc to 0.15 (Tc = Ts) places the pole
at the origin. When Tc = Ts/2 (eg 0.1, 0.2) the pole moves to -1 and the output is oscillatory. Furthur reduction in Tc produces an unstable output as the pole moves out of the unit circle. Because of the possibility of instability the Forward Difference approximation is not used to derive discrete transfer functions.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 1.0, 3, na, 0.15, 0.05) show the Impulse Invariant approximation showing good agreement with the continuous case. However to get the correct numerical result, the
output must be multiplied by the factor shown at the bottom. The pole remains on the +ve real axis and inside the unit circle as Tc, Ts are varied.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 1.0, 4, na, 0.15, 0.05) show the Step Invariant approximation showing good agreement with the continuous case. The pole remains on the +ve real axis and inside the unit circle as Tc, Ts are varied.
eg Parameters:-(1.0, 1.0, 1.0, 1.0, 80, 1.0, 5, na, 0.15, 0.05) show the Matched Z approximation showing good agreement with the continuous case. However to get the correct numerical result, the
output must be multiplied by the factor shown at the bottom. The pole remains on the +ve real axis and inside the unit circle as Tc, Ts are varied.
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© 2007 Cuthbert Nyack.