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Phase Shifts in RC Filters__
A tutorial on the
phase shift of a single pole RC filter (and a quick
lesson on the polar form of complex numbers).
The impedance of a capacitor of value C is
represented by
Where *w* is equal to
Therefore the transfer function of the low pass
filter the circuit below is
which equals
If plotted on a graph, all j terms (imaginary terms)
represent an excursion along the y axis and all non
j terms (real terms) represent an excursion along
the x axis. Positive values of j go UP the y axis
and negative values of j go DOWN the y axis.
Positive and negative real values go right and left
along the x axis. By picturing the real and
imaginary number on the x and y axes, it is easy to
convert a complex number into its polar form and
hence determine the phase shift of the circuit.
The polar form of a complex number can be found by
determining the magnitude of the complex number and
the angle between the real and imaginary parts.
Thus a complex number of 10 + j35 has a polar form
of magnitudeand
angle
or
Likewise a complex number of 7 – j12 has a polar
form of
If a fraction has a complex number on the numerator
and on the denominator, simply divide the
denominator magnitude into the numerator magnitude
and subtract the denominator angle from the
numerator angle.
Thus
can
be represented as
which
equals
It can also be seen that if the numerator has no
imaginary term (=0)
then the final angle is
just the angle of the denominator multiplied by
(-1).
The phase shift of the RC circuit above can be
evaluated at any frequency by replacing the *w*
term with
and
evaluating the real and imaginary numbers in the
numerator and the denominator and thus finding the
polar form.
For the circuit above, the transfer function is
So at a frequency of, say, 100Hz,
this has a transfer function of
which has a polar form of
So we can see that at 100Hz, the output is 0.995
times the input and we have a phase *lag* of
5.7°.
Likewise at a frequency of 10kHz, the polar form of
the transfer function is
Indeed we can see that for a single order filter, at
frequencies of less than 1/10^{th} of the
break frequency, the phase shift is nearly zero and
for frequencies of higher than 10x the break
frequency the phase shift is nearly 90 degrees.
Although we are 5 degrees out, this is a good enough
approximation for most applications.
At the break frequency the real and imaginary terms
are equal so the angle between the real and
imaginary terms is 45 degrees. |