Resistors and Capacitors in Series
Introduction
In the DC analysis of resistor circuits we examined how to calculate the total circuit resistance
of series components. In this section we will use this approach to analyse circuits containing
series resistors and capacitors. To do this we use the capacitative reactance as the effective
'resistance' of the capacitor and then proceed in a similar manner to before.
Analysing Series RC Circuits
You will recall that a series circuit provides only one route for the current to flow between two
points in a circuit, so for example the diagram below shows a resistor in series with a capacitor
between the points A andB.
The total impedance (resistance) of this circuit is the contribution from both the capacitor and
resistor. From the previous section we have seen that the capacitative reactance Xc is shifted by - 90o from the perturbing voltage signal and
therefore is expressed in complex form as
Xc = -j
Xc
The total circuit impedance is therefore
Z = R - j Xc
In ac analysis both the resistor and capacitor are treated as phasor quantities, so Xc is -
90o out of phase with respect to the resistor. Since Z is a phasor sum the result is presented on a phasor
diagram (or complex plane)
The magnitude of the impedance - the length of vector can be calculated using
For example if we take the above circuit with a resistor of 100
Ohms and Capacitor of 1x10-6
F and apply an sinusoidal voltage at a frequency of 10
Hz, the capacitative reactance can be calculated as:
Xc = 1/(2 x 3.1415 x 10 x
1x10-6) = 15.9x103Ohms
and so the total impedance in rectangular form is
Z = R - j Xc = 100 - j
15.9x103
We can convert this to polar form using the method above which gives a magnitude of 15.89x103 Ohms and angle of -89.6o. The magnitude is clearly dominated by
the capacitors reactance since we have considered a low operating frequency. As the frequency
is altered the following response is observed (in phasor and bode forms)
The analysis of more complex series circuits is carried out by summing the individual resistances
and capacitative reactances into single equivalent components and then performing the same
analysis as above
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