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The C-R (capacitance-resistance) circuit
The circuit shown in Figure 1 contains both resistance and capacitance, and therefore both the
component and the frequency of the supply voltage affect the current in the circuit.
The a.c. resistance of such a circuit is known as
the impedance of the circuit and is denoted by
the symbol Z. Impedance is measured in ohms.
We will now deduce the impedance of the circuit
using the vector treatment.
Consider the voltages round the circuit. The
supply voltage will be denoted by Vo and the
voltages across the resistor and capacitor by V R,0
and V C,0 respectively.
We know that for a resistor the current and
voltage are In phase, while for a capacitor the
current leads the voltage by 90o; vR,O therefore
leads vC,O by 90o, as shown in the vector diagram
in Figure 2.
C
R
vR
vC
Figure 1
vR0

vR0

t
vR
Time (t) = t
Time (t) = 0
v0
vC0
v0
v
vC
Figure 2
vC0
The resultant voltage v0 is given by
vo2 = vR,O 2 + vC,O 2 = i02R2 + i02X2C
Therefore the current in the circuit is given
by:
io =
vo
√[XC2 + R2]
and the impedance Z by:
Z = v0/i0 = √[XC2 + R2] = √[1/2C2 + R2]
Voltage
vC0
vR
vR0
Time
vC
Figure 3
The angle  that the resultant vector makes
with V is known as the phase angle of the voltage. You can see from Figure 3 that tan
vC0/vR0 = -1/CR
1