Series RC Circuits
In a series circuit, it is common to use the current as a reference, since it is constant throughout the circuit.
This applies equally to both DC and AC circuits.
Recall that in a purely resistive ac circuit, the current and voltage are in phase:
Vs,Vr,It
And that in a purely capacitive ac circuit, since the capacitor opposes a change in voltage, the current leads
the applied voltage by 90°:
It
Vs,Vc
When a circuit contains both a resistive and capacitive element, then the circuit behavior may be logically
analyzed as follows:
1. The current through the resistor is in phase with the voltage across it
Since the current is the same in a series circuit, it must also be the total circuit current
Therefore It must be in phase with Vr
2. Since the voltage across a capacitor lags the current by 90°, and the capacitor must be the same as the
resistor current, Vc must lag behind Vr by 90°
It,Vr
Vc
3. Vs will no longer be in phase with either the resistor or the capacitor, but may be found using
Kirchoff’s voltage law: Vs = Vr + Vc
remember, since Vr and Vc are out of phase, they must be added as vectors. This can be
accomplished using either complex or polar form:
Vs  Vr  jVc
OR
Vs    Vr0  Vc  90
It,Vr
Vs = Vr + Vc
Vc
Impedance (Z)
Impedance is defined as the total opposition to the flow of current in an ac circuit. In an RC circuit, it is
composed of two things: resistance and capacitive reactance.
Capacitive reactance has an angle of -90° associated with it, since according to Ohm’s Law, it must be
defined as:
Vc Vc Vc  90 Vc
in polar form it may be written as –jXc
Xc 



  90
Ic I T
I T 0
IT
Since the voltage and current associated with a resistive element are in phase, the resistive elements are
assigned a zero phase angle: R∟0°
The total circuit impedance is simply the sum of total resistance and total capacitive reactance:
Z = R –jXc

OR
 Xc 
Z  R 2  Xc 2   tan
 in polar form
 R 
Vr = It x R

Vs = It x Z
Vc = It x Xc

Vr
Vs = Vr - jVc
-jVc
R
Z = R - jXc
-jXc
Note that the circuit phase angle may be found by using either the impedance triangle OR the voltage
triangle. Thus:
 Vc 
 Xc 
   tan    tan 
 R 
 VR 
Series RL Circuits
The analysis for series RL circuits is similar to RC circuits except that since an inductor opposes a change in
current, the current lags behind the voltage by 90° in an inductive element. This means the the voltage
across the inductor leads the circuit current by 90°, and the supply voltage angle leads the circuit current by
an amount determined by the relative size of the total circuit resistance and inductive reactance.
VL = It x Xc
jVL
Vs = It x Z
Vs = Vr + jVL


Vr = It x R
Vr
jXL
Z = R + jXL

R
Z  R  jX L OR
X 
2
Z  R 2  X L  tan L  in polar form
 R 
X
Note once again that   tan L
 R
V

  tan L

 VR



Power in ac Circuits
Recall that Watt’s Law states the the power dissipated by a resistive element could be found by multiplying
the voltage across it by the current flowing through it:
P  V  I or P  I 2 R
In a DC ciruit there was no power associated with capacitive and inductive elements since after 5 time
constants either the voltage or current associated with the device was zero. Also, there was no resistance
associated with either device.
In an ac circuit this is no longer the case, since both capacitors and inductors have a reactance associated
with them due to the constantly changing circuit current.
The power associated with reactive elements is appropriately labeled reactive power, and is measured (or
calculated) in units of volt-ampere reactive (VARS). The power dissipated by all three circuit elements may
be calculated as follows:
Ptrue  I 2 R
Pr  I 2 X C
Pr  I 2 X L
Since the power associated with each element may be directly related to its resistance or reactance, a power
triangle may be derived based on the impedance triangles discussed earlier: