Electrical circuits lecture notes b. Tech

Instantaneous Voltages for a Series RLC Circuit: 
The phasor diagram for a series RLC circuit is produced by combining together the three 
individual phasors above and adding these voltages vectorially. Since the current flowing 
through the circuit is common to all three circuit elements we can use this as the reference 
vector with the three voltage vectors drawn relative to this at their corresponding angles. 
The resulting vector V
S
is obtained by adding together two of the vectors, V
L
and V
C
and then 
adding this sum to the remaining vector V
R
. The resulting angle obtained between V
S
and i 
will be the circuits phase angle as shown below. 
Phasor Diagram for a Series RLC Circuit: 

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We can see from the phasor diagram on the right hand side above that the voltage vectors 
produce a rectangular triangle, comprising of hypotenuse V
S
, horizontal axis V
R
and vertical 
axis V
L
– V
C
Hopefully you will notice then, that this forms our old favourite the Voltage 
Triangle and we can therefore use Pythagoras’s theorem on this voltage triangle to 
mathematically obtain the value of V
S
as shown.
 
Voltage Triangle for a Series RLC Circuit: 
Please note that when using the above equation, the final reactive voltage must always be 
positive in value, that is the smallest voltage must always be taken away from the largest 
voltage we cannot have a negative voltage added to V
R
so it is correct to have V
L
–
V
C
or V
C
– V
L
. The smallest value from the largest otherwise the calculation of V
S
will be 
incorrect.We know from above that the current has the same amplitude and phase in all the 
components of a series RLC circuit. Then the voltage across each component can also be 
described mathematically according to the current flowing through, and the voltage across 
each element as. 

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By substituting these values into Pythagoras’s equation above for the voltage triangle will 
give us: 
So we can see that the amplitude of the source voltage is proportional to the amplitude of the 
current flowing through the circuit. This proportionality constant is called the Impedance of 
the circuit which ultimately depends upon the resistance and the inductive and capacitive 
reactance’s. 
Then in the series RLC circuit above, it can be seen that the opposition to current flow is 
made up of three components, X
L
, X
C
and R with the reactance, X
T
of any series RLC circuit 
being defined as: X
T
= X
L
– X
C
or X
T
= X
C
– X
L
with the total impedance of the circuit 
being thought of as the voltage source required to drive a current through it.