Electrical circuits lecture notes b. Tech

The Impedance of a Series RLC Circuit 
As the three vector voltages are out-of-phase with each other, X
L
, X
C
and R must also be 
“out-of-phase” with each other with the relationship between R, X
L
and X
C
being the vector 
sum of these three components thereby giving us the circuits overall impedance, Z. These 
circuit impedance’s can be drawn and represented by an Impedance Triangle as shown below. 
The Impedance Triangle for a Series RLC Circuit 
The impedance Z of a series RLC circuit depends upon the angular frequency, ω as do X
L
and X
C
If the capacitive reactance is greater than the inductive reactance, X
C
> X
L
then the 
overall circuit reactance is capacitive giving a leading phase angle. 
Likewise, if the inductive reactance is greater than the capacitive reactance, X
L
> X
C
then the 
overall circuit reactance is inductive giving the series circuit a lagging phase angle. If the two 
reactance’s are the same and X
L
= X
C
then the angular frequency at which this occurs is 
called the resonant frequency and produces the effect of resonance

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Then the magnitude of the current depends upon the frequency applied to the series RLC 
circuit. When impedance, Z is at its maximum, the current is a minimum and likewise, when 
Z is at its minimum, the current is at maximum. So the above equation for impedance can be 
re-written as: 
The phase angle, θ between the source voltage, V
S
and the current, i is the same as for the 
angle between Z and R in the impedance triangle. This phase angle may be positive or 
negative in value depending on whether the source voltage leads or lags the circuit current 
and can be calculated mathematically from the ohmic values of the impedance triangle as: