Lecture 15
The analysis of the RLC series resonant circuit is continued.
This lecture covers:
• The dependence of impedance and current on frequency
• Concepts of band frequency, cut-off frequency and half
power frequency
• Expressions for band frequencies
• Relationship of quality factor with bandwidth
• Expressions for selectivity
Total impedance (ZT) versus frequency
R
L
I
Es
ZT
C
Fig. 15-1: Series resonant circuit
We have seen that the total impedance of a series RLC circuit at any
frequency is,
ZT = RL + JX L − JX C
The magnitude and phase angle of impedance as a function of
frequency can be written as,
ZT ( f ) = [ R( f )]2 + [ X L ( f ) − X C ( f )]2
Here, ZT (f) means the total impedance as a function of frequency.
The changes of the various terms R, XL , XC, total X, ZT as a function
of frequency are shown in the following figures.
R(f)
R
0
f
Fig. 15-2(a) R as function of frequency
We know the resistance of a circuit does not change with frequency
X L( f )
X L = 2π fL
∆y
∆x
2π L = ∆y = m
∆x
0
f
Fig. 15-2 (b) XL as function of frequency
Inductive reactance is 2 times pi times f times L. Hence the plot of X
sub L is a straight line as a function of frequency
XC ( f )
XC =
1
2π fC
0
f
Fig. 15-2 (c) XC as function of frequency
Capacitive reactance is reciprocal of 2 times pi times C. Hence plot of
X sub C is a hyperbolic curve
X
XL
XC
X L > XC
XC > X L
0
fs
f
Fig. 15-2 (d) Net X as function of frequency
At low frequency X sub C is very high and X sub L is very small. At
resonant frequency they are equal in magnitude. At high frequency X
sub C is very small and X sub L is very high.
ZT
b≠a
a
ZT ( f )
R
fs
f
Fig. 15-2 (e) ZT versus frequency
The total impedance is high at low frequency, then it decreases to a
minimum value, and then again is high at high frequency. At the
resonant frequency it assumes its minimum value.
The expression for the phase angle of the total impedance is,
θ = tan −1
(X L − XC )
R
The variation of angle θ with frequency is shown in Fig. 5(f)
θ
( E leads I )
90o
45o
0o
−45o
−90o
Circuit Capacitive
Leading Fp
Circuit Inductive
Lagging Fp
fs
f
Fig. 5(f) Impedance angle θ versus frequency
At low frequencies X sub C is greater than X sub L and theta will
approach -90 degrees. The circuit is capacitive. At resonant
frequency the angle of X sub C will cancel that of X sub L and the
angle is zero degree. At high frequency X sub L will be larger than X
sub C and the angle will approach +90 degrees.
Selectivity
If we plot the magnitude of current I=E/ZT versus frequency for a fixed
voltage E, we will obtain the curve shown in Fig. 6
I
I m ax = E
R
0.707 Imax
BW
0
f1
fs
f2
f
Fig. 6 Current versus frequency for the series resonant circuit
The current rises from zero to a maximum value of E over R, when Z
sub T is minimum, then drops toward zero again. The curve is
actually inverse of impedance, Z sub T, curve.
There is a range of frequencies at which the current is near its
maximum value and the impedance is at its minimum. Those
frequencies corresponding to 0.707 of the maximum current are
called,
• Band frequencies
• Cut-off frequencies
• Or, half power frequencies
They are indicated as f1 and f2 in Fig. 6.
Half power frequencies are those frequencies at which power
delivered is one half of that delivered at resonant frequency.
PHPF = I 2 R = (0.707 I max )2 R = (0.5)( I max )2 R = ( 1 ) Pmax
2
Determination of Band Frequencies
From Fig. 6, the maximum current
I m ax = E
R
The current at the cut-off frequencies, f1 and f2, is
I = 0.707 I max = 0.707( E ) = ( E )
R
2R
This implies that the magnitude of the impedance at the cut-off
frequencies is √2 R.
Equate the expressions for impedances at the cut-off frequencies,
2 R = R 2 + ( X L − X C )2
Square both sides and simplify to get,
R-XL + X C = 0
Consider the case XL>XC, which relates to ω2 or f2 . The above
equation is written as,
R − ω2 L + 1 = 0
ω2C
Solve the quadratic equation for positive frequency as,
ω 2 = R + ( R )2 + 1
2L
or
2L
LC
f 2 = 1 [ R + ( R )2 + 1 ]
2π 2L
2L
LC
For the condition XL<XC, ω1 and f1 are obtained from,
R − 1 + ω1L = 0
ω1C
Giving,
ω1 = − R + ( R )2 + 1
2L
2L
LC
or
f1 = 1 [− R + ( R )2 + 1 ]
2π 2L
2L
LC
The bandwidth in angular frequency,
BW (rad /sec) = ω2 −ω1 = R
L
= ωr
Qs
While in terms of Hz, it is,
f 2 − f1 = R
2π L
The ratio of resonant frequency (ωr ) to bandwidth is often termed as
Selectivity. For a series resonant circuit,
Selectivity= ωr = ωr = ωr L = Qs
ω2 −ω1 R / L R
Summary
In this lecture, we have covered the following items for a series
resonant circuit:
• Know the Dependence of impedance of a circuit on
frequency
• Know the Dependence of current and power on frequency
• Understand the definitions of half power frequency, cut-off
frequency and band frequency
• Know the expressions for half power of band frequencies
• Understand the definition of bandwidth
• Know about the selectivity of a resonant circuit
SELF-TEST (15)
Consider a series RLC resonant circuit.
™ If the supply frequency changes, the resistance of the circuit
a) Remains constant
b) Increases with increase in frequency
c) Decreases with increase in frequency
Ans: (a)
™ If the supply frequency changes, the inductive reactance (XL)
a) Remains constant
b) Increases with increase in frequency
c) Decreases with increase in frequency
Ans: (b)
™ The total impedance of a circuit at resonance is
a) 0
b) Infinity
c) Maximum possible value
d) Minimum possible value
Ans: (d)
™ The power factor of a circuit at resonance is
a) 0
b) 0.5 lagging
c) 1
d) 0.5 leading
Ans: (c)
™ The selectivity of a resonant circuit is high when
a) The quality factor (Qs) is low
b) Qs is zero
c) Qs is high
Ans: (c)
™ 6) At band frequency the power absorbed by the circuit is
a) Maximum
b) ¾ of maximum value
c) ½ of maximum value
d) ¼ of maximum value
e) 0
Ans: (c)