TITLE: RESONANT FREQUENCY OF SERIES LCR CIRCUIT
OBJECTIVE:
To find the resonant frequency of series LCR circuit.
APPARATUS REQUIRED:
Resistor
Inductor
Capacitor
Frequency generator
Multi-meter
Oscilloscope
THEORY:
A series LCR circuit includes a resistor, inductor, and capacitor connected in
series. Because they are in series, the current flowing through each
component is the same. This circuit can create harmonic oscillations similar
to an LC circuit. The resistor is added to increase the damping of these
oscillations, which means it helps the oscillations die out more quickly. The
resistor also lowers the peak resonant frequency. Resonance occurs when the
energy stored in the capacitor's electric field and the inductor's magnetic
field are in balance. The key point in an LCR circuit is understanding how the
inductive reactance (XL) and capacitive reactance (XC) change with different
frequencies. At a certain frequency, XL equals XC, making the circuit behave
as if it only has resistance. At this frequency, the impedance is at its
minimum, and the current is at its maximum. This condition is called
resonance, and the frequency at which this occurs is known as the resonant
frequency.
Resonant frequency:
We know,
V
I= ¿ Z∨¿ ¿ where z =
impedance
V
= √ R 2+( X −X )2 =
L C
√ R +( X −X
2
L C )2
Also we know,
XL = WL = 2πfL
1 1
XC = =
WL 2 π fC
For low frequency of input voltage,
XL << XC
And for higher frequency of input AC voltage
XC << XL
Thus, there exist on intermediate frequency such that
X L =X C
This particular frequency is known as resonant frequency Fo. Graphically we
can draw a relation between XL, XC, & R with reference to input frequency.
At resonant frequency (Fo)
X L =X C
1
WL =
WC
1
W2 =
LC
1
(2πFo)2 =
LC
1
Fo =
2 π √ LC
At this condition the current through the circuit is maximum and denoted by
I max.
Bandwidth:
Bandwidth of the circuit is defined as the range of frequencies for which
power delivered to the resistance is more than half of the maximum power.
Bandwidth (B.W.) = f2 -f1 where f1 = lower half power frequency
R
= f2 = higher half power frequency
2 πL
At any instance, magnitude of current is given by
V
I=
√ R +( X
2
C −X L )2
At frequency f1,
V
I1 =
√ R +( X 2
C −X L )2
I max .∗R
I1 =
√ R +( X2
L −X C )
2
I max I max .∗R
=
√2 √ R +( X 2
C −X L )
2
√ 2 R = √ R2 +( X C −X L )2
2R2 = R2 +( X C − X L )2
2
(X C −X L ) = R2
X C −X L = R
1
−W 1 L=R ……………..(i)
W 1C
Similarly for frequency f2
XL -Xc = R
1
W2 L - = R ……………..(ii)
W 2C
Equating (i) and (ii)
1 1
−W 1 L = W 2 L -
W 1C W 2C
1 1 + 1
( ) = L(W 1 +W 2)
C W1 W2
1 W 1 +W 2
( ) = L(W 1 +W 2)
C W 1W 2
1
W1.W2 = ……………………(iii)
LC
Also, we know
1
Fo =
2 π √ LC
1
(2πFo)2 =
LC
2 1
wo = ……………………..(iv)
LC
Adding equation (i) and (ii)
1 1
−W 1 L + W 2 L - = R+R
W 1C W 2C
1 1 + 1
( ) + L(W 1 +W 2) = 2R
C W1 W2
1 W 1 +W 2
( ) + L(W 1 +W 2) = 2R
C W 1W 2
1
(W 1 +W 2) [ + L ¿ = 2R
C . W 1W 2
LW 1 W 2
(W 1 +W 2) [ + L¿ = 2R
W 1W 2
2L(W 2 −W 1) = 2R
R
2πf1 - 2πf2 =
L
R
f2 – f1 = (i.e. bandwidth)
2 πL
Here,
f 2– f 1
f o – f1 =
2
R
f o – f1 =
2 πL∗2
R
fo = + f1
2 πL∗2
R
f1 = fo -
4 πL
R
Similarly, f2 = + fo
4 πL
OBSERVATION:
Resistance (R) = 111.51 Ω
Inductance (L) = 130.86 mH
Capacitance (C) = 1.03 μF
Observation table:
S.N frequen VP-P Vrms Irms
. cy
1. 100 1.72 0.60811 0.00545
2 3
2. 125 2.12 0.74953 0.00672
3 2
3. 150 2.52 0.89095 0.00799
5
4. 175 3 1.06066 0.00951
2
5. 200 3.56 1.2586 0.01128
5 7
6. 225 4 1.41421 0.01268
4 2
7. 250 4.52 1.59806 0.01433
1 1
8. 275 5 1.76776 0.01585
7 3
9. 300 5.52 1.95161 0.01750
5 2
10. 325 5.96 2.10717 0.01889
8 7
11. 350 6.36 2.2486 0.02016
5
12. 375 6.64 2.34759 0.02105
5 3
13. 400 6.88 2.43244 0.02181
7 4
14. 425 6.92 2.44658 0.02194
9 1
15. 450 (Fo) 6.96 2.46073 0.02206
2 7
16. 475 6.88 2.43244 0.02181
7 4
17. 500 6.72 2.37587 0.02130
9 6
18. 525 6.56 2.31931 0.02079
9
19. 550 6.32 2.23445 0.02003
7 8
20. 575 6.12 2.16374 0.01940
7 4
21. 600 5.88 2.07889 0.01864
4 3
22. 625 5.64 1.99404 0.01788
1 2
23. 650 5.44 1.92333 0.01724
8
24. 675 5.2 1.83847 0.01648
8 7
25. 700 5 1.76776 0.01585
7 3
26 725 4.84 1.71119 0.01534
8 6
27. 750 4.64 1.64048 0.01471
8 2
28. 775 4.48 1.58391 0.01420
9 4
29, 800 4.32 1.52735 0.01369
1 7
30. 825 4.2 1.48492 0.01331
4 7
31. 850 4.04 1.42835 0.01280
6 9
32. 875 3.92 1.38592 0.01242
9 9
33. 900 3.76 1.32936 0.01192
1 1
34 925 3.68 1.30107 0.01166
6 8
35. 950 3.56 1.25865 0.01128
7
36. 975 3.36 1.18793 0.01065
9 3
37. 1000 3.2 1.13137 0.01014
1 6
38. 1050 3.04 1.07480 0.00963
2 9
39. 1100 2.92 1.03237 0.00925
6 8
40. 1150 2.8 0.98994 0.00887
9 8
41. 1200 2.68 0.94752 0.00849
3 7
42. 1250 2.56 0.90509 0.00811
7 7
From Observation,
Fo = 450 Hz
Vrms = 2.460732 mV
Irms = 0.022067mA
We know, for theoretical value of fo
1
Fo =
2 π √ LC
Then,
Fo = 433.508 Hz
Bandwidth:
Calculated:
R
F1 = Fo - 4 πL = 365.697 HZ
R
F2 = Fo + 4 πL = 501.31 HZ
Bandwidth = F2 - F1 = 135.613 HZ
ERROR CACLCULATION:
We know,
Theoretical value−Observation value
Error = ∗100 %
Theoritical value
450−433.508
= ∗100 %
450
= 3.66 %
DISCUSSION:
In the experiment, we observed that for fixed values of resistance (R),
inductance (L), and capacitance (C) in an LCR circuit, there is a specific
frequency at which the current reaches its maximum value. This frequency is
known as the resonant frequency. As the frequency increases from a low
value to a high value, the inductive reactance ( X L) initially increases from
zero while the capacitive reactance ( X C ) decreases from infinity. At a certain
frequency, the inductive reactance equals the capacitive reactance,
effectively cancelling each other out. At this point, the phase of the current
aligns with the phase of the voltage, and the circuit behaves purely resistive.
CONCLUSION:
Hence from the experiment we found the resonant frequency of the circuit.
