Experiment no.

L-C-R seriesand parallel resonance

L-C-R series and

Obiect To study the frequency response and to find resonant frequencies of
to find the quality factor and band width in
L-C-R series circuit.
parallel circuits. Also
an a.c.
A non-inductive capacitor, an inductor,
resistor, a a frcquency generator,
Apparatus: -

milli- ammeter, a.c. voltmeter and the connecting wires.

Formula

1. The resonance frequency Hz

2T LC

where L =
self inductance in mH

C capaciatance in uF
Hz
2.Bandwidth ( - )
3. Quality factor f-f,
the frequencies at half-power points.
where f, and f are

Circuit diagram:

mA mA

Series Pavadlsl
 2

Description and theory (Series L-C-R)

circuit containing an inductance L

When an alternating voltage V is applied to a
can write the complex applied voltage V
capacitance C and resistance R in series then we

as follows:
V IR+j l»L -i
Or it can be written as
V-I|R+j (ol-)
Therefore thecomplex vectorimpedance, 2=R+j (oL wC

The modulus or magnitude of the impedance is, |Z| = R 2 + (wL - )2

wC

In case when oL = ,the applied voltage is in phase with the current. The potential
wC
differences across the inductance and capacitance are equal in magnitude but opposite in
phase and therefore cancel out each other and the whole voltage is dropped across the
resistance. The circuit is then said to be in resonance with the applied voltage. The
frequency at which resonance occurs is known as resonance frequency and .can be
obtained from condition 1
resonance a,l =

WC
giving
1
2TC
So, in a state of series resonance the impedance of the circuit is minimum equal to Z = R
and hence the current in the circuit at resonance will be maximum. The condition for
resonance may be achieved by varying either the frequency of the applied voltage or the
vaiues ofL or C.
The ratio of the potential difference across the inductance (or capacitance) at resonance
and the applied voltage is known as figure of merit or quality factor Q of the circuit and is
usually referred to as Q- factor of the circuit,

Q-a
R
The variation of current with
R is shown in
frequency for a series resonant cireuit with a given value of
figure below. It is clear from the curve that the series resonant circuit
shows maximum
response ( i.e., passes maximum
of the applied current) for one frequency component
voltage; in other words it shows selectivity
and often this circuit is referred
to as acceptor circuit.
The rapidity with which the current falls of the resonant value is known as the sharpness
of the resonance. It is measured in terms of the ratio of resonant frequency to the
difference of two frequencies, known as half- power frequencies, at which the current in
the circuit decreases t o o f its maximum value and the difference of half-power
frequencies is known as band-width.

0.707 1o

Frequency

Description and theory- Parallel LCR circuit:

admittance Y=; will be given as;
In this case the complex vector

+ jwC
This can be simplified to give;
R+ jwL

R wCR w'LC -
»lL
+j R2+wL2
R2wL2
admittance Y will be given as;
Therefore the magnitude of the

(wCR2 w'L2C -»L)?

1 /R2 +
R2 2L2

maximum when,
Y will be minimum
or impedance Z will be
The admittance

-wL = 0
wCR2+ w'L?C
 4

This happens at resonance frequency fr which can be obtained by solving above equality
and it gives;

For parallel resonance to occur,

should be reul

i.e. >0 R2 or
R
thus no parallel resonunce can occur if R is greater than

The freq
E
ncy at the parallel resonance becomes equal to the frequency at series
resonance ifR is very small so that R2/L2 becomes negligible as compared to the 1/LC
which is usually the case. As the impedance of parallel resonant circuit is maximum at
resonance frequency and is equal to Zmax (L/ RC) and hence the current in the circuit is
at minimum at parallel resonance. A parallel resonant circuit behaves as a rejector.circuit.

Frequency. f
Parallel Resonan ce
 Procedure: For L-C-R series, the circuit is connected as shown in the figure-1.The source
resistance and the series resistance should be small. The output voltage of the signal generator is
adjusted to be around 5v. The frequency of the signal generator is changed in steps and the
coresponding current values are noted from the a.c. milli ammeter. The readings are tabulated.
The current values increase with the increase of frequeney, up to the resonant frequency, further
increase of frequency causes the decrease of current. The L, C and R values are noted to
calculate the resonant frequency fr, band-width and Q factor using the above formulae.

Note: The experiment may be repeated with different values of 'R'. Here the f value is
unchanged, but Q- factor value is changed.
is drawn for current The frequency corresponding to
against frequency.
Graph: A graph
maximum current is noted and it is the resonant frequency f. The frequencies fi and f
it the bandwidth, (f2 fi) is noted.
corresponding to half power points is noted and from
-

in the figure-2. The frequency of the

For L-C-R parallel, the circuit is connected as shown
the corresponding current values are noted from the a.c.
signal generator is changed in steps and
the current values decrease with the
milli- ammeter. The readings are tabulated. But here,
further increase of frequency causes the
increase of frequency up to the anti- resonant frequency,
is noted corresponding to the minimum
increase of current. The anti- resonant frequency f
current in the circuit.

Graph:- A graph is drawn for

current against frequency. The frequency corresponding to
minimum current is noted and it is the anti- resonant frequency.

Observation table:
Table 1: Series LCR
R= R =

R =

Current Frequency Current

S.No. Frequency Current Frequency
(KHz) (mA) (KHz) (mA)
(KHz) (mA)
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Table 2: Parallel LCR

S.No. Frequency (KHz) Current (mA)_

Precautionss
series resistance should besmall.
1) The internal resistance of the source and from L and C
resonant frequency should be calculated
2) Before going to the experiment the
observation.
values so that to select the range of frequencies for

calculated values of resonant frequencies should

Result:: Experimentally and theoretically
be compared and percentage error should be reported.