1.SERIES AND PARALLEL LCR CIRCUITS
AIM: to study the frequency response of the series and parallel LCR circuits and hence to determine 1) the resonant frequency,
2) the quality factor,
3) bandwidth of the circuit and
4) the self inductance of the given inductor.
APPARATUS: Audio frequency oscillator, ac milliammeter, resistor, capacitor and inductor.
PRINCIPLE: When a capacitance C, inductance L and resistance R are connected in series with an ac source, it forms a series resonance circuit. The impedance of a series resonance circuit is given by,
Z =Ö R2+(wL-1/wc)2 where w=2pf; f is the frequency of ac. When inductive reactance is equal to capacitive reactance (wL=1/wc), frequency of the applied emf is equal to the natural frequency of the LCR circuit. This is called resonance.
\At resonance, wL=1/wc
w2=1/LC
(2pf0)2=1/LC
f0=1/2pÖLC
f0 is called resonant frequency. At resonant frequency, current in the circuit is maximum.
The value of the inductor, L=1/4p2f02c
Quality factor Q=1/RÖL/C
From frequency response graph, Q=f0/b whereb=fb-fa is the bandwidth.
In a parallel resonant circuit, an ac source is connected across a parallel combination of an inductance L and capacitance C. Let R be the resistance in the inductance branch. Here, at resonance, the current is minimum.
PROCEDURE:
1) Series resonant circuit
The circuit connections are made as shown in figure 1. The oscillator is switched on and the output of the oscillator is adjusted to a maximum value which is kept constant throughout the experiment. The frequency is increased in steps and corresponding reading of the current (I) in mA is recorded. Care should be taken to locate resonant frequency (frequency corresponding to maximum current) with maximum accuracy. The graph of circuit current as a function of frequency is plotted (figure 2).
The value of inductance L is verified using the formula L= 1/4p2f02c
A straight line drawn parallel to frequency axis at the point Imax/Ö2 on the Y-axis, cuts the curve at two points corresponds to the frequencies fa and fb on the X-axis.
Q-factor is evaluated from the graph, Q=f0/ fb-fa
The theoretical value of the Q-factor is evaluated using the formula
Q=1/RÖL/C
2) Parallel resonant circuit
The circuit connections are made as shown in figure (3). The oscillator is switched on and the output of the oscillator is adjusted to a maximum value which is kept constant throughout the experiment. The frequency is increased in steps and corresponding reading of the current (I) in mA is recorded. Care should be taken to locate resonant frequency (frequency corresponding to minimum current) with maximum accuracy. The graph of circuit current as a function of frequency is plotted (figure 4).
The value of inductance L is verified using the formula L= 1/4p2f02c
A straight line drawn parallel to frequency axis at the point Imin/Ö2 on the Y-axis, cuts the curve at two points corresponds to the frequencies fa and fb on the X-axis.
Q-factor is evaluated from the graph, Q=f0/ fb-fa
The theoretical value of the Q-factor is evaluated using the formula
Q=1/RÖL/C
2) the quality factor,
3) bandwidth of the circuit and
4) the self inductance of the given inductor.
APPARATUS: Audio frequency oscillator, ac milliammeter, resistor, capacitor and inductor.
PRINCIPLE: When a capacitance C, inductance L and resistance R are connected in series with an ac source, it forms a series resonance circuit. The impedance of a series resonance circuit is given by,
Z =Ö R2+(wL-1/wc)2 where w=2pf; f is the frequency of ac. When inductive reactance is equal to capacitive reactance (wL=1/wc), frequency of the applied emf is equal to the natural frequency of the LCR circuit. This is called resonance.
\At resonance, wL=1/wc
w2=1/LC
(2pf0)2=1/LC
f0=1/2pÖLC
f0 is called resonant frequency. At resonant frequency, current in the circuit is maximum.
The value of the inductor, L=1/4p2f02c
Quality factor Q=1/RÖL/C
From frequency response graph, Q=f0/b whereb=fb-fa is the bandwidth.
In a parallel resonant circuit, an ac source is connected across a parallel combination of an inductance L and capacitance C. Let R be the resistance in the inductance branch. Here, at resonance, the current is minimum.
PROCEDURE:
1) Series resonant circuit
The circuit connections are made as shown in figure 1. The oscillator is switched on and the output of the oscillator is adjusted to a maximum value which is kept constant throughout the experiment. The frequency is increased in steps and corresponding reading of the current (I) in mA is recorded. Care should be taken to locate resonant frequency (frequency corresponding to maximum current) with maximum accuracy. The graph of circuit current as a function of frequency is plotted (figure 2).
The value of inductance L is verified using the formula L= 1/4p2f02c
A straight line drawn parallel to frequency axis at the point Imax/Ö2 on the Y-axis, cuts the curve at two points corresponds to the frequencies fa and fb on the X-axis.
Q-factor is evaluated from the graph, Q=f0/ fb-fa
The theoretical value of the Q-factor is evaluated using the formula
Q=1/RÖL/C
2) Parallel resonant circuit
The circuit connections are made as shown in figure (3). The oscillator is switched on and the output of the oscillator is adjusted to a maximum value which is kept constant throughout the experiment. The frequency is increased in steps and corresponding reading of the current (I) in mA is recorded. Care should be taken to locate resonant frequency (frequency corresponding to minimum current) with maximum accuracy. The graph of circuit current as a function of frequency is plotted (figure 4).
The value of inductance L is verified using the formula L= 1/4p2f02c
A straight line drawn parallel to frequency axis at the point Imin/Ö2 on the Y-axis, cuts the curve at two points corresponds to the frequencies fa and fb on the X-axis.
Q-factor is evaluated from the graph, Q=f0/ fb-fa
The theoretical value of the Q-factor is evaluated using the formula
Q=1/RÖL/C
CALCULATIONS:
1) Series resonant circuit
Resonant frequency f0 =
Inductance L=1/4p2f02c=
From graph, fa = fb=
Bandwidth Df = fb-fa=
Quality factor Q = f0/Df=
Theoretical value Q=1/RÖL/C=
2) Parallel resonant circuit
Resonant frequency f0 =
Inductance L=1/4p2f02c=
From graph, fa = fb=
Bandwidth Df = fb-fa=
Quality factor Q = f0/Df=
Theoretical value Q=1/RÖL/C=
RESULTS:
1) Series resonant circuit
Resonant frequency (a) theoretically =
(b) graphically =
Inductance, L (theoretically) =
Bandwidth =
Q-factor (a) theoretically =
(b) graphically =
2) Parallel resonant circuit
Resonant frequency (a) theoretically =
(b) graphically =
Inductance, L (theoretically) =
Bandwidth =
Q-factor (a) theoretically =
(b) graphically =
1) Series resonant circuit
Resonant frequency f0 =
Inductance L=1/4p2f02c=
From graph, fa = fb=
Bandwidth Df = fb-fa=
Quality factor Q = f0/Df=
Theoretical value Q=1/RÖL/C=
2) Parallel resonant circuit
Resonant frequency f0 =
Inductance L=1/4p2f02c=
From graph, fa = fb=
Bandwidth Df = fb-fa=
Quality factor Q = f0/Df=
Theoretical value Q=1/RÖL/C=
RESULTS:
1) Series resonant circuit
Resonant frequency (a) theoretically =
(b) graphically =
Inductance, L (theoretically) =
Bandwidth =
Q-factor (a) theoretically =
(b) graphically =
2) Parallel resonant circuit
Resonant frequency (a) theoretically =
(b) graphically =
Inductance, L (theoretically) =
Bandwidth =
Q-factor (a) theoretically =
(b) graphically =
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