An a.c. source of voltage V = Vm sin wt is connected, one-by-one, to three circuit elements X, Y and Z. It is observed that the current flowing in them, (i) is in phase with applied voltage for element X. (ii) lags the applied voltage, in phase, by /2 for element Y.
(iii) leads the applied voltage, in phase, by /2 for element Z. Identify the three circuit elements. Find an expression for the (a) current flowing in the circuit, (b) net impedance of the circuit, when the same a.c. source is connected across a series combination of the elements X, Y and Z. (c) If the frequency of the applied voltage is varied, set up the condition of frequency when the current amplitude in the circuit is maximum. Write the expression for this current amplitude.
Answers
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Aꪀswer:-
a) i) circuit element X is Resistance R
⠀ii) circuit element Y is capacitance C
⠀iii) circuit element Z is capacitance L
➪ I = Iₘ sinwt ( For R )
➪ I = Iₘ sin( wt- π/2 ) ( For L )
➪ I = Iₘ sin( wt + π/2 ) ( For C )
b) let a resistance R , capacitance C and inductance L be connected in series to a source of alternating EMF ., as shown in figure (a) .since R,L and C are in series ,therefore ,current at any instant through three elements has the same amplitude and phase let it be given as I = I₀
However, voltage across each element bears a different phase relationship with the current
Now ,
i) the maximum voltage across R is
➪ Vᵣ = I₀ R
____________ Fig (a) _____________
in figure (b) current Phasor I₀ is the represent along OX
____________ Fig (b) _____________
as Vᵣ is in phase with current ,it is represented by the vector OA, along OX
ii) the maximum voltage across L is
➪ Vₗ = I₀ Xₗ
As voltage across the inductor leads the current by 90°, it is represented by a OB Along, 90° ahead of I₀
iii) the maximum voltage across C is
➪ Vc = I₀Xc
as voltage across the capacitor lags behind the alternating current by 90° ,it is represented by OC rotated clockwise through 90° from the direction of I₀ , OC is along a OY' is along OY'
As voltage across the capacitor lags behind the alternating current by 90° ,it is represented by rotated clockwise through 90 ° from the direction of I₀ ,OC is along OY' is along OY'
As the voltage across L and C have a phase difference of 180 degree, the net reactive voltage is ( Vₗ - Vc ) , assuming that Vₗ > Vc
In figure (a) and (b) it is represented by OB' . the resultant of the resultant of OA and OB' is the diagonal OK of the rectangle OAKB' .hence the vector sum of Vᵣ ,Vₗ and Vc is phasor E₀
representation by OK, making and angle ϕ with current Phasor I₀
As ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀OK = √OA² + OB²
➪ E₀ = Vᵣ² + ( Vₗ - Vc )²
➪ E₀ = ( I₀ R ) ² + (I₀ Xₗ - I₀ Xc )²
➪ E₀ = I₀ R² + (Xₗ - Xc)²
The total effective resistance of RLC circuit is called impedance of the circuit .it is represented by Z ,where
c) When the current amplitude in the circuit is maximum then Xₗ = Xc
➪ 2π f₀ L = 1/2πf₀ C
➪ f_0 = 1/2π√LC
⛥ where f₀ is called Resonant Frequency
@Aꪀցꫀl_
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Answer:
An a.c. source of voltage V = Vm sin wt is connected, one-by-one, to three circuit elements X, Y and Z. It is observed that the current flowing in them, (i) is in phase with applied voltage for element X. (ii) lags the applied voltage, in phase, by /2 for element Y.
(iii) leads the applied voltage, in phase, by /2 for element Z. Identify the three circuit elements. Find an expression for the (a) current flowing in the circuit, (b) net impedance of the circuit, when the same a.c. source is connected across a series combination of the elements X, Y and Z. (c) If the frequency of the applied voltage is varied, set up the condition of frequency when the current amplitude in the circuit is maximum. Write the expression for this current amplitude.