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CBSE Class 12-science Answered

EXPLAIN  LC   OSCILLATIONS  AND  ALSO  WATTLESS  AND  WATTFULL  CURRENT  WITH  THE  HELP  OF  DIAGRAM 
Asked by pardeepkumar2281 | 04 Oct, 2018, 09:44: PM
answered-by-expert Expert Answer
Capacitor stores electrical energy. Inductor stores magentic energy.
when a initially charged capacitor is connected to inductor, as shown in figure, the charge on the capacitor and
current in the circuit exhibit the phenomenon of electrical oscillations similar to oscillation in mechanical system.

Let a capacitor be charged qm  and connected to inductor at t=0.  The charge on the capacitor keep on decreasing
giving rise to currentin the circuit. Let q be the charge at capacitor and i be the current in the circuit at given time t.
 
Since di/dt is positive, induced emf in inductor will have polarity as shown in figure.
 
By Kirchoffs rule we have
 
begin mathsize 12px style q over c space minus space L fraction numerator d space i over denominator d space t end fraction space equals space 0 end style ...........................(1)
i = -(dq/dt) in this case ( as charge decreases current increases ), then we have
 
begin mathsize 12px style fraction numerator d squared q over denominator d t squared end fraction space plus space fraction numerator q over denominator L C end fraction space equals space 0 space end style  ............................(2)
Eqn.(2) is similar to Simple Harmonic Motion, hence charge and current oscillates with natural frequency begin mathsize 12px style omega space equals space bevelled fraction numerator 1 over denominator square root of L C end root end fraction space end style
Solution to eqn.(2) gives the charge at a given time t ,
 
begin mathsize 12px style q left parenthesis t right parenthesis space equals space q subscript m cos open parentheses omega t close parentheses space
c u r r e n t space i space equals space minus fraction numerator d q over denominator d t end fraction space equals space i subscript m sin left parenthesis omega t right parenthesis end style
 
Initially all the energy is stored in the form of electrical energy in the capacitor.
Then charge keep on decreasing and current will be established in the circuit.
Now inductor start storing the energy in the form of magnetic field.
When all the energy is transferred and no charge is present in the capacitor, capacitor starts charging in the
reverse direction using the stored energy of inductor till all the energy is transferred from inductor to capacitor.
Then the cycle repeats and oscillation is established in the circuit in this way.
-----------------------------
In RLC circuit, applied voltage v = vm sin(ωt) drives the current i = im sin(ωt+φ)
 
power in the circuit, p = v×i = vm im sin(ωt) sin(ωt+φ)  ....................(2)
 
where φ is phase angle given by, begin mathsize 12px style phi equals tan to the power of negative 1 end exponent open parentheses fraction numerator X subscript C minus X subscript L over denominator R end fraction close parentheses end style, XC and XL are capacitive and inductive reactance
 
simplifying by trignometry and averaging eqn.(2) over one cycle, we get average power as

begin mathsize 12px style p space equals space fraction numerator v subscript m i subscript m over denominator 2 end fraction cos phi space equals space fraction numerator v subscript m over denominator square root of 2 end fraction fraction numerator i subscript m over denominator square root of 2 end fraction cos phi space equals space V cross times I cross times cos phi space.............. left parenthesis 3 right parenthesis end style
average power dissipation depends on phase angle φ in addition to current and voltage.
In a purely inductive and capacitive circuit, phase angle φ=90º, that makes power dissipation zero though
there is a current circulating in the circuit. This current is known as wattless current.
 
In RLC circuit, I×cosφ is known as wattfull current

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