Alternating Current sin L v ( . ) This is like the equation for a forced, damped oscillator, [see Eq. { . (b)} in Class XI Physics Textbook].
Let us assume a solution q = q m sin ( ω t + θ ) [ . (a)] so that cos( θ [ . (b)] and sin( θ = − [ . (c)] Substituting these values in Eq.
( . ), we get [ ] cos( ( )sin( L X X θ θ sin v ( . ) where we have used the relation X c = / ω C , X L = ω L . Multiplying and dividing Eq.
( . ) by ( c L Z X X , we have ( cos( sin( L X X Z Z Z θ θ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ sin v ( . ) Now, let cos Z φ and ( sin L X X Z φ so that tan L X X φ ( . ) Substituting this in Eq.
( . ) and simplifying, we get: cos( sin Z v θ φ ( . ) Comparing the two sides of this equation, we see that v Z i Z where i q ω [ . (a)] and θ φ = − or θ φ = − [ .
(b)] Therefore, the current in the circuit is cos( i θ = i m cos ( ω t + θ ) or i = i m sin( ω t + φ ) ( . ) where ( L v v i Z X X [ . (a)] and tan L X X φ Thus, the analytical solution for the amplitude and phase of the current in the circuit agrees with that obtained by the technique of phasors. .
. Resonance An interesting characteristic of the series RLC circuit is the phenomenon of resonance. The phenomenon of resonance is common among systems that have a tendency to oscillate at a particular frequency. This frequency is called the system’s natural frequency .
If such a system is driven by an energy source at a frequency that is near the natural frequency, the amplitude of oscillation is found to be large. A familiar example of this phenomenon is a child on a swing. The the swing has a natural frequency for swinging back and forth like a pendulum. If the child pulls on the rope at regular intervals and the frequency of the pulls is almost the same as the frequency of swinging, the amplitude of the swinging will be large (Chapter , Class XI).
For an RLC circuit driven with voltage of amplitude v m and frequency ω , we found that the current amplitude is given by ( L v v i Z X X with X c = / ω C and X L = ω L . So if ω is varied, then at a particular frequency