# Sinusoidal Response of RLC Circuit

## Sinusoidal Response of RLC Circuit:

Consider a Sinusoidal Response of RLC Circuit consisting of resistance, inductance and capacitance in series as shown in Fig. 12.20. Switch S is closed at t = 0. At t= 0, a sinusoidal voltage V cos (ωt + θ) is applied to the RLC series circuit, where V is the amplitude of the wave and θ is the phase angle. Application of Kirchhoff s voltage law to the Sinusoidal Response of RLC Circuit results in the following differential equation. Differentiating the above equation, we get The particular solution can be obtained by using undetermined coefficients. By assuming Substituting ip, i′p and i″p in Eq. 12.33, we have Comparing both sides, we have

Sine coefficients. Cosine coefficients Solving Eqs 12.38 and 12.39, we get Substituting the values of A and B in Eq. 12.34, we get Putting To find M and Φ we divide one equation by the other Squaring both equations and adding, we get  The particular current becomes The complementary function is similar to that of DC series RLC circuit. To find out the complementary function, we have the characteristic equation The roots of Eq. 12.42, are By assuming K2 becomes positive, when (R/2L)2 > 1/LC

The roots are real and unequal, which gives an overdamped response. Then Eq. 12.42 becomes The complementary function for the above equation is

ic = c1 e(K1 + K2)t  + c2 e(K1 – K2)t

Therefore, the complete solution is

i = ic + ip

= c1 e(K1 + K2)t  + c2 e(K1 – K2)t

K2 becomes negative, when (R/2L)2 < 1/LC

Then the roots are complex conjugate, which gives an underdamped response. Equation 12.42 becomes

[D – (K1 + jK2)] [D – (K1 – jK2)]i = 0

The solution for the above equation is

ic = eK1t [c1 cos K2t + c2 sin K2t]

Therefore, the complete solution is

i = ic + ip

i = eK1t [c1 cos K2t + c2 sin K2t] K2 becomes zero, when (R/2L)2 = 1/LC

Then the roots are equal which gives critically damped response. Then, Eq. 12.42 becomes (D – K1) (D – K1) i = 0.

The complementary function for the above equation is

ic = eK1t (c1 + c2t)

Therefore, the complete solution is

i = ic + ip

i = eK1t (c1 + c2t)

Updated: February 13, 2020 — 10:34 pm