Basic Circuit Theory
Woo-Young Choi
Dept. of Electrical and Electronic EngineeringYonsei University
Lecture 14,15:
Natural Response of RLC Circuits
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
iL = I0
t < 0
t > 0v(t) = ?
KCL iR + iL + iC = 0
(N&R 8.1, 8.2, 8.4)
v = V0
Second-order linear differential equation
Homogeneous solutions (Natural response)
Assume
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
t < 0
t > 0 v(t) = ?
[Neper (attenuation) frequency]
(Characteristic Equation)
(resonance frequency, natural frequency)
iL = I0v = V0
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
t < 0
t > 0 v(t) = ?
Different types of solutions
(1) a > w0
iL = I0v = V0
Over-damped
(2) a < w0 Under-damped
(3) a = w0 Critically damped
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
t < 0
t > 0 v(t) = ?
v(t) = A1exp[(- a + sqrt(a2 - w02)t] + A2exp[(- a - sqrt(a2 - w0
2)t]
A1 =
iL = I0v = V0
(1) a > w0 Over-damped
Initial conditions v(t=0) = V0
iR + iL + iC = 0 v/R + iL + C dv/dt = 0
dv/dt (t=0) = - (I0+V0/R) /C
At t=0 V0/R + I0 + Cdv/dt (t=0) = 0
A2 = V0 - A1
Different types of solutions
iL (t=0) = I0
{- (I0+V0/R)/C + V0 [a + sqrt(a2 - w02)]}/ 2sqrt(a2 - w0
2) ]
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
t < 0
t > 0 v(t) = ?
j = sqrt(-1)
(2) a < w0
sqrt(a2 - w02 ) = j sqrt(w0
2 - a2)
v(t) = A1exp(-at+jwdt) + A2exp(- at – jwdt)
Different types of solutions
= j wd
= exp(- at) [A1exp(+jwdt) + A2exp(-jwdt)]
= exp(- at) [A1(coswdt +jsinwdt)+ A2(coswdt - jsinwdt)]
exp(jwdt) =
= exp(- at) [(A1+A2)coswdt +(A1-A2)jsinwdt]
= exp(- at) (B1coswdt +B2sinwdt)
IL = 0 v = V0
Under-damped
(Euler’s Formula)cos(wdt)+jsin(wdt)
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
t < 0
t > 0 v(t) = ?
IL = 0 v = V0
(2) a < w0
Different types of solutions
B1=V0
v(t) = exp(- at) (B1coswdt +B2sinwdt)
v(t=0) = V0Initial conditions
B2= [aV0 - (I0+V0/R) / C] /wd
dv/dt (t=0) =
Under-damped
- (I0+V0/R) /C
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
t < 0
t > 0 v(t) = ?
(3) a = w0
Different types of solutions
v(t) = D1 t exp(- at) + D2 exp(- at)
v(t=0) = V0 dv/dt(t=0) = - (I0+V0/R) / C
iL = I0v = V0
D1= - (I0+V0/R) / C + aV0 D2 = V0
Critically Damped
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
= 10,000 (1/s)= 12,500 (1/s)
a > w0 Over-damped
v(t) = A1exp[(- a + sqrt(a2 - w02)t] + A2exp[(- a - sqrt(a2 - w0
2)t]
A1 = -14(V) A2 = 26 (V)
A1 = {- (I0+V0/R)/C + V0 [a + sqrt(a2 - w02)]}/ 2sqrt(a2 - w0
2) ] A2 = V0 - A1
s1 = -5000 (1/s) s2 = -20000 (1/s)
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
= 1,000 (1/s)= 200 (1/s)
a < w0 Under-damped
B1 = 0 (V) B2 = 100 (V)
v(t) = exp(- at) (B1coswdt +B2sinwdt)
= 979.8 (1/s)
B1=V0 B2= [aV0 - (I0+V0/R) / C] /wd
sqrt(w02 - a0
2)
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
1,000 (1/s)= 1,000 (1/s)
a = w0 Critically Damped
R = 4kW
1/ (2x4000x0.125x10-6 )
D1=98,000
v(t) = D1 t exp(- at) + D2 exp(- at)
D2= 0
D1= - (I0+V0/R) / C + aV0 D2 = V0
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
R = 4kW
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
(https://www.youtube.com/watch?v=99ZE2RGwqSM&ab_channel=xmdemo)
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
Homework
A. (Optional but strongly suggest. No need to hand- in)
- Assessment Problems: 8.3 8.4, 8.5
B. (Due on 4/ 26pm. Hand in through LearnUs) (Mid-term exam covers up to Lecture 13)
Consider a series RLC circuit shown below. Initially (t=0), the inductor has I0 and the capacitor has V0.
(a) Show
(b) What is the characteristic equation for the above differential equation?
(c) What are the initial conditions for i(t) and di(t)/dt?
(Hint: See Section 8.4 in R&L) (d) What are the three different types of solutions for the above differential equation?
Basic Circuit Theory (2021/1) W.-Y. Choi
Lecture 14,15: Natural Response of RLC Circuits
Homework
(e) Determine i(t) when R= 560W, L=100mH, C= 0.1mF, I0=0, and V0=100V
B. (Due on 4/ 26pm. Hand in through LearnUs)