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)