Chapter 5

Question 1

Resonant Electric Circuits

Differential Equation

Answer 1

LI’’ + RI’ + (1/C)I = E’

Question 2

Resonant Electric Circuits

Attenuation Factor

Answer 2

α = R/2L

Question 3

Resonant Electric Circuits

Resonant Frequency

Answer 3

ω0 = 1 / √(LC)

Question 4

Resonant Electric Circuits

Complimentary Function

Answer 4

Icf = e^(-αt) (C1cos(tωd) + C2sin(tωd))

Question 5

Resonant Electric Circuits

Damped Resonant Frequency

Answer 5

ωd = √(ω0² - α²)

Question 6

Resonant Electric Circuits

Particular Integral

Answer 6

-to calculate the particular integral you need to know the form of E(t)
-if Ipi is trigonometric
-Ipi = Acosωt + Bsinωt
can also be written as
Ipi = sin(ωt + arctan(A/B)) * √(A² + B²)

Question 7

Resonant Electric Circuits

General Solution

Answer 7

I = Icf + Ipi

• but Icf tends to 0 as t tends to ∞
• so only Ipi remains after Icf has decayed away

Question 8

Resonant Electric Circuits

Amplitude of Particular Integral

Answer 8

Amplitude = √(A² + B²)
= E0/L * ω / (√[(ω² - ω0²)² + 4α²ω²])
-amplitude is maximum when ω = ω0