The current, I, in an RLC circuit is given by the equation: R d’I dI + 2a +®;I = 0 (1) dt? dt V C where the damping factor, a, and resonant frequency, w, =Vo² +a² , are dependent on the values of the resistor, R, capacitor, C, and inductor L: 1 R and @. 2L a = VLC A solution of equation (1) is given by I(t) = e -at sin ot (2) (a) Determine the first and second derivatives of I(t) and hence show that /(t) is a solution of equation (1). (b) Show that the stationary points of I(t) =e -at sin ot occur at time t: tan ot = (c) Determine the partial derivatives ôw, ôw.
The current, I, in an RLC circuit is given by the equation: R d’I dI + 2a +®;I = 0 (1) dt? dt V C where the damping factor, a, and resonant frequency, w, =Vo² +a² , are dependent on the values of the resistor, R, capacitor, C, and inductor L: 1 R and @. 2L a = VLC A solution of equation (1) is given by I(t) = e -at sin ot (2) (a) Determine the first and second derivatives of I(t) and hence show that /(t) is a solution of equation (1). (b) Show that the stationary points of I(t) =e -at sin ot occur at time t: tan ot = (c) Determine the partial derivatives ôw, ôw.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hi, can you help me with all parts of this question, parts a,b,and c?
thank you
Transcribed Image Text:Question 1
The current, I, in an RLC circuit is given by the equation:
R
d²I
dI
+ 2a
+ wI = 0
(1)
dt?
dt
where the damping factor, a, and resonant frequency, o, = Vo? +a? ,
are dependent on the values of the resistor, R, capacitor, C, and
inductor L:
1
R
and @o
2L
a =
VIC
A solution of equation (1) is given by
-at
I(t) = e-" sin ot
(2)
(a) Determine the first and second derivatives of /(t) and hence show that /(t) is a
solution of equation (1).
(b) Show that the stationary points of I(t) = e" sin wt occur at time t:
tan ot =
(c) Determine the partial derivatives
De , Te
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