Network Analysis

Muhammad Irfan Yousuf (Peon of Holy Prophet (P.B.U.H)) cos ω t: ω A + [1/CR1]B = 0 ⇒ A = -B/ω CR1 sin ω t: -ω B + [1/CR1]A = I0/C -ω B + [1/CR1][-B/ω CR1] = I0/C B = I0/C[-ω - 1/C2R12] A = -[I0/C[-ω - 1/C2R12]]/ω CR1

2000-E-41

118

Complete solution V(t) = Ke-(1/CR1)t + [-[I0/C[-ω - 1/C2R12]]/ω CR1]sin ω t + [I0/C[-ω - 1/C2R12]]cos ω t At t = 0+ V(0+) = Ke-(1/CR1)0+ + [-[I0/C[-ω - 1/C2R12]]/ω CR1]sin ω (0+) + [I0/C[-ω - 1/C2R12]]cos ω (0+) I0sin ω t [R1 + R2] = K(1) + [I0/C[-ω - 1/C2R12]] I0sin ω t [R1 + R2] = K + [I0/C[-ω - 1/C2R12]] K = I0sin ω t [R1 + R2] - [I0/C[-ω - 1/C2R12]] V(t) = [I0sin ω t [R1 + R2] - [I0/C[-ω - 1/C2R12]]]e-(1/CR1)t + [-[I0/C[-ω 1/C2R12]]/ω CR1]sin ω t + [I0/C[-ω - 1/C2R12]]cos ω t Q#6.29: Consider a series RLC network which is excited by a voltage source. 1. Determine the characteristic equation. 2. Locus of the roots of the equation. 3. Plot the roots of the equation. Solution:

R V(t) i(t)

L C

For t ≥ 0 According to KVL di 1 L + ∫ idt + Ri = V(t) dt C Differentiating with respect to ‘t’ d2i i di

Muhammad Irfan Yousuf (Peon of Holy Prophet (P.B.U.H)) L

2000-E-41

119

+ +R =0 dt2 C dt Dividing both sides by ‘L’ d2i i Rdi + + =0 (i) dt2 LC Ldt The characteristic equation can be found by substituting the trial solution i = est or by the equivalent of substituting s2 for (d2i/dt2), and s for (di/dt); thus 1 s + 2) ζ =0 jω jω
2

R s=0

+ LC L

n

ζ =1 ζ→∞ σ

ζ →∞ -jω ζ =0

n

1 s2 + +

R s=0

LC L Characteristic equation: as2 + bs + c = 0 Here a b c -b ± √b2 – 4ac 2a

1 R L 1 LC

s1, s2 =

Muhammad Irfan Yousuf (Peon of Holy Prophet (P.B.U.H))

2000-E-41

120

R L s1, s2 = ±

R L

2 4(1)

1 LC 2(1)

R L s1, s2 = 2 R L s1, s2 = 2 R = 2L ± ± ±

R L

2 4(1)

1 LC 2

R L

2 4(1)

1 LC 4

R 2L

2 4(1)

1 4LC

R = 2L ±

R 2L

2 -

1 LC radical term (ii)

Hint: 4 = 2 To convert equation (i) to a standard form, we define the value of resistance that causes