Question

A 215 Ω resistor, a 0.900 H inductor, and a 6.50 μF capacitor are connected in series across a voltage source that has voltage amplitude 29.5 Vand an angular frequency of 220 rad/s .

a) What is v at t= 18.0 ms ? ANS: -27.7V

b) What is vR at t= 18.0 ms ?

c)What is vL at t= 18.0 ms ?

d)What is  vC at t= 18.0 ms ?

e)What is VR?

f)What is VC?

g)What is VL?

Answer 1

a)

w = angular frequency = 220 rad/s

R = Resistance of the resistor = 215 ohm

L = Inductance = 0.900 H

X_L = Inductive reactance = wL = (220)(0.9) = 198 ohm

C = capacitance = 6.50 x 10^-6 F

X_C = Capacitive reactance = 1/wC = 1/((220)( 6.50 x 10^-6)) = 699.3 ohm

phase constant is given as

= tan^-1((X_L - X_C)/R) = tan^-1((198 - 699.3)/215) = - 1.166 rad

V_o = amplitude of the source voltage = 29.5 volts

Voltage at any time is given as

v = V_o Cos(wt + )

v = (29.5) Cos((220) t - 1.166)

at t = 18 ms = 0.018 s

v = (29.5) Cos((220) (0.018) - 1.166)

v = - 27.7 volts

b)

impedance of the circuit is given as

z = sqrt(R² + (X_L - X_C)²)

z = sqrt(215² + (198 - 699.3)²)

z = 545.5 ohm

amplitude of current through the circuit is given as

I_o = V_o/z = 29.5/545.5 = 0.054 A

current at any time through the resistor is given as

i = I_o Coswt

So voltage across the resistor is given as

v_R = iR

v_R = I_oR Coswt

v_R = (0.054) (215) Cos((220)(0.018))

v_R = 11.6 volts

c)

current at any time through the indutor is given as

i = I_o Cos(wt + /2)

So voltage across the inductor is given as

v_L = i X_L

v_L = I_o X_L Cos(wt + /2)

v_L = (0.054) (198) Cos((220)(0.018) + 1.57)

v_L = 10.6 volts

d)

current at any time through the capacitor is given as

i = I_o Cos(wt - /2)

So voltage across the inductor is given as

v_C = i X_C

v_C = I_o X_C Cos(wt - /2)

v_C = (0.054) (699.3) Cos((220)(0.018) - 1.57)

v_C = 37.7 volts

e)

V_R = I_oR = (0.054) (215) = 11.6 volts

f)

V_C = I_o X_C = (0.054) (699.3) = 37.8 volts

g)

V_L = I_o X_L = (0.054) (198) = 10.7 volts