Question

An ac circuit consists of a 1.00 x 102 Ω resistor, a 1.00 x 10-1 H inductor, and a 1.00 x 10-5 F capacitor connected in series with each other and with a generator that supplies a voltage ΔVT = 140 V sin (500t + φ ).

1. Find the following quantities: (a) the impedance, (b) the rms current, (c) the rms voltage drops across the: resistor, inductor and capacitor (d) the average power loss, (e) the phase angle, (f) the power factor, (g) the resonant frequency

Please Show Work:

Answers are: (a) 180 Ω (b) 0.55 A (c) 55.0 V, 27.5 V, 110 V (d) 30.3 W (e) -56.3° (f) 0.555 (g) 159 Hz

2. For the ac circuit above, draw a phasor diagram showing the following phasors and their position relative to each other: ΔVR, ΔVL, ΔVC, ΔVT. Also draw the current phasor i and the phase angle of the circuit φ.

Answer 1

The impedance of a series LCR circuit is√{R²+(x_l-x_c)^2}

Z=√100^2+(10^-2-10^-5)²

=√10000+2500×0.09999

=√32761=181ohms

RMS current=i_o/√2

  _{=(140/180)/√2=0.77777/√2=0.55A}

i_rms=0.55A

V_{rms across the resistance=}i_rms×100=55v

V_rms across the inductor=V_rms/2

55/2=27.5ohms

V_rms across the capacitor=V_rms×2=110ohms

Phase angle=tan^-1(v_l-v_c)/v_r

tan^-1(27.5-110)/55

Tan^-1(-1.5)

-56.3

Average power loss=v_o/√2.i_o/√2.cos¢

V_o I_{o cos(-56.3)/2}

_0.55wats

_{Resonant frequency=1/2π√LC}

   _{=1/2π×10^-3=1000/6.28=!59.2Hz}

_=v