An ac voltmeter, which displays the rms voltage between the two points touched b

Question

An ac voltmeter, which displays the rms voltage between the two points touched by its leads, is used to measure voltages in the circuit shown in the figure. In this circuit, the ac generator has an rms voltage of 16.00 V and a frequency of 15.0 kHz. C1 = 65nF, C2 = 45 nF, R1 = 7.53 , R2 = 6.55 · (a) what should be the inductance of the coil that sets the circuit to resonate with the power supply? (b) What is the power delivered by the power supply? L& CL R 2. Now, the ac power generator is set to 48.0 V and frequency, f = 50.0 kHz; A coil of L= 0.35-mH inductance is connected between A and B, a capacitor having a 42.0-nF capacitance between B and C, and a resistor with a resistance of 550- between C and D. (c) Calculate the voltage across the coil, Vas, (d) across the capacitance, Vsc, (e) across the resistor, Vco, and (f) the voltage between A and D, VAD. (g) What is the power supplied by the circuit? (h) What should the frequency of the power supply be so that it will be at resonance with the circuit? (i) Deduce the current at resonance.

Explanation / Answer

equivalent capacitance

1/C =1/65+ 1/45

C=26.59 nF

Resonant frequency is given

fr=1/2pi*sqrt(LC)

L=1/4pi2f2C =1/4pi2*150002*(26.59*10-9)

L=4.23*10-3 H or 4.23 mH

b)

equivalent resistance

1/R =1+/7.53 + 1/6.55

R=3.503 ohms

Power delivered to the supply

P=EI =16*(16/3.503) =73.04 watts

c)

Inductive reactance

XL=2pifL =2pi*(50000)*(0.35*10-3) =109.956 ohms

Capacitive reactance

XC=1/2pifC =1/2pi*50000*(42*10-9) =0.0132 ohms

Impedance

Z=sqrt[R2+(XL-XC)2]=sqrt[552+(109.956-0.0132)2]=122.932 ohms

Total current

I=V/Z =48/122.932=0.39 A

Voltage across the coil

VAB=IXL=0.39*109.956 =42.93 Volts

d)

Voltage across capcitor

VBC=IXC=0.39*(0.0132) =0.00515 Volts

e)

Voltage across resistor

VCD =IR =0.39*55=21.5 Volts

f)

VAD=48 Volts

g)

Phase angle

o=tan-1(109.956-0.0132/55)=63.42o

Power supplied

P=VICos(o) =48*0.3904*Cos63.42=8.384 Watts

h)

fr=1/2pisqrt(LC) =1/2pisqrt(0.35*10-3*42*10-9)

fr=41511 Hz =41.51 kHZ

i)

I=48/55 =0.8727 A

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