An alternating emf source with a variable frequency f d is connected in series w

Question

An alternating emf source with a variable frequency fd is connected in series with an 82.0 ? resistor and a 42.2 mH inductor. The emf amplitude is 6.06 V. Consider a phasor diagram for phasor VR (the potential across the resistor) and phasor VL (the potential across the inductor). (a) At what driving frequency fd do the two phasors have the same length? At that driving frequency, what are (b) the phase angle in degrees, (c) the angular speed at which the phasors rotate, and (d) the current amplitude?

Explanation / Answer

a)

Since VR and VC have same length

VR=Vc

I*R=I*XL

R=2pi*fd*L

=>fd=R/2pi*L=82/(2pi*42.2*10-3)

fd=309.26 Hz

b)

Phase angle

o=tan-1(XL/R) =tan-1(2pi*309.26*42.2*10-3/82)

o=45o

c)

angular frequency

W=2pi*309.26=1943.14 rad/s

d)

Impedance

Z=sqrt[R2+XL2]=sqrt[822+(2pi*309.26*42.2*10-3)2]

Z=115.97 ohms

Current in the circuit

I=V/Z =6.06/115.97

I=0.05226 or 52.26 mA

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