Question

You have a 59.0 µH inductor, a 59.0 µF capacitor, and a variable frequency AC source. Determine the resonant frequency at which the inductor and capacitor have the same reactance.

	a.	2698
	b.	3460
	c.	10610
	d.	5895
	e.	4974
	f.	none of the answers

Answer 1

The resonant frequency is the one at which the impedence of the circuit becomes minimum. This impedence becomes minimum when reactance due to capacitor (Xc) is equal to that of the reactance of the inductor(XL) i.e

Xc = XL ..........eqn 1

Xc = 1/wC and XL = wL

where w is the angular frequency, L is the magnitude of inductance and C is the magnitude of capacitance

Using values of Xc and XL in eqn 1, we get

1/wC = wL

or 1/LC = w2

Writing w = wR = resonant frequency; we get

1/LC = wR2

or wR = (1/LC)1/2

Here, L =59.0 uH = 59*10-6 H ( 1 uH = 10-6 H )

Also, C = 59.0 uF = 59*10-6 F

Using values of L and C in expression for wR, we get

wR = (1/59*10-6*59*10-6)1/2

wR = (1/3481*10-12)1/2

= (0.00028727*1012)1/2

wR = 0.016949*106 Hz

Also, wR = 2pie fR

where fR is the resonant frequency.

Hence, 2*pie*fR = 0.016949*106

fR = 0.016949*106 / 2*3.142

= 0.00269716*106 Hz

fR = 2697.16 Hz .............required resonant frequency

The closest option to this result is option a .

Hence, option a is the correct answer.