AC Circuits?

Question

A particular inductor has appreciable resistance. When the inductor is connected to a 16 V battery, the current in the inductor is 3 A. When it is connected to AC source with an rms output of 16 V and a frequency of 62 Hz, the current drops to 1 A. Find the impedance at 62 Hz. Answer in ohms.

Answer 1

The resistive component of the impedance with DC excitation is R =E/I = 16/3 = 5.333… ohms.
Impedance is calculated the same way with AC excitation at 62 Hz: Z = E/I = 16/1 = 16 ohms.
Impedance, being a complex number, has a polar angle equal to arc cos |R/Z| = arc cos (16/3) /(16) = arc cos 1/3 = 70.53 degrees.

You didn’t ask, but from these two measurements you can now calculate the inductive reactance, the imaginary component of the complex impedance that is at right angles to the resistive component:

X = square root (Z^2 – R^2)
= sqrt (16^2 – (16/3)^2)
= sqrt (16^2 (1 – 1/9))
= sqrt (16^2 (8/9))
= 16 sqrt (8/9)
= 16/3 sqrt (8)
= 16/3 srrt (2 x 4)
= (16)(2) /(3) sqrt 2
= (10.667)(1.414)
= 15.085 ohms (everything rounded to three decimal places for no particular reason)

Knowing the inductive reactance at 62 Hz, you can calculate the inductance, which is the usual reason for making the two separate AC and DC measurements:

X = (2)(pi)(f)(L) or, after solving for L:

L = X / (2)(pi)(f) = 15.085 / (2) (3.1416) (62) = 0.0387 Hy

Answer 2

Z = R of L+X of L and Z = e / i then Z =16/1 = 16 ohms. This is the total impedance of the inductor. One third of this impedance is resistive and two thirds is inductive reactance.We know this since the current dropped from 3 amps to 1 amp when we switched from DC to 62 Hz ac. Resistance stays the same for all frequencys therefore the added impedance had to be inductive reactance.

Answer 3

When the DC source is connected to the inductor, its inductance has got no effect on the current. It will act just like any other resistance.
DC Voltage = 16 V
The current = 3 Amps
Hence the ohmic resistance of the material of the inductor will be = V / I = 16 / 3 = 5.3Ω
When AC is connected the inductive reactance too will come into play. So the combined effect of ohmic resistance and the inductive reactance will come into play. This combined opposing force is called IMPEDANCE. Its unit is also 'ohms'
RMS value of the AC voltage = 16V
Frequency of the AC voltage = 62 Hz
The current in the AC circuit = 1 Amps
The impedance at 62 hertz = V / I = 16 / 1 = 16 ohms
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The impedance "Z" = √(R² + (XL)², where X = 2 x π x f x L, f,the frequency in hertz, L, the inductance in henries(H)'

Answer 4

Super-simple.

Z =V / I = 16 V / 1 A = 16 Ω.