Ac current

Alternating Current

Resistance, Inductance and Capacitance

AC through resistance, inductance and capacitance

• AC through ohmic resistance alone

Let the voltage v(t) = Vm sin t is applied to a pure resistor R

Let i = instantaneous current.

Applied voltage has to supply ohmic voltage drop only.

Hence, for equilibrium, v = iR = Vm sin t

•Current, i is maximum when sin t is unity. So

• Hence i = Im sin t

• Alternating voltage and current are in phase with each other


• Power in ohmic resistance alone

Instantaneous power, p = v i = (Vm sin t)(Im sin t) = Vm Im sin2t = (Vm Im/2) (1- cos 2t) = Vm Im/2 - (Vm Im/2) cos 2t

• Power consists of a constant part = Vm Im/2 and

• a fluctuating part (Vm Im/2) cos 2t of double frequency

For a complete cycle, the average value of (Vm Im/2) cos 2t = 0 Hence, power for the whole cycle is P = (Vm Im/2) = (Vm/2) Im/2)

P = V x I watt

Where V = rms value of applied voltage and I = rms value of the current





Ex: A 60 Hz voltage of 115 V (r m s) is impressed on a 10 Ω resistor.(i) Write down the time expression (equation) of voltage v(t) and the resulting current i(t). Sketch v(t) and i(t).


AC through pure inductance alone

• Whenever an alternating voltage is applied to a purely inductive coil, a back emf is produced due to the self inductance of the coil.

• The back emf at every step, opposes the rise or fall of current through the coil.

• The applied voltage has to overcome this self induced emf only as there is no ohmic voltage drop,

So at every step



Now

So

Hence,



If applied voltage Then resulting current

Vectorially

• Clearly, the current lags behind the applied voltage by a quarter cycle (as shown, or the phase difference between is /2 with voltage leading.

• Here, Im = (Vm/L) = (Vm/XL) . Hence L plays the part of resistance. It is called the inductive reactance XL of the coil and is given in ohms if L is in henry and is in radian/second.



•current lags behind the applied voltage by /2 • inductive reactance XL= L

Power

• It is clear that the average demand of power from the supply for a complete cycle is zero. Power wave is a sine wave of double frequency. Maximum instantaneous power is



•current lags behind the applied voltage by /2 • inductive reactance XL= L • average demand of power from the supply is zero• Energy is stored in one half cycle and • releases in the next half cycle

AC through pure inductance

Ex: A sinusoidal voltage of v = 100 Sin 20t isApplied across a 0.5 H inductor. What is the sinusoidal expression for the current?

Inductive reactance,

Ex: A sinusoidal voltage of v = 100 Sin 20t isApplied across a 0.5 H inductor. What is the sinusoidal expression for the current?

Inductive reactance,

AC through pure inductance


AC through pure capacitance

Whenever an alternating voltage is applied to the plates of a capacitor, the capacitor is charged first in one direction and then in the opposite direction

We know,





In Ohm’s law format

• Capacitive reactance is the opposition to the flow of charge, whichresults in the continual interchange of energy between the source andthe electric field of the capacitor. • Like the inductor, the capacitor does not dissipate energy in any form







Ex: A voltage of v = 3- Sin 400t is applied across a 1 µF capacitor.What is the sinusoidal expression for the current?



Ex: A current I = 40 Sin (500t +60 ) is impressed on a 100 µF capacitor. What is the sinusoidal expression for voltage?


AC through pure capacitance: Power

Instantaneous power p = vC iC = Vm sin t.Im sin(t + /2).

= Vm Im sint cos t

= ½ Vm Im sin 2t Power for the whole cycle = ½ Vm Im sin 2t dt = 0 So, in a purely capacitive circuit, the average power demand from power supply is zero. Also power wave is a sine wave of frequency double that of the voltage and current waves. Maximum value of the instantaneous power is = Vm Im/2.

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Ac current