The document discusses alternating current (AC) through resistors, inductors, and capacitors, detailing their phase relationships and the resulting power calculations. It explains how current behaves in resistive, inductive, and capacitive circuits, noting that power averages to zero in purely inductive or capacitive scenarios, while powers fluctuate in resistive circuits. Examples illustrate how to determine voltage and current equations in these types of circuits.

1. Alternating Current
Resistance, Inductance and
Capacitance

2. AC through resistance, inductance and capacitance
• AC through ohmic resistance alone
Let the voltage v(t) = Vm sin ωt 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 ωt
•Current, i is maximum when sin ωt is unity. So
• Hence i = Im sin ωt
• Alternating voltage and current are in phase with each other

3. AC through resistance, inductance and capacitance
• Power in ohmic resistance alone
Instantaneous power, p = v i = (Vm sin ωt)(Im sin ωt) = Vm Im sin2
ωt
= (Vm Im/2) (1- cos 2ωt) = Vm Im/2 - (Vm Im/2) cos 2ωt
• Power consists of a constant part = Vm Im/2 and
• a fluctuating part (Vm Im/2) cos 2ωt of double frequency
For a complete cycle, the average value of (Vm Im/2) cos 2ωt = 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

4. AC through resistance, inductance and capacitance
• AC through ohmic resistance alone

5. AC through resistance, inductance and capacitance
• AC through ohmic resistance alone
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).

6. AC through resistance, inductance and capacitance
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

7. AC through resistance, inductance and capacitance
AC through pure inductance alone
Now
So
Hence,

8. AC through resistance, inductance and capacitance
AC through pure inductance alone
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 π/2 with voltage leading.
• Here, Im = (Vm/ωL) = (Vm/XL) . Hence ωL 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 ω is in radian/second.

9. AC through resistance, inductance and capacitance
AC through pure inductance alone
•current lags behind the applied voltage by π/2
• inductive reactance XL= ωL
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

10. AC through resistance, inductance and capacitance
AC through pure inductance alone
•current lags behind the applied voltage by π/2
• inductive reactance XL= ωL
• average demand of power from the supply is zero
• Energy is stored in one half cycle and
• releases in the next half cycle

11. AC through pure inductance
Ex: A sinusoidal voltage of v = 100 Sin 20t is
Applied 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 is
Applied across a 0.5 H inductor. What is the
sinusoidal expression for the current?
Inductive reactance,

12. AC through pure inductance

13. AC through resistance, inductance and capacitance
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,

14. AC through resistance, inductance and capacitance
AC through pure capacitance

15. AC through resistance, inductance and capacitance
AC through pure capacitance
In Ohm’s law format
• Capacitive reactance is the opposition to the flow of charge, which
results in the continual interchange of energy between the source and
the electric field of the capacitor.
• Like the inductor, the capacitor does not dissipate energy in any form

16. AC through resistance, inductance and capacitance
AC through pure capacitance

17. AC through resistance, inductance and capacitance
AC through pure capacitance

18. AC through resistance, inductance and capacitance
AC through pure capacitance
Ex: A voltage of v = 3- Sin 400t is applied across a 1 µF capacitor.
What is the sinusoidal expression for the current?

19. AC through resistance, inductance and capacitance
AC through pure capacitance
Ex: A current I = 40 Sin (500t +60 ) is impressed on a 100 µF
capacitor. What is the sinusoidal expression for voltage?

20. AC through resistance, inductance and capacitance
AC through pure capacitance: Power
Instantaneous power
p = vC iC = Vm sin ωt.Im sin(ωt + π/2).
= Vm Im sinωt cos ωt
= ½ Vm Im sin 2ωt
Power for the whole cycle = ½ Vm Im sin 2ωt 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.