20.4 Phase of A.C. | Alternating Current | Physics 12

What is Phase?

The phase of a sinusoidal AC quantity is the angle $θ = ω t$ that appears in its mathematical expression. For an alternating voltage:

$V = V_{o} sin (ω t)$

The phase $ω t$ specifies the instantaneous state of the quantity — both its magnitude and direction — at any given moment. The angular frequency $ω = 2 π f$ determines how rapidly the phase advances with time.

Phase Difference

When two AC quantities (e.g., voltage and current) vary sinusoidally at the same frequency but do not reach their peak values at the same instant, they are said to have a phase difference $ϕ$ .

If $V = V_{o} sin (ω t)$ and $I = I_{o} sin (ω t - ϕ)$ , the current lags the voltage by $ϕ$ .
If $I = I_{o} sin (ω t + ϕ)$ , the current leads the voltage by $ϕ$ .

Phasors and Phasor Diagrams

A phasor is a rotating vector:

Its length represents the peak value ( $V_{o}$ or $I_{o}$ ).
Its angular position represents the phase at a given instant.

A phasor diagram shows the relative phase relationships between different AC quantities in a circuit at a glance.

Phase Relationships in Basic AC Circuits

A.C. Through a Resistor

For a resistor $R$ connected to $V = V_{o} sin (ω t)$ :

$I = \frac{V _{o}}{R} sin (ω t) = I_{o} sin (ω t)$

Voltage and current are in phase: $ϕ = 0°$ .
They reach their maximum and minimum values simultaneously.
Power factor $= cos (0°) = 1$ .
Average power: $P = \frac{1}{2} I_{o} V_{o} = I_{r m s} V_{r m s}$ .

A.C. Through an Inductor

For a pure inductor $L$ connected to $V = V_{o} sin (ω t)$ , the back-emf opposes the change in current. The result is:

$I = I_{o} sin (ω t - \frac{π}{2})$

The voltage leads the current by $90°$ (or $π /2$ radians), equivalently the current lags the voltage by $90°$ .
Power factor $= cos (90°) = 0$ .
Average power dissipated $= 0$ (energy is stored and returned each cycle).

A.C. Through a Capacitor

For a pure capacitor $C$ connected to $V = V_{o} sin (ω t)$ , the charge $q = C V$ must build up before the voltage rises. Differentiating:

$I = \frac{d q}{d t} = ω C V_{o} cos (ω t) = I_{o} sin (ω t + \frac{π}{2})$

The current leads the voltage by $90°$ (or $π /2$ radians).
Power factor $= cos (90°) = 0$ .
Average power dissipated $= 0$ .

Summary Table

Circuit Element	Phase of $I$ relative to $V$	Phase Difference $ϕ$	Power Factor
Resistor $R$	In phase	$0°$	$1$
Inductor $L$	$I$ lags $V$	$90°$	$0$
Capacitor $C$	$I$ leads $V$	$90°$	$0$

Key Flashcards

What is Phase?

The phase of a sinusoidal AC quantity is the angle

θ = ω t

that appears in its mathematical expression. For an alternating voltage:

V = V_{o} sin (ω t)

The phase

ω t

specifies the instantaneous state of the quantity — both its magnitude and direction — at any given moment. The angular frequency

ω = 2 π f

determines how rapidly the phase advances with time.

Phase Difference

When two AC quantities (e.g., voltage and current) vary sinusoidally at the same frequency but do not reach their peak values at the same instant, they are said to have a phase difference

ϕ

V = V_{o} sin (ω t)

and

I = I_{o} sin (ω t - ϕ)

, the current lags the voltage by

ϕ

I = I_{o} sin (ω t + ϕ)

, the current leads the voltage by

ϕ

Phase Relationships in Basic AC Circuits

For a resistor

R

connected to

V = V_{o} sin (ω t)

I = \frac{V _{o}}{R} sin (ω t) = I_{o} sin (ω t)

Voltage and current are in phase:

ϕ = 0°

They reach their maximum and minimum values simultaneously.

Power factor

= cos (0°) = 1

Average power:

P = \frac{1}{2} I_{o} V_{o} = I_{r m s} V_{r m s}

For a pure inductor

L

connected to

V = V_{o} sin (ω t)

, the back-emf opposes the change in current. The result is:

I = I_{o} sin (ω t - \frac{π}{2})

The voltage leads the current by $90°$ (or

π /2

radians), equivalently the current lags the voltage by

90°

Power factor

= cos (90°) = 0

Average power dissipated

= 0

(energy is stored and returned each cycle).

For a pure capacitor

C

connected to

V = V_{o} sin (ω t)

, the charge

q = C V

must build up before the voltage rises. Differentiating:

I = \frac{d q}{d t} = ω C V_{o} cos (ω t) = I_{o} sin (ω t + \frac{π}{2})

The current leads the voltage by $90°$ (or

π /2

radians).

Power factor

= cos (90°) = 0

Average power dissipated

= 0

Circuit Element

Phase of

I

relative to

V

Phase Difference

ϕ

Power Factor

Resistor

R

In phase

0°

1

Inductor

L

I

lags

V

90°

0

Capacitor

C

I

leads

V

90°

0