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# Voltage and Current Relationship [AC Circuit]

Introduction

At constant temperature, the current flowing through the conductor is directly proportional to the potential difference (voltage) between the two ends of the conductor. This is ohm’s law.
The relation may be written as,

V = IR

Where V and I now vary with time. When a sinusoidal if I = Imsinώt is applied to a resistor the potential across it.
E = Im R sinώt
= Vm sinώt

VOLTAGE AND CURRENT RELATIONSHIP [A.C.CIRCUIT]

A.C through pure resistor only

A pure resistive circuit is shown in fig, the alternating voltage applied across the resister is
V= Vm sinώt
Let ‘I’ be the alternating current through the circuit
VR= voltage drop across the resister.
V = I ×R
I = V / R
I = Vmsinώ t /R
I = Im sinώ t ------------------------ 2
Im sinώ t = Vmsinώ t / R
Im = Vm sinώ t / R sinώ t,
Im = Vm/ R
From the equ. 1 and 2, the phase angle voltage and current is zero.
Ө = 0

In pure resistive circuit, the circuit is in phase with the voltage .the waveform and vector diagram are shown in the fig (a) and (b)
Power
Instantaneous, power, P = V x I
= Vmsinώ t × Im sinώ t
= Vm Im sin2ώ t

Power = VI watts

Where V and I are R.M.S values

Phase Angle

Phase angle is an angle between the voltage and current.
In a pure resistive current, the voltage and the current in phase with each other hence the phase angle is zero.

Power factor

Power factor = cos
= cos θ [θ=0]
= 1 (Unity)
Power factor is also defined as the ratio of resistance to impedance
cos θ R/Z
Its value cannot be more than one

A.C through pure inductor only

Pure inductive current is shown in fig

When an alternating current flows through a pur inductive coil a back e.m.f is induced due to the inductance of coil. This e.m.f opposes the aplied voltage at every instant since there is no resistance the induced e.m.f will be equal and opposite to the applied voltage.

i.e., Applied voltage = back e.m.f

Applied voltage, V = Vm sinώ t ------------------- 1

Im = Vm/ώL

from equ.2 and 3, we find that the current. Lags behind the applied voltage by 90˚.
the waveform and vector diagram are shown in fig (a) and (b)

Power (P)

V= Vmsinώ t, i= Im sin (ώt-90)
Instantaneous power= V×I= Vmsinώ t× Im sin (ώt-90) = VmIm sin2ώ t

Phase angle

Phase angle is an angle between the voltage and current

In inductive circuit, the angle between the voltage and current. is 90˚
i.e. θ =90˚ The current is always lagging the voltage by 90˚

Power factor

Power factor = cosθ
= cos (90˚)
= 0 (lag) ( θ=90˚ )

AC through pure capacitor only

The circuit shown in the fig is a pure capacitive circuit.

When an alternating voltage is applied to a capacitor the capacitor is charged first in one direction and then in the opposite direction.
V= Vmsinώ t ------------- 1
Q = CV= C Vmsinώ t
Current ,i = dQ/dt
= d(C Vm sinώ t)/dt
= C Vm sinώt . ώ =ώ C Vm sinώ t
= ώ C Vm sinώ t (ώt-90)
I= Im sin (ώt-90) ------------- 2
Im sin (ώt-90) = ώ C Vm sinώ t (ώt-90)
Im = ώ C Vm (or)
Im = Vm/(1/ ώ C)
From the equ, 1 and 2, we find that the current leads the voltage by 90˚ the waveform and vector diagram. Are shown in fig. (a) and (b)

Power (p)

V= Vmsinώ t,
I = Im sin (ώt+90)
Instantaneous power= V x I = Vmsinώ t x Im sin (ώt+90)

Phase angle

Phase angle is an angle between the voltage and current
In inductive circuit, the angle between the voltage and current. is 90˚
i.e. θ =90˚
The current is always lagging the voltage by 90˚

Power factor

Power factor = cosθ
= cos (90˚)
= 0 (lead)

Important Terms:

1. Impedance

It is ratio of the applied voltage to the resulting current.
It is represented by the letter ‘Z’.
Its unit is Ohm.( Ω)

i.e. Z = V/ohm.

2. Admittance

It is defined as the reciprocal of impedance.
It is represented by the letter ‘Y’.
Its unit is mho.

i.e. Y = 1/ I/V mho

3. Reactance

Inductive reactance, Xl =2πfl
Where f =freq,
L=inductance of the coil
Capacitive reactance, Xc = 1 / 2πfc
Where C= capacitance of the capacitor
Unit of the reactance is Ohm

4. Susceptance

It is defined as the reciprocal of reactance.
It is represented by the letter ‘b’ Its unit is mho

B = 1 / X mho

5. Conductance

It is defined as the reciprocal of resistance.
It is represented by the letter ’g’. Its unit is ‘mho’.

G= 1/R mho