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ALTERNATING CURRENT

 What is alternating current?

Alternating current is described as an electric current, that reverses its direction with a constant frequency. The graph of the current against time in the case of AC current is a sine curve.

Thus, we see that alternating current, unlike direct current, is a continuously varying. The direction of alternating current also reverses periodically. Its magnitude is given either, as its peak value I_0, or its root mean square value I₀/2^½.

An alternating current is represented, as a function of time as, i = i₀sinwt. Where i is the instantaneous value of the current at time t and i_o is the peak value of the current. Here w is called the angular frequency of the alternating current. w is given by 2p/t = 2pf.

Alternating emf is the emf or the voltage, which produces an alternating current. Alternating emf is thus defined as the emf or voltage, whose magnitude change continuously wit time between zero and a positive value. The direction of alternating emf also reverses periodically. Like alternating current its instantaneous value is given by

E = E0Sinwt

REACTANCE

Reactance is a property of a circuit, containing an inductance, a capacitor or both, that together with any resistance makes up its impedance. The impedance Z is given by

Z2 = R2 + X2

 

Where R is the resistance and X is the reactance. For a pure capacitor, X is represented by X_C and is given by X_C = 1/Cw. For a pure inductive circuit it is represented by X_L and is given by X_L = Lw. For a circuit which contains both a capacitor and an inductor, X is given by X = X_L - X_C . The unit of reactance is the same as resistance. It is expressed in ohms.

IMPEDANCE

It is described as the quantity that measures the opposition of a circuit to the passage of current and therefore determines the amplitude of current. It can be taken as a resistance in a DC circuit. In an AC circuit it is given by

Z² = R² + X²

The impedance in the complex form is given by Z = R + iX, here i is -1^½. The real part of the complex impedance is the resistance, and it represents the loss of power according to Joule’s Law. The ratio of the real part to the imaginary part is an indication of the difference in phase between the voltage and current.

NOTE: In different types of AC circuits containing a induction, a capacitor, a resistance or a combination of these the impedance is used in the same way as resistance is used in a DC circuit.

PHASOR DIAGRAMS

What is a phasor?

A phasor is a rotating vector, used to represent a varying, sinusoidal quantity. The length of a phasor represents the amplitude of the quantity and it is imagined to rotate with angular velocity equal to the angular frequency of the quantity. The instantaneous value of the quantity is represented by its projection upward on a fixed axis.

In the study of AC circuits, if we treat alternating current and the alternating emf as vectors, it simplifies our understanding. The phase difference between the current and emf is represented as the angle between these vectors. These vectors are thus called phasors, and the diagram representing alternating current and alternating voltage as vectors with the phase angle between them is called a phasor diagram.

DIFFERENT TYPES OF AC CIRCUITS

1. CIRCUIT CONTAINING ONLY A RESISTANCE

   If the Alternating emf is given by E = E₀Sinwt , then the corresponding instantaneous current in the circuit is given by

    

   i = E0Sinwt /R

   Where R is the resistance of the circuit. The peak value of current in this type of circuit is given by

    

   i = E0/R

    

   In this case, the current is always in phase with the applied emf.

2. CIRCUIT CONTAINING ONLY AN INDUCTANCE

If the Alternating emf is given by E = E₀Sinwt, then the corresponding instantaneous current in the circuit is given by

 

i = (E0 /wL) Sin(wt - p/2)

Where L is the inductance of the circuit. The peak value of current in this type of circuit is given by

 

i = i0sinwt - p/2

 

In this case, the current lags behind the emf by a phase angle of p/2. We can also say that the voltage leads the current by a phase difference of p/2.

3. CIRCUIT CONTAINING ONLY A CAPACITANCE

If the Alternating emf is given by E = E₀Sinwt, then the corresponding instantaneous current in the circuit is given by

 

i = (E0 /1/wC) Sin(wt + p/2)

Where L is the inductance of the circuit. The peak value of current in this type of circuit is given by

 

i = i0sinwt + p/2

 

In this case, the current leads the emf by a phase angle of p/2. We can also say that the voltage lags behind the current by a phase difference of p/2.

POWER IN AN AC CIRCUIT

If the electric current in an AC circuit is given as i = i₀ sinwt and the corresponding voltage be given by V = V₀ sin(wt + f)

Then the power in an AC circuit is given by

 

P = irmsVrmscosf

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