# Alternating Current and Electrical Devices

### Alternating Current

An alternating current is one which changes in magnitude and direction periodically and is abbreviated as a.c.

Alternating current

The source of alternating emf may be a dyamo or an electronic oscillator. The alternating emf E at any instant may be expressed as:

$E = E_0 sin \omega t \cdots equation \, \, 1$

Where $\omega$ is angular frequency of alternating emf and $E_o$ is the peak value or amplitude of alternating emf.

The frequency of alternating emf, $f = \dfrac{ \omega}{2 \pi}$

And time period of alternating emf,

$T = \dfrac{1}{f} = \dfrac{2 \pi}{ \omega}$

The alternating current in a circuit, fed by an alternating source of emf may be controlled by inductance L, resistance R and capacitance C. Due to presence of element L and C, the current is not necessarily in phase with the applied emf. Therefore alternating current is, in general, expressed as:

$I = I_o \, sin ( \omega t + \phi ) \cdots Equation \, \, 1$

Where $\phi$ is the phase which may be positive, zero or negative depending on the values of reactive components L and C.

The average and rms value:

The average value of AC over full cycle is zero since there are equal positive and negative half cycles.

$( I_{av} ) _{full \, cycle} = 0 \cdots equation \, \, 3$

The average value of AC over half cycle is given by:

$( I_{av} ) _{half \, cycle} = \dfrac{2 I_o}{\pi} \cdots Equation \, \, 4$

The root mean square value of AC is:

$I_{rms} = \sqrt{ [ ( I^2 ) _{av} ]} = \dfrac{I_o}{ \sqrt{2}} \cdots Equation \, \, 5$

Similarly the average value of alternating voltages are given by:

$( E_{av} ) _{full \, \, cycle} = 0 \cdots equation \, 6$

$( E_{av} ) _{half \, \, cycle} = \dfrac{2 E_0}{\pi} \cdots equation \, 7$

$E_{rms} = \dfrac{e_o}{ \sqrt{2}} \cdots Equation \, \, 8$

The rms value of alternating current can also defined as the direct current which produces the same heating effect in a given resistor. Due to this reason the rms value of current is also known as effective or apparent value of current:

$\therefore I_{effective} = I_{virtual} = \dfrac{I_o}{ \sqrt{2}}$

Similarly the rms value of alternating voltage is called the effective or virtual value of alternating voltage.

$E_{virtual} = E_{rms} = \dfrac{E}{ \sqrt{2}}$

Alternating current shows heating effect only. The ac meters are based n heating effect and measure rms values.

### Impedance and Reactance

Alternating current in a circuit may be controlled by resistance, inductance and capacitance, while the direct current may be controlled only by resistance.

Impedance (Z): In alternating current circuit, the ratio of emf applied and consequent current produced is called the impedance and is denoted by Z.

$Z = \dfrac{E}{I}$

Physically impedance of ac circuit is the hindrance offered by the circuit to the flow of ac thought it.

Reactance (X): The hindrance offered by inductance and capacitance to the flow of ac in an ac circuit is called reactance and is denoted by X. thus when there is no ohmic resistance in the circuit; the reactance is equal to impedance. The reactance due to inductance alone is called inductive reactance and is denoted by $X_L$ while the reactance due to capacitance alone is called the capacitive reactance and is denoted by $X_c$ .

### Impedances and Phases of Ac circuits containing different elements

In an ac circuit the current and applied emfs are not necessarily in same phase. The applied emf (E) and current produed (I) may be expressed as:

$E = E_o \, sin \omega t \cdots Equation \, 1$ and

$I = I_o sin ( \omega t + \varphi )$ with $I_o = \dfrac{E_o}{Z} \cdots equation \, 2$

Where $E_o \text{and} \, \, I_o$ are peak values of alternating emf and current.

(i) Circuit containing Pure Resistance: If a circuit, fed by an alternating emf E = $E_o \, sin \omega t ,$ contains pure resistance R, then current is given by:

$I = \dfrac{E}{R} = \dfrac{E_o sin \omega t}{R} = I_o sin \omega t$

Where, $I_o = \dfrac{E_o}{R} \cdots Equation \, \, 3$

Circuit consisting pure resistance

Circuit consisting pure resistance

Compare this with standard equation (2), we note that Impedance of circuit, Z = R and phase lead of current over emf, $\phi = o$ ,

That is in a purely resistive a.c. circuit the current and voltage are in same phase and impedance of circuit is equal to the ohmic resistance.

(ii) Circuit Containing Pure Inductance: If a circuit, fed by an alternating emf E= $E_o \, \, sin \, \, \omega t$ , contains pure inductance L, then current is given by:

$I =I_o sin ( \omega t - \dfrac{ \pi}{2} ) \cdots Equation \, \, 4$

Where, $I_o = \dfrac{E-o}{ \omega L}$

Circuit Containing Pure Inductance

Comparing this with standard equation (2), we note that:

$Z = \omega L \text{and} \, \, \phi = - \dfrac{ \pi}{2} \cdots Equation \, \, 5$

That is in a purely inductive circuit the current lags behind the applied voltage by an angle $\dfrac{ \pi}{2}$ and impedance of the circuit is $\omega L$ and obviously that is inductive reactance,

$Z-l = X_l = \omega L \cdots Equation \, \, 6$

Circuit Containing Pure Inductance

(iii) Circuit Containing Pure Capacitance: Let a circuit pure capacitance and the applied alternating e.m.f. be:

$E = E_o sin \omega t$

The current in circuit is given by:

$I = I_o sin ( \omega t + \dfrac{ \pi}{2} ) \cdots equation \, \, 7$

Where,

$I_o = \dfrac{E_o}{1 / \omega C}$

Circuit Containing Pure Capacitance

Comparing this with standard equation 2, we note that:

$Z = \dfrac{1}{ \omega C} \text{and} \varphi = + \dfrac{ \pi}{2} \cdots Equation \, \, 8$

That us in a purely capacitive circuit the current leads the applied emf by an angle $\dfrac{ \pi}{2}$ and the impedance of the circuit is $\dfrac{1}{ \omega C}$ and obviously this is capacitive reactance:

$Z_C = X_C = \dfrac{1}{ \omega C}$

Circuit Containing Pure Capacitance

### Circuit Containing Resistance and Inductance in series

Let a circuit containing resistance R and inductance L in series be fed with an alternating emf $E_o sin \omega t$ .

Let I be the current flowing in the circuit and $V_R ( = IR )$ the potential difference across resistance and $V_l ( = \omega L I )$the potential difference $V_L$ across inductance leads the current I by an angle $\dfrac{ \pi}{2}$ .

Circuit Containing Resistance and inductance

Circuit Containing Resistance and inductance

Therefore, resultant voltage is given by:

$E = \sqrt{ ( VR^2 + VL^2 ) } = \sqrt{ ( RI ) ^2 + ( \omega L I ) ^2}$

$\therefore \dfrac{E}{I} = \sqrt{R^2 + ( \omega L ) ^2}$

$= \sqrt{ R^2 + XL ^2}$

Therefore, Impedance of R – L circuit,

$Z = \dfrac{E}{I} = \sqrt{R^2 + XL^2}$

It is obvious that the current lags behind the emf by an angle $\phi$ given by:

$tan \phi = \dfrac{V_L}{V_R} = \dfrac{X_L I}{RI} = \dfrac{X_L}{R}$

### Circuit Containing Resistance and Capacitance in series

Let a circuit containing resistance R and capacitance C in series be fed with an alternating emf $E = E_o sin \omega t$ . Let I be the current flowing in the circuit, VR the potential difference across resistance and Vc the potential difference across capacitance.

Circuit Containing Resistance and Capacitance in series

Circuit Containing Resistance and Capacitance in series

The potential difference $V_R$ and current I are in same phase and potential difference lags behind the current I (and hence $V_R$ ) by angle $\dfrac{\pi}{2}$ .

The resultant emf is given by:

$E = \sqrt{ ( VR^2 + VC^2 )} = \sqrt{ (RI) ^2 + (XCL) ^2}$

$\therefore , Z = \dfrac{E}{I} = \sqrt{ (R^2 + Xc^2)}$

The current leads the applied emf by an angle $\phi$ given by:

$tan \phi = \dfrac{V_C}{V_R} = \dfrac{X_C I}{RI} = \dfrac{X_C}{R}$

### Circuit Containing Inductance and Capacitance in series

Let a circuit containing inductance L and capacitance C in series be fed with an alternating emf $E_o sin \omega t$ . Let I be the current flowing in circuit, $V_L$ the potential difference across inductance L and $V_c$ the p.d. across capacitance C.

Circuit Containing Inductance and Capacitance in series

Circuit Containing Inductance and Capacitance in series

The potential difference $V_c$ lags behind the current by angle $\dfrac{ \pi}{2}$ and the potential difference $V_L$ leads the current by angle $\dfrac{ \pi}{2}$ .

$\therefore Resultant \, \, \, applied \, \, \, emf , E = V_c - V_l$

$= X_C I - X-L I$

$\therefore Impedance \, \, of \, \, circuit \, Z = \dfrac{E}{I} = X_c - X_L = ( \dfrac{1}{ \omega C} - \omega L )$

The leading of current over applied emf $\phi = \dfrac{pi}{2}$ .

In case $X_c = X_L , Z = 0 , then$

$\dfrac{1}{ \omega C} = \omega L$

$\therefore Frequency \, \, f = \dfrac{ \omega}{2 \pi} = \dfrac{1}{ 2 \pi \sqrt{LC}}$

This frequency is called the resonant frequency.

### Circuit containing Resistance, Inductance and Capacitance in series

Let a circuit containing a resistance R, inductance L and capacitance C in series be fed with an alternating emf $E = E_o sin \omega t$. Let I be the current flowing in circuit, $V_R , V_L \, \, and \, \, V_C$ , and the respective potential differences across resistance R, inductance L and capacitance C. The p.d. $V_c$ is in phase with current I. The p.d. Vc lags behind the current by angle $\dfrac{ \pi}{2}$ . The p.d. $V_L$ leads the current by angle $\dfrac{ \pi}{2}$ .

Therefore ,

Resultant applied emf,

Circuit containing Resistance, Inductance and Capacitance in series

Circuit containing Resistance, Inductance and Capacitance in series

$E = \sqrt{ (VR^2 + ( V_C - V_L ) ^2}$

The phase lead of current over applied emf.

$tan \phi = \dfrac{V_C - V_L}{V_R} = \dfrac{ X_C I - X_L I}{RI} = \dfrac{X_C - X_L}{R}$

I.E.

$\phi = tan ^{-1} ( \dfrac{X_C - X_L}{R} )$

It is obvious that:

(i) If $X_C > X_L$ , the value of $\phi$ is negative i.e, current leads the applied emf.

(ii) If $X_C < X_L$ the value of $\phi$ is negative i.e, current lags behind the applied emf.

(iii) If $X_C = X_L$ , the value of $\phi$ is zero i.e, current and emf are in same phase. This is called the case of resonance and resonant frequency is given by the condition $X_C = X_L$ .

$f_r = \dfrac{ \omega r}{2 \pi} = \dfrac{1}{ 2 \pi \sqrt{LC}}$

At resonance impedance is minimum: $Z_{min} = R$ and current is maximum $I_{max} = \dfrac{E}{Z_{min}} = \dfrac{E}{R}$

Circuit containing Resistance, Inductance and Capacitance in series

### Power in an AC circuit

The power is defined as the rate at which work is being done in the circuit. In ac circuit, the current and emf are not necessarily in the same phase, therefore we write:

$E = E_o sin \omega t , I = I_0 sin ( \omega t + \phi )$

The instantaneous power,

$P = EI = E_o sin \omega t , I_o sin ( \omega t + \phi )$

$= \dfrac{1}{2} E_o I_0 2 sin \omega t sin ( \omega t + \phi )$

### Transformer

A transformer is a device for converting high voltage into low voltage and vice versa.

There are two types of transformer.

1. Step up transformer: It converts low voltage into high voltage.

2. Step down Transformer: It converts high voltage into low voltage.

Transformer

The principle of a transformer is based on mutual induction and a transformer always works on AC. The input is applied across primary terminals and output across secondary terminals.

The ratio of number of turns in secondary and primary is called the turn ratio i.e,

$\dfrac{n_s}{n_p} = turn \, \, ratio \, \, n$

If $E_p$ and $E_s$ are alternating voltage, $i_p$ and $i_s$ the alternating currents across primary and secondary terminals respectively, we have:

$\dfrac{E_s}{E_p} = \dfrac{i_p}{i_s} = \dfrac{n_s}{n_p} = n$

Efficiency of transformer:

$\eta = \dfrac{output \, \, power}{input \, \, power} = \dfrac{P_{out}}{p_{in}} = \dfrac{E_s i_s}{E_p i_p}$

The step up transformer is used to transmit power at high voltage to reduce line loss appreciably.

### Induction Coil

An induction coil is based on the phenomenon of mutual induction and is used to produce a large emf from a source of low emf. An emf of the order of 50,000 V (but feeble current) may be achieved from 12 V battery (but high current).

Induction Coil

It consists of a primary coil $P_1 P_2$ containing a few turns of thick insulated copper wire wound on a laminated soft iron core. A secondary coil is wound on the primary coil and contains a large number of turns of thin copper wire. The make and break arrangement is provided by screw and soft iron hammer arrangement DH. When the current is passed by the help of battery in the primary coil, the iron core is magnetized and attracts the hammer, so the contact between screw and hammer is broken. The current in primary thus stops and iron core is demagnetized and the hammer is released and the contact is established again. A capacitor is connected across the air gap to prevent undue sparking and save the surfaces from being damaged.

When current is established in primary coil the flux linked with secondary increases and so a large emf is induced in the secondary. When the current breaks, the resistance of circuit becomes infinite and so time constant $\dfrac{L}{R}$ becomes infinitely small and so the rate of fall of current becomes very much and consequently a very large emf is induced across the secondary. Thus two opposite emfs are induced, one at make and the other at break, but the induced emf at break is very much higher than that at make. The emf at make is insufficient to break the insulation of air gap across $S_1 \, \, and \, \, S_2$ , and discharge passes only at break. Hence current in the secondary is intermittent and unidirectional.

Induction coil is used in discharge tube to cause discharge of air/gas filled in it.

### Generator or Dynamo

It is to device to convert mechanical energy into electrical energy. The generator is based on the principle of electromagnetic induction. According to which when a coil rotates in a uniform magnetic field, an alternating emf is induced in it.

An ac generator consists of:

(i) Armature: It is rectangular coil (ABCD) carrying a large number of turns and wound on a soft iron core. The soft iron core is used to increase the magnetic flux.

Armature

(ii) Field magnet: It is a strong magnet having two poles N and S. The armature is rotated between the poles so that the axis armature is perpendicular to the magnetic field lines.

(iii) Slip Rings $C_1$ and $C_2$ : The leads of armature-coil are connected to two rings $C_1 \, \, and \, \, C2$ called the slip rings. The slip rings also rotate with the coil.

(iv) Brushes: The two brushes $B_1 \, \, and \, \, B_2$ are made of graphite and they touch the slip rings $C_1 \, \, and \, \, C_2$ permanently. As the rings rotate, the brushes remain in constant touch with the rings, the brushes are connected to the two terminals, $T_1 \, \, and \, \, T_2$ . The external circuit is connected to these terminals. The emf induced in the coil $e = NBA \omega sin \omega t = e_o sin \omega t$ where $e-o = NBA \omega$ is the peak voltage.

### Motor

A motor is a device which converts electrical energy into mechanical energy. The principle of a dc motor is based on interaction of current and magnetic field i.e. a current carrying coil placed in a uniform magnetic field experiences a torque.

A dc motor consists of a field magnet, an armature, the slip rings and brushes. The arrangement is same as that of a dynamo.

When current is passed in the armature coil through the brushes, the coil experiences a torque. This torque rotates the coil which is on the shaft to which the mechanical load is attached. .

When the coil rotates, the induced emf e is produced which opposes the applied emf E. That is why the induced emf is also called the back emf.

So net emf = E – e

If R is the resistance of circuit, then current at any instant $I = \dfrac{E - e}{r}$

The induced emf is proportional to angular speed $( \omega )$ of coil. When motor is at full speed, the back emf is very high and a low current flows through the armature.

### Choke

The Choke coil is a coil of high inductance and low resistance. It is used to control current in ac circuit without any appreciable power loss. The use of choke coil is preferred over resistance because the powerless in choke coil is negligible. The average power in ac is given by:

$P = E_{rms} I_{rms} cos \phi$

Where,

$cos \phi = \dfrac{R}{Z}$

In a choke coil $R \rightarrow 0$ . This implies that $cos \phi = 0$ or power loss in choke coil is nearly zero.

### Starter

The armature-resistance of a motor is low to reduce energy losses. At the start of motor, the back emf (e) is zero, so current may be too heavy (I=E\R) and may break the insulation of coil. For this reason, a large variable resistance called the starter is introduced in series with the armature.

Therefore at the start, a resistance is included in the circuit and moderate current flows. As motor picks up speed, the variable resistance is gradually withdrawn so that moderate current flows. If the electric supply suddenly fails, a large current flows due to back emf which may damage the armature. Therefore the starter resistance is automatically introduce at ‘on’ and ‘off’ of a motor.

### Skin effect

When a wire carries a direct current (dc), it is distributed uniformly throughout the whole cross-section of the wire. But if the wire carries alternating current of high frequency, it is merely concentrated on outer layers and if the frequency of ac is very high, the current is almost wholly confined to the surface layer of the wire.

This phenomenon is called the skin effect. Due to reduction of effective cross-sectional area a conductor offers high resistance to ac. Hence in the case of ac, current carrying conductor is in the form of strands of fine wire connected in parallel at their ends.

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