In this article, we shall study resonating circuit: a) LCR Series Circuit and b) LC parallel circuit.

Series Resonating Circuit:

Consider LCR circuit containing Resistance R, Inductance L and capacitance C connected in series. Let the combination be connected to an A.C. source supplying current i = i_o Sinωt.

The instantaneous e.m.f. of the circuit is equal to the sum of voltage drops across the resistor, the capacitor, and the inductor.

Dividing both sides of the equation by C

Substituting the values of equations (2) and (3) in equation (1)

Substituting these values in equation (4) we have

Thus the e.m.f. leads the current in phase  given by

In this condition the impedance of the circuit is minimum and the circuit acts as a purely resistive circuit. The current in the circuit is maximum. This condition is called a series resonance of the circuit. At resonance

The frequency f_r is called the resonating frequency. The frequency at which the resonance takes place and the maximum current flows through the circuit is called the resonating frequency.

The variation of the current with frequency for the series LCR circuit is as shown in the graph.

Parallel Resonating Circuit:

 Consider LC circuit containing Inductance L and capacitance C connected in parallel. Let the combination be connected to an A.C. source with e.m.f.  e = e_o Sin ωt.

The current in the inductance lags behind the e.m.f. by π/ 2. Thus the current through the inductance is given by

Where,  X_L = Inductive reactance

The current in the capacitance leads the e.m.f. by π/ 2. Thus the current through the inductance is given by

Where X_C = Capacitive reactance

The total current in the circuit is given by

When X_L = X_C i.e. ωC =  1/ ωL , the current in the circuit is zero. This condition is called parallel resonance of the circuit.

At resonance

The frequency f_r is called the resonating frequency. The frequency at which the resonance takes place and the zero current flows through the circuit is called the resonating frequency. At resonating frequency the impedance is maximum.

Science > Physics > Electromagnetic Induction > Resonating Circuit

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In this article, we shall study the concept of the reactance of inductive and capacitive circuits.

Concept of Reactance of a Circuit:

When A.C. current flows through a circuit, the resistance of the circuit opposes the flow of current. The circuit may contain other components like inductor or capacitor. These components also oppose the flow of current through them. This property of inductor and capacitor to oppose the flow of current is called reactance. The Reactance of inductor is called inductive reactance (XL) and that of the capacitor is called capacitive reactance (X_C).

Reactance of a circuit is defined as the ratio of the r.m.s. voltage across the component to the r.m.s. current passing through it. Its unit is ohm.

Concept of Impedance of a Circuit:

An A.C. circuit may contain resistor, inductor and capacitor. Thus besides the resistance, the circuit has reactance. The combined effect of the resistance and reactance is called impedance (Z) of the circuit. The impedance of a circuit is defined as the ratio of r.m.s. the voltage across the circuit to r.m.s. current passing through it. The unit of impedance is ohm.

The Expression for Inductive Reactance:

Consider a purely inductive circuit containing inductor only of inductance L. Let the inductor be connected across an A.C. source sending current i = i_o Sinωt. The instantaneous value of e.m.f. induced in the inductance due to alternating current is given by

This equation shows that the induced e.m.f. in the inductor leads the current in the inductor by the phase of π/2. In other words the current in the inductor lags the e.m.f. induced in the inductor by the phase of π/2.

This is an expression for inductive reactance

The Expression for Capacitive Reactance:

Consider a purely capacitive circuit containing capacitor only of capacitance C. Let the capacitor be connected across an A.C. source sending current i = i_o Sinωt

Let the charge on capacitor be q and thus the potential difference across the capacitor is q/C. The rate of flow of charge is called the current

The total charge can be obtained by integrating the above expression.

But q = e C

This equation shows that the induced e.m.f. in the capacitor lags the current in the capacitor by the phase of π/2.In other words, the current in the capacitor leads the e.m.f. induced in the capacitor by the phase of  π/2.

This is an expression for capacitive reactance

Expression for Power Dissipated in LCR Circuit:

Consider LCR circuit containing Resistance R, Inductance L and capacitance C connected in series. Let the combination be connected across an A.C. source of e.m.f. e = e_o Sinωt. Let the phase difference between the e.m.f. and current be Φ. Therefore the instantaneous current is given by

i = i_o Sin (ωt  ± Φ)

The sign of Φ is positive for capacitive reactance and negative for inductive reactance.

The instantaneous power in the circuit is given by

P = e I

∴ P = e_o Sinωt. i_o Sin (ωt  ± Φ)

∴ P = e_o i_o Sinωt. [Sin ωt CosΦ ± Cos ωt Sin Φ]

∴ P = e_o i_o. [Sin²ωt CosΦ ± Sinωt.Cos ωt SinΦ]

Therefore average power dissipated is given by

Knowing

The quantity cosΦ is called power factor of the circuit. The quantity e r.m.s.  x I r.m.s. is called the apparent power.

The apparent power is calculated from the readings of voltmeter and ammeter connected in the circuit. If Z is the impedance of the circuit, then

The power factor, cosΦ = R / Z, Thus we have

True power dissipated in the circuit = Apparent power X Power factor.

Science > Physics > Electromagnetic Induction > The Reactance of a Circuit

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In this article, we shall study the rotation of coil in uniform magnetic field, the concept of alternating current.

The principle of Electrical Generator:

Consider a circular coil of ‘n’ turns of each area A placed with its plane perpendicular to a uniform magnetic field of induction B. At time t = 0, the plane of the coil is normal to field. Therefore, the magnetic flux passing through it is given by

ø = nAB.

Where n = number of turns

A = Area of the coil

B = Uniform magnetic induction

The coil is now rotated with constant angular speed ‘ω’ about a diameter. In a time ‘t’, the plane of the coil makes an angle θ with the initial position. Therefore, the angle traced by the coil in time ‘t’ is given by

θ = ωt

To find the new flux, the magnetic induction is resolved into two mutually perpendicular components. The component B sin θ is along the plane of the coil and hence it does not contribute flux. However, component B cos θ is perpendicular to the plane of the coil. Therefore, the flux through the coil at time t is given by

ø =  nAB cos θ= nAB cos ωt

Therefore the induced e.m.f. in the coil is given by

Also ω= 2 π f, where f is the frequency of rotation.

Where e = instantaneous induced e.m.f.

e₀  =  peak e.m.f.

n  = number of turns of the coil

A  = Area of the coil

B = Uniform magnetic induction

Since sin ωt changes from +1 to -1, the e.m.f. induced changes periodically. The e.m.f. given by the above equation i.e. e = e₀ sinωt is known as alternating e.m.f.

When        sin ωt = ±1

e = e₀ (±1) = ± e₀ = Peak e.m.f.

The peak value of induced e.m.f. (e) is defined as the maximum value of alternating e.m.f.

Peak e.m.f.  = e₀ =  2πfnAB

Graphically an alternating e.m.f. e = e₀ sin ωt is as shown. The graph shows that the induced e.m.f. changes its magnitude and direction periodically.

Simple A.C. Circuit With Resistance Only (Alternating Current):

Consider an A.C. Circuit consisting of a resistance R connected to a source of e.m.f.,

e = e₀ sin ωt.

By Ohm’s law, the current through the circuit is given by,

Peak current (I0) is defined as the maximum value of alternating current.

Comparing the equations of A.C. e.m.f. and A.C. current it is obvious that they are in phase.

Terminology of AC Circuits:

Peak value of alternating e.m.f.:

The magnitude of the maximum value of alternating e.m.f. is called the peak value of e.m.f. It is denoted by e₀.

Peak Value of Alternating Current:

The magnitude of the maximum value of alternating current is called the peak value of the current. It is denoted by I₀.

Average value of Alternating e.m.f.:

The constant value shown by a D.C. voltmeter during one period of alternating e.m.f. (T) is called the average value of alternating e.m.f.  It is denoted by e_av. The average alternating e.m.f. is zero.

Average Value of Alternating Current:

The constant value shown by a D.C. ammeter during one period of alternating current (T) is called the average value of alternating current. It is denoted by I_av.  As the average alternating e.m.f. is zero the average alternating current in the circuit is also zero.

R.M.S. Value of e.m.f.:

The root mean square (r.m.s.) value of e.m.f. is that steady e.m.f., which is to be applied across the given resistor to produce the same amount of heat in a given time as it is produced by given alternating e.m.f. It is denoted by e_r.m.s.

R.M.S. Value of Current:

The root mean square (r.m.s.) value of current is that steady current which is passed through given resistor to produce the same amount of heat in a given time as it is produced by given alternating current. It is denoted by I_r.m.s..

Expression for Power in AC Circuit:

An alternating current of peak value ‘I’ is equivalent to a constant direct current of value I₀/√2. This is known as the effective value, virtual value or r.m.s. value of current.

I_r.m.s. = I₀/√2

The power generated with the D.C. circuit containing resistance ‘R’ is given by

P  =  I²R

Therefore, the power generated in an A.C. circuit through which current I_r.m.s. is flowing through resistance ‘R’ is given by

But according to Ohms Law,

This is an expression for the power dissipated in A.C. circuit.

Science > Physics > Electromagnetic Induction > Rotation of a Coil in Uniform Magnetic Field

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