ALTERNATING CURRENTS

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Presentation by

AJAI P

PGT PHYSICS, JNV THRISSUR

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Alternating EMF

• The output of a generator with slip rings will be changing its direction periodically, ie after every half cycle.
• Such an emf, which changes its magnitude continuously and changes its direction periodically is called an alternating emf.
• This has many special features compared to steady current supplied by cells.
• In this class we are going to study about AC and its special features.

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Alternating current

Alternating current –� Graphical representation

V = V_m sin ωt OR V = V_m cos ωt

Average value of a time varying quantity

Average value of AC

• The standard equation for AC voltage is given by

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• Where V_m is the peak voltage.
• This, being a sine function , the average value <V> over a complete cycle is zero.
• The fact that the average current is zero, however, does not mean that the average power consumed is zero and that there is no dissipation of electrical energy.
• How to overcome this crisis produced by the mathematical operation?

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Root Mean Square

• These values can be expressed in a different way by the following process:
• Square all terms – this will make all positive.
• Take average of squares
• Find the square root
• This value is now called as RMS value which means “root mean square”. This gives the effective values of voltage or current, which correctly account for the work they are doing.

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Finding RMS value

RMS values

Mean value of AC over T/2

RMS value , Average value and Peak value

AC parameters

AC applied to a resistor

The equation and the graph indicates that current is following voltage without any lagging or leading. Both are reaching peak value simultaneously, and also the zero value.

So in a resitor, current is in phase with voltage.

AC applied to an inductor

• Consider an AC voltage applied to a pure inductor with inductance L.
• From Kirchoff’s law,

Put

Current in an inductor

AC applied to an inductor-

AC applied to an inductor-

AC applied to a capacitor-

• Consider an ac voltage applied to a capacitor of capacitance C.

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Put

AC applied to a capacitor-

AC applied to a capacitor-

Phasor diagrams

• Phasors are rotating vectors.
• Harmonically varying physical quantities like AC voltage, AC current etc are represented by phasors. Even though they are scalars , they are added like vectors.
• AC in resistors
• Wave form Phasor diagram

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AC in an Inductor

AC applied to a capacitor-

Series LCR Circuit

Consider an ac circuit consisting of an inductor L , capacitor C and resistor R connected in series with an AC voltage.

Since the components are in series, current I is same in all ,but voltage across different components are different.

The voltages are

V_l = I X_l

V_C = I X_c

V_R = I R

Voltage across Resistance is in phase with current. Voltage across capacitor lags behind current whereas voltage across inductor leads the current.

Phasor diagram of LCR

Source voltage=V V_l = I X_l

V_C = I X_c

V_R = I R

V_l - V_c = I (X_l-X_c)

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The magnitude of the vector sum is equal to Vc-V_l

Series LCR circuit-expression for current�Phasor diagram method

V_l = I X_l

V_C = I X_c

V_R = I R

V_l - V_c = I (X_l-X_c)

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Series LCR circuit-expression for current�

• it is clear that the voltage V_R and hence the current I leads the voltage by a phase angle φ. Hence the current

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• Where the phase difference φ is given by

or equivalently

Impedance triangle

Types of circuits

• Every circuit can be assumed as containing L,C and R, even though not deliberately included.
• In a circuit, if Xl > Xc , it is an inductive circuit. I lags behind V.
• In a circuit, if Xc > Xl , it is a capacitive circuit. I leads V.

Resonance

All are driven by same source. But the natural frequency of is equal to driver frequency

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Resonance

Sharpness of resonance

Resonance curve🡪 A plot of i_max v/s ω

Sharpness of resonance - The quality factor Q

TV tuner circuit

Just for illustration .Not to be studied

LC oscillations

• When an inductor is connected parallel to a charged capacitor , charge starts oscillating between the two components.

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Here the energy continuously shuttles back and forth from electrostatic energy to magnetic energy.

The magic of LC circuits

Power in an LCR circuit

Sin A Sin B = ½ [cos (A-B)-cos (A+B)]

Power consumed by an inductor

Wattless current

Choke coil -Wattless current

Power consumed by a capacitor

Transformer

• Transformer is a device used to alter AC voltage / current without changing frequency. This is essential for transportation of electrical energy to long distances.
• A transformer consists of two sets of coils, insulated from each other.
• They are wound on a soft-iron core, either one on top of the other or on separate limbs of the core.
• One of the coils called the primary coil has Np turns. The other coil is called the secondary coil; it has Ns turns. Often the primary coil is the input coil and the secondary coil is the output coil of the transformer.

Transformer

Working of a transformer

• A transformer works based on mutual induction.
• The primary coil and secondary coil are wound over same core. So same flux passes through the coils.
• But the flux linkage through the coils will be proportional to the number of turns.
• A fluctuating emf applied in the primary will induce an emf in the secondary.
• The emf induced is proportional to the flux linkage of secondary.
• The fluctuating emf induced in secondary , in turn will induce an emf in primary which is proportional to its flux linkage.

Working of a transformer

Faraday’s law:

Emf induced is directly proportional to rate of change of flux

Working of a transformer

Step up v/s Step down

Step Up

Step down

• Ns<Np
• Vs <Vp
• Is>Ip
• Thickness of secondary coil is more to handle large current.

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• Ns>Np
• Vs >Vp
• Is<Ip
• Thickness of primary coil is more to handle large current.

Losses in transformer

• (i.)Flux Leakage: There is always some flux leakage; that is, not all of the flux due to primary passes through the secondary due to poor design of the core or the air gaps in the core. It can be reduced by winding the primary and secondary coils one over the other.
• (ii) Resistance of the windings: The wire used for the windings has some resistance and so, energy is lost due to heat produced in the wire (I ²R). In high current, low voltage windings, these are minimised by using thick wire.
• (iii) Eddy currents: The alternating magnetic flux induces eddy currents in the iron core and causes heating. The effect is reduced by having a laminated core.
• (iv) Hysteresis: The magnetisation of the core is repeatedly reversed by the alternating magnetic field. The resulting expenditure of energy in the core appears as heat and is kept to a minimum by using a magnetic material which has a low hysteresis loss.

Advantages of AC v/s DC

AC

• AC voltage can be easily stepped up /down with transformers.
• AC at high voltages can be sent to long distances with minimum power loss
• The cost of large scale production of AC is less.
• AC can be easily converted to DC with simple rectifier circuits.
• AC can be controlled without power loss by choke coils.

DC

• Transformers do not work with DC.

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• There are limits for DC voltage

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• Batteries are expensive.

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• Complex Oscillator circuits are required to convert DC to AC
• DC is controlled by resistors which results in power loss.

Dis advantages of AC

• AC is more dangerous than DC. Shocks from AC is fatal due to its capacitive nature with earth.
• Comparing DC voltage with AC of same rms voltage, peak value of AC is higher than the steady value of DC.
• AC cannot be used electroplating ,electro refining etc.
• AC frequency is not pure. It contains higher harmonics too.

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