08(T)%20-%20Alternating%20Current

08(T)%20-%20Alternating%20Current - 8 ALTERNATING CURRENT...

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8 - ALTERNATING CURRENT Page 1 8.1 Alternating Voltage and Alternating Current ( A. C. ) The following figure shows N turns of a coil of conducting wire PQRS rotating with a uniform angular speed ϖ with respect to the X-axis in a uniform magnetic field along the Y-axis. Let the angle between the direction of the area vector of the coil and the magnetic field direction be zero at time, t = 0 and θ θθ = ϖϖ t at time t = t. The magnetic flux, Φ 0 , associated with the coil at time t = 0 and Φ ΦΦ t at time t = t are given by Φ 0 = N B A = N A B cos 0 = N A B and Φ ΦΦ t = N A B cos ϖ t The emf induced in the coil according to Faraday’s law is V = - dt t d Φ = N A B ϖ sin ϖ t = V m sin ϖ t ... ... ... ... ... ( 1 ) where V m = N A B ϖ is the maximum value of the induced emf. Equation ( 1 ) shows that the induced emf versus time is a sine curve. This emf is obtained between the brushes B 1 and B 2 which are in contact with the slip rings A 1 and A 2 as shown in the above figure. The voltage is zero at time t = 0 and varies as per the function sin ϖ t reaching maximum value V m at time t = π ππ / // 2 ϖ and again zero at time t = π ππ / // ϖ . B 2 being at greater potential than B 1 acts like a positive end of the voltage source during this time interval. After time t = π ππ / // ϖ , the potential of B 1 starts to rise with respect to B 2 till time t = 3 ππ ππ / // 2 ϖ, ϖ, ϖ, reaches maximum in the reverse direction and again becomes zero at time t = 2 π ππ / // ϖ . This cycle keeps repeating in every time interval of T = 2 π ππ / // ϖ . The voltage so developed is known as alternating voltage and its graph versus time is shown by a continuous line in the following figure. The arrangement to produce such a voltage is
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8 - ALTERNATING CURRENT Page 2 known as A.C. ( alternating current ) dynamo or generator. In the adjoining figure, an alternating voltage source is connected in a circuit with resistor, R. The voltage between a and b is zero at time t = 0. Zero current will flow at this time. The voltage in the circuit varies according as V = V m sin ϖ t. The current as per Ohm’s law will be Ι Ι Ι = R t sin V m ϖ . The current changes sinusoidally in the same way as the voltage as shown by the broken line in the above graph. Here, voltage was considered changing with time as per sin ϖ t. Both the voltage and the current can be considered to be changing as per cos ϖ t. It is not necessary that the voltage and current should change as above. There are also other ways in which both can change periodically with time. 8.2 A. C. Circuit with Series Combination of Resistor, Inductor and Capacitor The following figure shows a series combination of resistor having resistance R, inductor of inductance L and a capacitor of capacitance C with an alternating voltage source of voltage changing with time as V = V m cos ϖ t. It is assumed that the resistor has zero inductance and the inductor has zero resistance.
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