SINGLE PHASE AC CIRCUITS Definition of Alternating Quantity e +Em 0 π/2 π 3π/2 2π ωt -Em T An alternating quantity changes continuously in magnitude and alternates in direction at regular intervals of time. Important terms associated with an alternating quantity are defined below. 1. Amplitude It is the maximum value attained by an alternating quantity. Also called as maximum or peak value 2. Time Period (T) It is the Time Taken in seconds to complete one cycle of an alternating quantity 3. Instantaneous Value It is the value of the quantity at any instant 4. Frequency (f) It is the number of cycles that occur in one second. The unit for frequency is Hz or cycles/sec. The relationship between frequency and time period can be derived as follows. Time taken to complete f cycles = 1 second Time taken to complete 1 cycle = 1/f second T = 1/f
Advantages of AC system over DC system 1. AC voltages can be efficiently stepped up/down using transformer 2. AC motors are cheaper and simpler in construction than DC motors 3. Switchgear for AC system is simpler than DC system Generation of sinusoidal AC voltage Consider a rectangular coil of N turns placed in a uniform magnetic field as shown in the figure. The coil is rotating in the anticlockwise direction at an uniform angular velocity of ω rad/sec. When the coil is in the vertical position, the flux linking the coil is zero because the plane of the coil is parallel to the direction of the magnetic field. Hence at this position, the emf induced in the coil is zero. When the coil moves by some angle in the anticlockwise direction, there is a rate of change of flux linking the coil and hence an emf is induced in the coil. When the coil reaches the horizontal position, the flux linking the coil is maximum, and hence the emf induced is also maximum. When the coil further moves in the anticlockwise direction, the emf induced in the coil reduces. Next when the coil comes to the vertical position, the emf induced becomes zero. After that the same cycle repeats and the emf is induced in the opposite direction. When the coil completes one complete revolution, one cycle of AC voltage is generated.
The generation of sinusoidal AC voltage can also be explained using mathematical equations. Consider a rectangular coil of N turns placed in a uniform magnetic field in the position shown in the figure. The maximum flux linking the coil is in the downward direction as shown in the figure. This flux can be divided into two components, one component acting along the plane of the coil Φmaxsinωt and another component acting perpendicular to the plane of the coil Φmaxcosωt. ω rad/sec θ x Фmaxsinωt Фmax Фmaxcosωt The component of flux acting along the plane of the coil does not induce any flux in the coil. Only the component acting perpendicular to the plane of the coil ie Φmaxcosωt induces an emf in the coil. Φ = Φ max cos ωt dΦ dt d e = − N Φ max cos ωt dt e = NΦ maxω sin ωt e = −N e = E m sin ωt Hence the emf induced in the coil is a sinusoidal emf. This will induce a sinusoidal current in the circuit given by i = I m sin ωt
Angular Frequency (ω) Angular frequency is defined as the number of radians covered in one second(ie the angle covered by the rotating coil). The unit of angular frequency is rad/sec. ω= 2π = 2πf T Problem 1 An alternating current i is given by i = 141.4 sin 314t Find i) The maximum value ii) Frequency iii) Time Period iv) The instantaneous value when t=3ms i = 141.4 sin 314t i = I m sin ωt i) Maximum value Im=141.4 V ii) ω = 314 rad/sec f = ω/2π = 50 Hz iii) T=1/f = 0.02 sec iv) i=141.4 sin(314x0.003) = 114.35A Average Value The arithmetic average of all the values of an alternating quantity over one cycle is called its average value Average value = Area under one cycle Base 1 Vav = 2π 2π ∫ vd (ωt ) 0