1 AC Circuit Theory

2 Sinusoidal AC Voltage Waveform: The path traced by a quantity, such as voltage, plotted as a function of some variable such as time, position, degree, radius, temperature and so on. The +ve polarity and current direction will be for an instant in time in the positive portion of the sinusoidal waveform. Periodic waveform : A waveform that continually repeats itself after the same time interval.

3 (+ve Peak) (-ve Peak)

4 Sinusoidal AC Voltage Period (T): The time interval between successive repetitions of a periodic waveform (the period T 1 = T 2 = T 3 ), as long as successive similar points of the periodic waveform are used in determining T. Cycle: The portion of a waveform contained in one period of time. Frequency: (Hertz) the number of cycles that occur in 1 s.

5 Sinusoidal AC Voltage Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase letters ( e,  and i ) Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value (0 V for a sinusoidal ac voltage), denoted by the uppercase letters E m or E p (source of voltage) and V m or V p (voltage drop across a load) Peak-to-peak value: Denoted by E p-p or V p-p, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks

6 Sinusoidal AC Voltage The average value of any current or voltage is the value indicated on a dc meter – over a complete cycle the average value is the equivalent dc value. For a sinusoidal ac current or voltage, the equivalent dc value over a complete cycle is zero. Effective Value: Irrespective of direction, current of any magnitude through a resistor will deliver power to that resistor – during the positive and negative portions of a sinusoidal ac current, power is being delivered at each instant of time to the resistor. The equivalent dc value is called the effective value, or root mean square (rms) value of the sinusoidal quantity.

7 Sinusoidal AC Voltage Generation An ac generator (or alternator) powered by water, wind, gas, or steam is the primary component in the energy-conversion process. The energy source turns a rotor (constructed of alternating magnetic poles) inside a set of windings housed in the stator (the stationary part of the dynamo) and will induce voltage across the windings of the stator. A function generator, as used in the lab, can generate and control alternating waveforms.

8

9 The Sine Wave The sinusoidal waveform is the only alternating waveform whose shape is unaffected by the response characteristics of R, C, and L elements. The voltage across (or current through) a resistor, coil, or capacitor is sinusoidal in nature. The unit of measurement for the horizontal axis is the degree or radian (rad).

o  radians 2  radians 180 o 360 o 

11 The Sine Wave The quantity of  is the ratio of the _________of a circle to its ________ Degrees and radians are related by:

12 The Sine Wave The angular velocity (  ) {Greek letter omega} is: –Since (  ) is typically provided in radians per second, the angle  obtained using  =  t is usually in radians. The time required to complete one revolution is equal to the period (T) of the sinusoidal waveform. The radians subtended in this time interval are 2 . or

13 Sine and Cosine A Asin  t Acos  t =t=t Vector A is rotating anti-clockwise at angular velocity of .

14 Sinusoidal Voltage or Current The basic mathematical format for the sinusoidal waveform is: –where: *A is the peak value (amplitude) of the waveform. *  is the unit of measure for the horizontal axis.  The equation  =  t states that the angle  through which the rotating vector will pass is determined by the angular velocity of the rotating vector and the length of time the vector rotates.

15 Sinusoidal Voltage or Current For electrical quantities such as current and voltage, the general format is: i = I p sin  t = I p sin  e = E p sin  t = E p sin  v = V p sin  t = V p sin  where: the capital letters with the subscript p represent the amplitude, The lower case letters i, e and v represent the instantaneous value of current and voltage, respectively, at any time t.

16 Phase Relations If the waveform is shifted to the right or left of 0°, the expression becomes: –where:  is the angle (in radians) that the waveform has been shifted. If the waveform passes through the horizontal axis with a positive-going (increasing with the time) slope before 0°. If the waveform passes through the horizontal axis with a positive-going slope after 0°.

17 Asin       Asin  (  )(  ) (2  )(2  ) (  )(  ) (2  )(2  )

18   

19 Phasors The addition of sinusoidal voltages and currents will frequently be required in the analysis of ac circuits. –One lengthy but valid method of performing this operation is to place both sinusoidal waveforms on the same set of axes and add algebraically the magnitudes of each at every point along the axis. –Long and tedious process with limited accuracy. –A shorter method uses the rotating radius vector. –The radius vector, having a constant magnitude (length) with one end fixed at the origin, is called a phasor when applied to electric circuits.

20 Phasors Phasor algebra for sinusoidal quantities is applicable only for waveforms having the same frequency. j

21 j I2I2 I1I1 I 1 + I 2 Phasors Waveforms

22 Impedance and the Phasor Resistive Elements –Using  R = 0° in the following polar format to ensure the proper phase relationship between the voltage and the current –Z R, having both magnitude and an associate angle, is referred to as the impedance of a resistive element. –Z R is not a phasor since it does not vary with time. *Even though the format R  0° is very similar to the phasor notation for sinusoidal current and voltage, R and its associated angle of 0° are fixed, non- varying quantities.

23

24

25 Capacitors in AC Circuits Capacitor –Capacitance is a measure of the rate at which a capacitor will store charge on its plates, –Therefore, for a particular change in voltage across a capacitor, the greater the value of capacitance, the greater will be the resulting capacitive current. –The fundamental equation relating the voltage across a capacitor to the current of a capacitor [i = C(dv/dt)] indicates that for a particular capacitance, the greater the rate of change of voltage across the capacitor, the greater the capacitive current.

26 Voltage across and Current through a Capacitor  C tt

27

28 Capacitors in AC Circuits –An increase in frequency corresponds to an increase in the rate of change of voltage across the capacitor and to an increase in the current of the capacitor. –The quantity 1/  C, called the reactance magnitude of a capacitor, is symbolically represented by X c and is measured in ohms; i.e., V C and I C are phasors. –Capacitive reactance (not a phasor) is the opposition to the flow of charge, which results in the continual interchange of energy between the source and the electric field of the capacitor.

29 ICIC VCVC p p + v c = V p.sinωt -

30 Impedance and the Phasor Capacitive Reactance –Using  C = – 90° in the following polar format for capacitive reactance to ensure the proper phase relationship between the voltage and current of an capacitor. –Z C, having both magnitude and an associated angle, is referred to as the impedance of a capacitive element. –Be aware that Z C is not a phasor quantity for the same reason indicated for a resistive element.

31

32

33 Impedance and the Phasor Inductive Reactance –Using  L = 90° in the following polar format for inductive reactance to ensure the proper phase relationship between the voltage and the current of an inductor. –Z L, having both magnitude and an associated angle, in referred to as the impedance of an inductive element. –Be aware that Z L is not a phasor quantity for the same reason indicated for a resistive element.

34

35 Series RC Circuits R-C –Phasor notation –i = 7.07 sin (  t °) I = 5 A  53.13°

36 Series RC Circuits Phasor diagram –In the phasor diagram the current I is in phase with the voltage across the resistor V R and leads the voltage across the capacitor V C by 90°.

37

38 Admittance and Susceptance In dc, conductance (G) was defined as being equal to 1/R. The total conductance of a parallel circuit was then found by adding the conductance of each branch. In ac circuit, we define admittance (Y) as being equal to 1/Z. The SI unit of measure for admittance is siemens (S). Admittance is a measure of how well an ac circuit will admit, or allow, current flow in the circuit. The larger its value the greater the current flow, for the same applied potential. Total impedance Z T of the circuit is then 1/ Y T.

39 Admittance and Susceptance The reciprocal of reactance (1/X) is called susceptance (B) and is a measure of how susceptible an element is to the passage of current through it. Susceptance is measured in siemens (S). For inductance, an increase in frequency or inductance will result in a decrease in susceptance or in admittance. For the capacitor, therefore, an increase in frequency or capacitance will result in an increase in its susceptibitly.

40 Admittance and Susceptance For any configuration (series, parallel, series-parallel), the angle associated with the total admittance is the angle by which the source current leads the applied voltage. For inductive networks,  T is negative, whereas for capacitive networks,  T is positive.

41 Parallel RC Circuits In a parallel ac network the total impedance or admittance is determined, then the source current is determined by Ohm’s law : E = V S Kirchhoff’s current law can then be applied in the same manner as employed for dc networks.

42

43 90 o o

44