Page 1 AC Theory â€“ Professor J R Lucas 1 November 2001 Alternating Current Theory - J R Lucas An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = V m sin(? t + f ) If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = A m sin(? t+ f ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = A m (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. Mean value A mean = ? +T t t o o dt t a T ). ( 1 The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. For a sinusoid, Mean value = ? + + T t t m o o dt t A T ). ( sin 1 f ? = 0 (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value A avg = ? ? ? ? ? ? ? ? ? ? - ? ? fcycle hal negative fcycle hal positive dt t a dt t a T ). ( ). ( 1 For a sinusoid, Average value = ? ? ? ? ? ? ? ? - ? ? T T m T m dt t A dt t A T 2 2 0 . sin . sin 1 ? ? = m m A A T p ? 2 2 1 = · The average value is defined in the above manner in electrical engineering as otherwise all alternating waveforms would have zero value and would be indistinguishable. a(t) -A m A mean v(t) t T T V m f/? A avg v rect t T T Page 2 AC Theory â€“ Professor J R Lucas 1 November 2001 Alternating Current Theory - J R Lucas An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = V m sin(? t + f ) If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = A m sin(? t+ f ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = A m (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. Mean value A mean = ? +T t t o o dt t a T ). ( 1 The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. For a sinusoid, Mean value = ? + + T t t m o o dt t A T ). ( sin 1 f ? = 0 (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value A avg = ? ? ? ? ? ? ? ? ? ? - ? ? fcycle hal negative fcycle hal positive dt t a dt t a T ). ( ). ( 1 For a sinusoid, Average value = ? ? ? ? ? ? ? ? - ? ? T T m T m dt t A dt t A T 2 2 0 . sin . sin 1 ? ? = m m A A T p ? 2 2 1 = · The average value is defined in the above manner in electrical engineering as otherwise all alternating waveforms would have zero value and would be indistinguishable. a(t) -A m A mean v(t) t T T V m f/? A avg v rect t T T AC Theory â€“ Professor J R Lucas 2 November 2001 (e) Effective value or r.m.s. value: Neither the peak value, nor the mean value, nor the average value defines the useful value of the waveform with regard to the power or energy. Thus the effective value is defined based on the power equivalent of the quantity. The effective value is thus defined as the constant value which would cause the same power dissipation as the original quantity over one complete period. Thus considering current Power dissipation = I eff 2 . R = ? +T t t o o dt R t i T . ). ( 1 2 giving I eff = ? +T t t o o dt t i T ). ( 1 2 or in general Effective value A eff = ? +T t t o o dt t a T ). ( 1 2 It is to be noted that the method of calculating the effective value involves the following processes. Taking the root of the mean of the squared waveform over one complete cycle. Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform. i.e. r.m.s. value A rms = ? +T t t o o dt t a T ). ( 1 2 This is defined for both current as well as voltage. For a sinusoid, r.m.s. value = ? + + T t t m o o dt t A T ). ( sin 1 2 2 f ? = 2 m A Unless otherwise specified, the rms value is the value that is always specified for ac waveforms, whether it be a voltage or a current. For example, 230 V in the mains supply is an rms value of the voltage. Similarly when we talk about a 5 A, 13 A or 15A socket outlet (plug point), we are again talking about the rms value of the rated current of the socket outlet. For a given waveform, such as the sinusoid, the peak value, average value and the rms value are dependant on each other. The peak factor and the form factor are the two factors that are most commonly defined. value average value rms = Factor Form and for a sinusoidal waveform, Form Factor = 111 . 1 1107 . 1 2 2 ? = p m m V V The form factor is useful such as when the average value has been measured using a rectifier type moving coil meter and the rms value is required to be found. [Note: You will be studying about these meters later] Peak Factor = peak value rms value and for a sinusoidal waveform, Peak Factor = The peak factor is useful when defining highly distorted waveforms such as the current waveform of compact fluorescent lamps. V V m m 2 2 1 4142 == . v 2 (t) t T T t i Page 3 AC Theory â€“ Professor J R Lucas 1 November 2001 Alternating Current Theory - J R Lucas An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = V m sin(? t + f ) If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = A m sin(? t+ f ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = A m (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. Mean value A mean = ? +T t t o o dt t a T ). ( 1 The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. For a sinusoid, Mean value = ? + + T t t m o o dt t A T ). ( sin 1 f ? = 0 (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value A avg = ? ? ? ? ? ? ? ? ? ? - ? ? fcycle hal negative fcycle hal positive dt t a dt t a T ). ( ). ( 1 For a sinusoid, Average value = ? ? ? ? ? ? ? ? - ? ? T T m T m dt t A dt t A T 2 2 0 . sin . sin 1 ? ? = m m A A T p ? 2 2 1 = · The average value is defined in the above manner in electrical engineering as otherwise all alternating waveforms would have zero value and would be indistinguishable. a(t) -A m A mean v(t) t T T V m f/? A avg v rect t T T AC Theory â€“ Professor J R Lucas 2 November 2001 (e) Effective value or r.m.s. value: Neither the peak value, nor the mean value, nor the average value defines the useful value of the waveform with regard to the power or energy. Thus the effective value is defined based on the power equivalent of the quantity. The effective value is thus defined as the constant value which would cause the same power dissipation as the original quantity over one complete period. Thus considering current Power dissipation = I eff 2 . R = ? +T t t o o dt R t i T . ). ( 1 2 giving I eff = ? +T t t o o dt t i T ). ( 1 2 or in general Effective value A eff = ? +T t t o o dt t a T ). ( 1 2 It is to be noted that the method of calculating the effective value involves the following processes. Taking the root of the mean of the squared waveform over one complete cycle. Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform. i.e. r.m.s. value A rms = ? +T t t o o dt t a T ). ( 1 2 This is defined for both current as well as voltage. For a sinusoid, r.m.s. value = ? + + T t t m o o dt t A T ). ( sin 1 2 2 f ? = 2 m A Unless otherwise specified, the rms value is the value that is always specified for ac waveforms, whether it be a voltage or a current. For example, 230 V in the mains supply is an rms value of the voltage. Similarly when we talk about a 5 A, 13 A or 15A socket outlet (plug point), we are again talking about the rms value of the rated current of the socket outlet. For a given waveform, such as the sinusoid, the peak value, average value and the rms value are dependant on each other. The peak factor and the form factor are the two factors that are most commonly defined. value average value rms = Factor Form and for a sinusoidal waveform, Form Factor = 111 . 1 1107 . 1 2 2 ? = p m m V V The form factor is useful such as when the average value has been measured using a rectifier type moving coil meter and the rms value is required to be found. [Note: You will be studying about these meters later] Peak Factor = peak value rms value and for a sinusoidal waveform, Peak Factor = The peak factor is useful when defining highly distorted waveforms such as the current waveform of compact fluorescent lamps. V V m m 2 2 1 4142 == . v 2 (t) t T T t i AC Theory â€“ Professor J R Lucas 3 November 2001 Some advantages of the sinusoidal waveform for electrical power applications a. Sinusoidally varying voltages are easily generated by rotating machines b. Differentiation or integration of a sinusoidal waveform produces a sinusoidal waveform of the same frequency, differing only in magnitude and phase angle. Thus when a sinusoidal current is passed through (or a sinusoidal voltage applied across) a resistor, inductor or a capacitor a sinusoidal voltage waveform (or current waveform) of the same frequency, differing only in magnitude and phase angle, is obtained. If i(t) = I m sin (?t+f), for a resistor, v(t) = R.i(t) = R.I m sin (?t+f) = V m sin (?t+f) - magnitude changed by R but no phase shift for an inductor, vt L di dt L d dt It LI t LIt mm m () ..(sin()) ..cos() ..sin( /) == + = + = ++ ?f ? ? f ? ?f p 2 - magnitude changed by L? and phase angle changed by p/2 for a capacitor, vt C idt C It dt C It C It mm m () . sin() .cos() .sin( /) =·= +·= - · += · +- ?? 11 1 1 2 ?f ? ?f ? ?f p - magnitude changed by 1/C? and phase angle changed by -p/2 c. Sinusoidal waveforms have the property of remaining unaltered in shape and frequency when other sinusoids having the same frequency but different in magnitude and phase are added to them. A sin (?t+a) + B sin (?t+ß) = A sin ?t. cos a + ? cos ?t. sin a + B sin ?t. cos ß + B cos ?t. sin ß = (A. cos a + B. cos ß) sin ?t + (A. sin a + B. sin ß) cos ?t = C sin (?t + ?), where C = ) cos( 2 2 2 ß a - + + AB B A , and ? = ? ? ? ? cos sin sin cos tan 1 B A B A + + - are constants. d. Periodic, but non-sinusoidal waveforms can be broken up to direct terms, its fundamental and harmonics. f(t) = F o + F 1 sin(?t + ? 1 ) + F 2 sin(2?t + ? 2 ) + F 3 sin(3?t + ? 3 ) + F 4 sin(4?t + ? 4 ) + ........ where F n and ? n are constants dependant on the function f(t). e. Sinusoidal waveforms can be represented by the projections of a rotating phasor. Phasor Representation of Sinusoids You may be aware that sin ? can be written in terms of exponentials and complex numbers. i.e. ej j? ?? =+ cos sin or etj t jt ? ?? =+ cos sin a(t) t T A m sin? t ? ? O O P X Page 4 AC Theory â€“ Professor J R Lucas 1 November 2001 Alternating Current Theory - J R Lucas An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = V m sin(? t + f ) If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = A m sin(? t+ f ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = A m (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. Mean value A mean = ? +T t t o o dt t a T ). ( 1 The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. For a sinusoid, Mean value = ? + + T t t m o o dt t A T ). ( sin 1 f ? = 0 (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value A avg = ? ? ? ? ? ? ? ? ? ? - ? ? fcycle hal negative fcycle hal positive dt t a dt t a T ). ( ). ( 1 For a sinusoid, Average value = ? ? ? ? ? ? ? ? - ? ? T T m T m dt t A dt t A T 2 2 0 . sin . sin 1 ? ? = m m A A T p ? 2 2 1 = · The average value is defined in the above manner in electrical engineering as otherwise all alternating waveforms would have zero value and would be indistinguishable. a(t) -A m A mean v(t) t T T V m f/? A avg v rect t T T AC Theory â€“ Professor J R Lucas 2 November 2001 (e) Effective value or r.m.s. value: Neither the peak value, nor the mean value, nor the average value defines the useful value of the waveform with regard to the power or energy. Thus the effective value is defined based on the power equivalent of the quantity. The effective value is thus defined as the constant value which would cause the same power dissipation as the original quantity over one complete period. Thus considering current Power dissipation = I eff 2 . R = ? +T t t o o dt R t i T . ). ( 1 2 giving I eff = ? +T t t o o dt t i T ). ( 1 2 or in general Effective value A eff = ? +T t t o o dt t a T ). ( 1 2 It is to be noted that the method of calculating the effective value involves the following processes. Taking the root of the mean of the squared waveform over one complete cycle. Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform. i.e. r.m.s. value A rms = ? +T t t o o dt t a T ). ( 1 2 This is defined for both current as well as voltage. For a sinusoid, r.m.s. value = ? + + T t t m o o dt t A T ). ( sin 1 2 2 f ? = 2 m A Unless otherwise specified, the rms value is the value that is always specified for ac waveforms, whether it be a voltage or a current. For example, 230 V in the mains supply is an rms value of the voltage. Similarly when we talk about a 5 A, 13 A or 15A socket outlet (plug point), we are again talking about the rms value of the rated current of the socket outlet. For a given waveform, such as the sinusoid, the peak value, average value and the rms value are dependant on each other. The peak factor and the form factor are the two factors that are most commonly defined. value average value rms = Factor Form and for a sinusoidal waveform, Form Factor = 111 . 1 1107 . 1 2 2 ? = p m m V V The form factor is useful such as when the average value has been measured using a rectifier type moving coil meter and the rms value is required to be found. [Note: You will be studying about these meters later] Peak Factor = peak value rms value and for a sinusoidal waveform, Peak Factor = The peak factor is useful when defining highly distorted waveforms such as the current waveform of compact fluorescent lamps. V V m m 2 2 1 4142 == . v 2 (t) t T T t i AC Theory â€“ Professor J R Lucas 3 November 2001 Some advantages of the sinusoidal waveform for electrical power applications a. Sinusoidally varying voltages are easily generated by rotating machines b. Differentiation or integration of a sinusoidal waveform produces a sinusoidal waveform of the same frequency, differing only in magnitude and phase angle. Thus when a sinusoidal current is passed through (or a sinusoidal voltage applied across) a resistor, inductor or a capacitor a sinusoidal voltage waveform (or current waveform) of the same frequency, differing only in magnitude and phase angle, is obtained. If i(t) = I m sin (?t+f), for a resistor, v(t) = R.i(t) = R.I m sin (?t+f) = V m sin (?t+f) - magnitude changed by R but no phase shift for an inductor, vt L di dt L d dt It LI t LIt mm m () ..(sin()) ..cos() ..sin( /) == + = + = ++ ?f ? ? f ? ?f p 2 - magnitude changed by L? and phase angle changed by p/2 for a capacitor, vt C idt C It dt C It C It mm m () . sin() .cos() .sin( /) =·= +·= - · += · +- ?? 11 1 1 2 ?f ? ?f ? ?f p - magnitude changed by 1/C? and phase angle changed by -p/2 c. Sinusoidal waveforms have the property of remaining unaltered in shape and frequency when other sinusoids having the same frequency but different in magnitude and phase are added to them. A sin (?t+a) + B sin (?t+ß) = A sin ?t. cos a + ? cos ?t. sin a + B sin ?t. cos ß + B cos ?t. sin ß = (A. cos a + B. cos ß) sin ?t + (A. sin a + B. sin ß) cos ?t = C sin (?t + ?), where C = ) cos( 2 2 2 ß a - + + AB B A , and ? = ? ? ? ? cos sin sin cos tan 1 B A B A + + - are constants. d. Periodic, but non-sinusoidal waveforms can be broken up to direct terms, its fundamental and harmonics. f(t) = F o + F 1 sin(?t + ? 1 ) + F 2 sin(2?t + ? 2 ) + F 3 sin(3?t + ? 3 ) + F 4 sin(4?t + ? 4 ) + ........ where F n and ? n are constants dependant on the function f(t). e. Sinusoidal waveforms can be represented by the projections of a rotating phasor. Phasor Representation of Sinusoids You may be aware that sin ? can be written in terms of exponentials and complex numbers. i.e. ej j? ?? =+ cos sin or etj t jt ? ?? =+ cos sin a(t) t T A m sin? t ? ? O O P X AC Theory â€“ Professor J R Lucas 4 November 2001 Consider a line OP of length A m which is in the horizontal direction OX at time t=0. If OP rotates at an angular velocity ? , then in time t its position would correspond to an angle of ? t. The projection of this rotating phasor OP (a phasor is somewhat similar to a vector, except that it does not have a physical direction in space but a phase angle) on the y-axis would correspond to OP sin ?t or A m sin ? t and on the x-axis would correspond to A m cos ? t. Thus the sinusoidal waveform can be thought of being the projection on a particular direction of the complex exponential e j?t . If we consider more than one phasor, and each phasor rotates at the same angular frequency, then there is no relative motion between the phasors. Thus if we fix the reference phasor OR in a particular reference direction (without showing its rotation), then all others phasors moving at the same angular frequency would also be fixed at a relative position. Usually this reference direction is chosen as horizontal on the diagram for convenience. It is also usual to draw the Phasor diagram using the rms value A of the sinusoidal waveform, rather than with the peak value A m . This is shown on an enlarged diagram. Thus unless otherwise specified it is the rms value that is drawn on a phasor diagram. It should be noted that the values on the phasor diagram are no longer time variables. The phasor A is characterised by its magnitude ?A? and its phase angle f. These are also the polar co-ordinates of the phasor and is commonly written as ?A? f . The phasor A can also be characterised by its cartesian co-ordinates A x and A y and usually written using complex numbers as A = A x + j A y . Note: In electrical engineering, the letter j is always used for the complex operator - 1 because the letter i is regularly used for electric current. It is worth noting that AA A xy =+ 22 and that tanf = A A y x or f = ? ? ? ? ? ? - tan 1 A A y x Also, A x = ?A? cos f, A y = ?A? sin f and ?A? e jf = ?A? cos f + j?A? sin f = A x + j A y Note: If the period of a sinusoidal waveform is T, then the corresponding angle would be ? T. Also, the period of a waveform corresponds to 1 complete cycle or 2 p radians or 360 0 . ? ? T = 2p R P f 0 A m R P f 0 A a(t) t T A m sin (? t+f) ? ? 0 0 R X P f Rotating Phasor diagram reference direction f 0 A A m = 2 Phasor diagram A A Page 5 AC Theory â€“ Professor J R Lucas 1 November 2001 Alternating Current Theory - J R Lucas An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = V m sin(? t + f ) If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = A m sin(? t+ f ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = A m (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. Mean value A mean = ? +T t t o o dt t a T ). ( 1 The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. For a sinusoid, Mean value = ? + + T t t m o o dt t A T ). ( sin 1 f ? = 0 (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value A avg = ? ? ? ? ? ? ? ? ? ? - ? ? fcycle hal negative fcycle hal positive dt t a dt t a T ). ( ). ( 1 For a sinusoid, Average value = ? ? ? ? ? ? ? ? - ? ? T T m T m dt t A dt t A T 2 2 0 . sin . sin 1 ? ? = m m A A T p ? 2 2 1 = · The average value is defined in the above manner in electrical engineering as otherwise all alternating waveforms would have zero value and would be indistinguishable. a(t) -A m A mean v(t) t T T V m f/? A avg v rect t T T AC Theory â€“ Professor J R Lucas 2 November 2001 (e) Effective value or r.m.s. value: Neither the peak value, nor the mean value, nor the average value defines the useful value of the waveform with regard to the power or energy. Thus the effective value is defined based on the power equivalent of the quantity. The effective value is thus defined as the constant value which would cause the same power dissipation as the original quantity over one complete period. Thus considering current Power dissipation = I eff 2 . R = ? +T t t o o dt R t i T . ). ( 1 2 giving I eff = ? +T t t o o dt t i T ). ( 1 2 or in general Effective value A eff = ? +T t t o o dt t a T ). ( 1 2 It is to be noted that the method of calculating the effective value involves the following processes. Taking the root of the mean of the squared waveform over one complete cycle. Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform. i.e. r.m.s. value A rms = ? +T t t o o dt t a T ). ( 1 2 This is defined for both current as well as voltage. For a sinusoid, r.m.s. value = ? + + T t t m o o dt t A T ). ( sin 1 2 2 f ? = 2 m A Unless otherwise specified, the rms value is the value that is always specified for ac waveforms, whether it be a voltage or a current. For example, 230 V in the mains supply is an rms value of the voltage. Similarly when we talk about a 5 A, 13 A or 15A socket outlet (plug point), we are again talking about the rms value of the rated current of the socket outlet. For a given waveform, such as the sinusoid, the peak value, average value and the rms value are dependant on each other. The peak factor and the form factor are the two factors that are most commonly defined. value average value rms = Factor Form and for a sinusoidal waveform, Form Factor = 111 . 1 1107 . 1 2 2 ? = p m m V V The form factor is useful such as when the average value has been measured using a rectifier type moving coil meter and the rms value is required to be found. [Note: You will be studying about these meters later] Peak Factor = peak value rms value and for a sinusoidal waveform, Peak Factor = The peak factor is useful when defining highly distorted waveforms such as the current waveform of compact fluorescent lamps. V V m m 2 2 1 4142 == . v 2 (t) t T T t i AC Theory â€“ Professor J R Lucas 3 November 2001 Some advantages of the sinusoidal waveform for electrical power applications a. Sinusoidally varying voltages are easily generated by rotating machines b. Differentiation or integration of a sinusoidal waveform produces a sinusoidal waveform of the same frequency, differing only in magnitude and phase angle. Thus when a sinusoidal current is passed through (or a sinusoidal voltage applied across) a resistor, inductor or a capacitor a sinusoidal voltage waveform (or current waveform) of the same frequency, differing only in magnitude and phase angle, is obtained. If i(t) = I m sin (?t+f), for a resistor, v(t) = R.i(t) = R.I m sin (?t+f) = V m sin (?t+f) - magnitude changed by R but no phase shift for an inductor, vt L di dt L d dt It LI t LIt mm m () ..(sin()) ..cos() ..sin( /) == + = + = ++ ?f ? ? f ? ?f p 2 - magnitude changed by L? and phase angle changed by p/2 for a capacitor, vt C idt C It dt C It C It mm m () . sin() .cos() .sin( /) =·= +·= - · += · +- ?? 11 1 1 2 ?f ? ?f ? ?f p - magnitude changed by 1/C? and phase angle changed by -p/2 c. Sinusoidal waveforms have the property of remaining unaltered in shape and frequency when other sinusoids having the same frequency but different in magnitude and phase are added to them. A sin (?t+a) + B sin (?t+ß) = A sin ?t. cos a + ? cos ?t. sin a + B sin ?t. cos ß + B cos ?t. sin ß = (A. cos a + B. cos ß) sin ?t + (A. sin a + B. sin ß) cos ?t = C sin (?t + ?), where C = ) cos( 2 2 2 ß a - + + AB B A , and ? = ? ? ? ? cos sin sin cos tan 1 B A B A + + - are constants. d. Periodic, but non-sinusoidal waveforms can be broken up to direct terms, its fundamental and harmonics. f(t) = F o + F 1 sin(?t + ? 1 ) + F 2 sin(2?t + ? 2 ) + F 3 sin(3?t + ? 3 ) + F 4 sin(4?t + ? 4 ) + ........ where F n and ? n are constants dependant on the function f(t). e. Sinusoidal waveforms can be represented by the projections of a rotating phasor. Phasor Representation of Sinusoids You may be aware that sin ? can be written in terms of exponentials and complex numbers. i.e. ej j? ?? =+ cos sin or etj t jt ? ?? =+ cos sin a(t) t T A m sin? t ? ? O O P X AC Theory â€“ Professor J R Lucas 4 November 2001 Consider a line OP of length A m which is in the horizontal direction OX at time t=0. If OP rotates at an angular velocity ? , then in time t its position would correspond to an angle of ? t. The projection of this rotating phasor OP (a phasor is somewhat similar to a vector, except that it does not have a physical direction in space but a phase angle) on the y-axis would correspond to OP sin ?t or A m sin ? t and on the x-axis would correspond to A m cos ? t. Thus the sinusoidal waveform can be thought of being the projection on a particular direction of the complex exponential e j?t . If we consider more than one phasor, and each phasor rotates at the same angular frequency, then there is no relative motion between the phasors. Thus if we fix the reference phasor OR in a particular reference direction (without showing its rotation), then all others phasors moving at the same angular frequency would also be fixed at a relative position. Usually this reference direction is chosen as horizontal on the diagram for convenience. It is also usual to draw the Phasor diagram using the rms value A of the sinusoidal waveform, rather than with the peak value A m . This is shown on an enlarged diagram. Thus unless otherwise specified it is the rms value that is drawn on a phasor diagram. It should be noted that the values on the phasor diagram are no longer time variables. The phasor A is characterised by its magnitude ?A? and its phase angle f. These are also the polar co-ordinates of the phasor and is commonly written as ?A? f . The phasor A can also be characterised by its cartesian co-ordinates A x and A y and usually written using complex numbers as A = A x + j A y . Note: In electrical engineering, the letter j is always used for the complex operator - 1 because the letter i is regularly used for electric current. It is worth noting that AA A xy =+ 22 and that tanf = A A y x or f = ? ? ? ? ? ? - tan 1 A A y x Also, A x = ?A? cos f, A y = ?A? sin f and ?A? e jf = ?A? cos f + j?A? sin f = A x + j A y Note: If the period of a sinusoidal waveform is T, then the corresponding angle would be ? T. Also, the period of a waveform corresponds to 1 complete cycle or 2 p radians or 360 0 . ? ? T = 2p R P f 0 A m R P f 0 A a(t) t T A m sin (? t+f) ? ? 0 0 R X P f Rotating Phasor diagram reference direction f 0 A A m = 2 Phasor diagram A A AC Theory â€“ Professor J R Lucas 5 November 2001 Phase difference Consider the two waveforms A m sin (? t+f 1 ) and B m sin (? t+f 2 ) as shown in the figure. It can be seen that they have different amplitudes and different phase angles with respect to a common reference. These two waveforms can also be represented by either rotating phasors A m e j(?t+f 1 ) and B m e j (?t+f 2 ) with peak amplitudes A m and B m , or by a normal phasor diagram with complex values A and B with polar co-ordinates ?A? f 1 and ?B? f 2 as shown . Any particular value (such as positive peak, or zero) of a(t) is seen to occur at a time T after the corresponding value of b(t). i.e. the positive peak A m occurs after an angle (f 2 -f 1 ) after the positive peak B m . Similarly the zero of a(t) occurs after an angle (f 2 -f 1 ) after the corresponding zero of b(t). In such a case we say that the waveform b(t) leads the waveform a(t) by a phase angle of (f 2 -f 1 ). Similarly we could say that the waveform a(t) lags the waveform b(t) by a phase angle of (f 2 -f 1 ). [Note: Only the angle less than 180 o is used to specify whether a waveform leads or lags another waveform]. We could also define, lead and lag by simply referring to the phasor diagram. Since angles are always measured anticlockwise (convention), we can see from the phasor diagram, that B leads A by an angle of (f 2 -f 1 ) anticlockwise or that A lags B by an angle (f 2 -f 1 ). Addition and subtraction of phasors can be done using the same parallelogram and triangle laws as for vectors, generally using complex numbers. Thus the addition of phasor A and phasor B would be A + B = (A cos f 1 + j A sin f 1 ) + (B cos f 2 + j B sin f 2 ) = (A cos f 1 + B cos f 2 ) + j (A sin f 1 + B sin f 2 ) = C x + jC y = ?C? f c = C where CC C A B A B xy =+ = + + + 22 12 2 12 2 (cos cos ) ( sin sin ) f f f f and ? ? ? ? ? ? + + = ? ? ? ? ? ? = - - ) cos cos ( ) sin sin ( tan tan 2 1 2 1 1 1 f f f f f B A B A C C x y c Example 1 Find the addition and subtraction of the 2 complex numbers given by 10?30 o and 25? 48 o . Addition = 10 ?30 o + 25 ?48 o = 10(0.8660 + j 0.5000) + 25(0.6691 + j 0.7431) = (8.660 + 16.728) + j (5.000 + 18.577) = 25.388 + j 23.577 = 34.647 ? 42.9 o Subtraction = 10?30 o - 25?48 o = (8.660 - 16.728) + j (5.000 - 18.577) = - 8.068 - j 13.577 = 15.793?239.3 o f 1 0 A A m = 2 f 2 -f 1 B B m = 2 f 1 0 f 2 B C A f 1 O f 2 y(t) ? t A m sin (? t+f 1 ) B m sin (? t+f 2 ) ?T f 2 -f 1 f 2 - f 1 A m B m ?Read More

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