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# Chapter 9 - Chapter 9 Sinusoids and Phasors SJTU 1...

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SJTU 1 Chapter 9 Sinusoids and Phasors

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SJTU 2 Sinusoids A sinusoid is a signal that has the form of the sine or cosine function. angle phase ument t frequency angular amplitude Vm where t V v m = = + = = + = φ ω arg ) cos(
SJTU 3 ) cos( φ ω+ = t V v m φ ϖ t f T π ϖ 2 2 = = radians/second (rad/s) f is in hertz(Hz)

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SJTU 4 ) cos( ) ( ) cos( ) ( 2 2 2 1 1 1 φ ϖ + = + = t V t v t V t v m m Phase difference: θ by v lags v by v leads v phase in are v and v phase of out are v and v if t t 2 1 0 2 1 0 2 1 0 2 1 0 ) ( ) ( 2 1 2 1 < = - = + - + =
SJTU 5 Complex Number form l exponentia form sinusoidal form polar form r rectangula φ j re z jrsin rcos z r z jy x z = + = = + = φ

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SJTU 6 Phasor a phasor is a complex number representing the amplitude and phase angle of a sinusoidal voltage or current. Eq.(8-1) and Eq. (8-2) Eq.(8-3)
SJTU 7 When Eq.(8-2) is applied to the general sinusoid we obtain E q.(8-4) The phasor V is written as Eq.(8-5)

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SJTU 8 Fig. 8-1 shows a graphical representation commonly called a phasor diagram. Fig. 8-1: Phasor diagram Two features of the phasor concept need emphasis: 1. Phasors are written in boldface type like V or I 1 to distinguish them from signal waveforms such as v(t) and i1(t) . 2. A phasor is determined by amplitude and phase angle and does not contain any information about the frequency of the sinusoid.
SJTU 9 In summary, given a sinusoidal signal , the corresponding phasor representation is . Conversely, given the phasor , the corresponding sinusoid is found by multiplying the phasor by and reversing the steps in Eq. (8-4) as follows: E q.(8-6) ) cos( ) ( φ ω+ = t V t v m = Vm V Time domain representation Phase-domain representation

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SJTU 10 Properties of Phasors additive property Eq.(8- 7) Eq.(8-8) Eq.(8-9)
SJTU 11 derivative property Eq.(8- 10) V j dt dv ϖ Time domain representation Phase-domain representation

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SJTU 12 Integral property Time domain representation Phase-domain representation ϖ j V vdt The differences between v(t) and V: 1.
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Chapter 9 - Chapter 9 Sinusoids and Phasors SJTU 1...

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