3-Lesson_Notes_Lecture_20 - Sinusoidal Circuits - Impedance...

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EE 204 Lecture 20 Sinusoidal Circuits - Impedance and Admittance Sinusoidal Circuits: A sinusoidal circuit is characterized by: 1) All sources vary sinusoidally with time 2) All currents and voltages vary sinusoidally with time 3) If all sources have the same ω All currents and voltages vary at that Figure 1 If all sources have the same all currents and voltages are described by: () cos ( ) ft A t θ =+ where is fixed for all currents and voltages However, A and are generally different An example of a sinusoidal circuit is shown: 1) All sources are sinusoidal: ( ) 120cos(100 ) [ ] s vt t V = & ( ) 5cos(100 ) [ ] s it t A = 2) All sources have the same angular frequency 100[ / ] rad s = 3) Analysis of the circuit reveals that: 1 ( ) 6.986cos(100 103.241 ) [ ] o t A =− & ( ) 152.06cos(100 26.565 ) [ ] o L t V 1 ( ) 69.86cos(100 103.241 ) [ ] o t V & ( ) 7.603cos(100 63.435 ) [ ] o L t A
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Thus: 1) 100[ / ] rad s ω = is common to all currents and voltages in the circuit 2) A & θ are generally different for each current and voltage 3) only A & are needed in order to determine a particular voltage or current Figure 2 New concepts will be introduced next, which are needed for the analysis of sinusoidal circuits The Instantaneous and the Phasor Representations: Given the general sinusoidal function () cos( ) ft A t = + It can be rewritten as: ( ) cos( ) Re[ cos( ) sin( )] Re[ ] jt A t A t jA t A e ωθ + =+ = + + + = , where Re = real part cos( ) Re[( ) ] jj t A tA e e += The complex constant j Ae is called the phasor representation of cos( ) At + To determine the phasor j Ae , we need to know the amplitude A and the phase angle Given () f t we can find its phasor F (and vice versa) using the following rule: cos ( ) A t j FA e = ( ) A t [ formally called the instantaneous representation or the time representation] j e = [ formally called the phasor representation]
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The Amplitude-Angle Representation of a Complex Number: The exponential form j Ae θ can be written as A [called the amplitude-angle representation] It is more economical to use A as compared to j Ae j A eA [Identical representations of the exponential form of a complex number] Thus: 12 1 2 () 1 2 1 1 2 2 12 1 2 ( ) ( ) jj j Ae AAe A A AA θθ + ×= = + 1 2 11 1 22 2 2 2 j j j A A A e A A A =≡ = cos ( ) ft A t ω =+ j FA e = ( ) cos( ) A t = + = Example 1: Find the phasor representation of: a) ( ) 3cos(20 30 ) o t b) ( ) 12cos(10 60 ) o vt t =− c) ( )
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3-Lesson_Notes_Lecture_20 - Sinusoidal Circuits - Impedance...

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