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EE40_Fall08_Lecture16-22

EE40_Fall08_Lecture16-22 - EE40 Lecture 16-22 Connie...

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Slide 1 EE40 Fall 08 Connie Chang-Hasnain EE40 Lecture 16-22 Connie Chang-Hasnain October 6, 8, 10, 13, 15, 17, 20 Reading: Chap. 5, 6, Supplementary Reader Ch. 1, appendix on complex numbers
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Slide 2 EE40 Fall 08 Connie Chang-Hasnain Chapter 5 • OUTLINE – Phasors as notation for Sinusoids – Arithmetic with Complex Numbers – Complex impedances – Circuit analysis using complex impdenaces – Dervative/Integration as multiplication/division – Phasor Relationship for Circuit Elements • Reading – Chap 5 – Appendix A
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Slide 3 EE40 Fall 08 Connie Chang-Hasnain Example 1: 2nd Order RLC Circuit R + - C V s L t=0
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Slide 4 EE40 Fall 08 Connie Chang-Hasnain Example 2: 2nd Order RLC Circuit R + - C V s L t=0
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Slide 5 EE40 Fall 08 Connie Chang-Hasnain Sinusoidal Sources Create Too Much Algebra ) cos( ) sin( ) ( wt B wt A t x P + = ) cos( ) sin( ) ( ) ( wt F wt F dt t dx t x B A P P + = + τ ) cos( ) sin( )) cos( ) sin( ( )) cos( ) sin( ( wt F wt F dt wt B wt A d wt B wt A B A + = + + + τ Guess a solution Equation holds for all time and time variations are independent and thus each time variation coefficient is individually zero 0 ) cos( ) ( ) sin( ) ( = - + + - - wt F A B wt F B A B A τ τ 0 ) ( = - + B F A B τ 0 ) ( = - - A F B A τ 1 2 + + = τ τ B A F F A 1 2 + - - = τ τ B A F F B Dervatives Addition Two terms to be general Phasors (vectors that rotate in the complex plane) are a clever alternative.
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Slide 6 EE40 Fall 08 Connie Chang-Hasnain Complex Numbers (1) x is the real part y is the imaginary part z is the magnitude θ is the phase ( 1) j = - θ z x y real axis imaginary axis Rectangular Coordinates Z = x + jy Polar Coordinates: Z = z ∠ θ Exponential Form: θ cos z x = θ sin z y = 2 2 y x z + = x y 1 tan - = θ (cos sin ) z j θ θ = + Z j j e ze θ θ = = Z Z 0 2 1 1 1 0 1 1 90 j j e j e π = = ∠ ° = = ∠ °
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Slide 7 EE40 Fall 08 Connie Chang-Hasnain Complex Numbers (2) 2 2 cos 2 sin 2 cos sin cos sin 1 j j j j j j e e e e j e j e θ θ θ θ θ θ θ θ θ θ θ θ - - + = - = = + = + = j j e ze z θ θ θ = = = Z Z Euler’s Identities Exponential Form of a complex number
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Slide 8 EE40 Fall 08 Connie Chang-Hasnain Arithmetic With Complex Numbers To compute phasor voltages and currents, we need to be able to perform computation with complex numbers. – Addition – Subtraction – Multiplication – Division (And later use multiplication by j ω to replace – Diffrentiation – Integration
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Slide 9 EE40 Fall 08 Connie Chang-Hasnain Addition Addition is most easily performed in rectangular coordinates: A = x + jy B = z + jw A + B = ( x + z ) + j ( y + w ) Real Axis Imaginary Axis A B A + B
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Slide 10 EE40 Fall 08 Connie Chang-Hasnain Subtraction Subtraction is most easily performed in rectangular coordinates: A = x + jy B = z + jw A - B = ( x - z ) + j ( y - w ) Real Axis Imaginary Axis A B A - B
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Slide 11 EE40 Fall 08 Connie Chang-Hasnain Multiplication Multiplication is most easily performed in polar coordinates: A = A M ∠ θ B = B M ∠ φ A × B = ( A M × B M ) ( θ + φ ) Real Axis Imaginary Axis A B A × B
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Slide 12 EE40 Fall 08 Connie Chang-Hasnain Division Division is most easily performed in polar coordinates: A = A M ∠ θ B = B M ∠ φ A / B = ( A M / B M ) ( θ - φ ) Real Axis Imaginary Axis A B A / B
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