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Unformatted text preview: Slide 1 EE40 Fall 08 Connie ChangHasnain EE40 Lecture 1622 Connie ChangHasnain October 6, 8, 10, 13, 15, 17, 20 Reading: Chap. 5, 6, Supplementary Reader Ch. 1, appendix on complex numbers Slide 2 EE40 Fall 08 Connie ChangHasnain 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 Slide 3 EE40 Fall 08 Connie ChangHasnain Example 1: 2nd Order RLC Circuit R + C V s L t=0 Slide 4 EE40 Fall 08 Connie ChangHasnain Example 2: 2nd Order RLC Circuit R + C V s L t=0 Slide 5 EE40 Fall 08 Connie ChangHasnain 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 ) cos( ) ( ) sin( ) ( = + + wt F A B wt F B A B A τ τ ) ( = + B F A B τ ) ( = 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. Slide 6 EE40 Fall 08 Connie ChangHasnain 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 2 1 1 1 0 1 1 90 j j e j e π = = ∠ ° = = ∠ ° Slide 7 EE40 Fall 08 Connie ChangHasnain 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 Slide 8 EE40 Fall 08 Connie ChangHasnain 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 Slide 9 EE40 Fall 08 Connie ChangHasnain 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 Slide 10 EE40 Fall 08 Connie ChangHasnain 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 Slide 11 EE40 Fall 08 Connie ChangHasnain 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 Slide 12...
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This note was uploaded on 10/24/2009 for the course EE 40 taught by Professor Changhasnain during the Fall '07 term at University of California, Berkeley.
 Fall '07
 ChangHasnain

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