Announcements HW 2 due on Tuesday at 6pm in Cory Midterm 1 on 7 12 12 00 1 30 Location TBD EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 1 Lecture 6 OUTLINE Chap 5 Phasors Complex Impedances Reading Chap 5 1 5 4 EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 2 1 Complex Numbers x is the real part y is the imaginary part z is the magnitude is the phase imaginary axis z y x EE40 Summer 2005 Lecture 6 real axis Instructor Octavian Florescu 3 More Complex Numbers Polar Coordinates A z Rectangular Coordinates A x jy x z cos z x2 y2 EE40 Summer 2005 Lecture 6 y z sin tan 1 Instructor Octavian Florescu y x 4 2 Summary of Phasors Phasor frequency domain is a complex number X z x jy Sinusoid is a time function x t z cos t EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 5 Examples Find the time domain representations of X 1 j2 V 104V j60V A 1mA j3mA EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 6 3 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 EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 7 Addition Addition is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 8 4 Addition Imaginary Axis A B B EE40 Summer 2005 Lecture 6 A Real Axis Instructor Octavian Florescu 9 Subtraction Subtraction is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 10 5 Subtraction Imaginary Axis B A A B EE40 Summer 2005 Lecture 6 Real Axis Instructor Octavian Florescu 11 Multiplication Multiplication is most easily performed in polar coordinates A AM B BM A B AM BM EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 12 6 Multiplication A B Imaginary Axis B A EE40 Summer 2005 Lecture 6 Real Axis Instructor Octavian Florescu 13 Division Division is most easily performed in polar coordinates A AM B BM A B AM BM EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 14 7 Division Imaginary Axis B A Real Axis A B EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 15 Complex Exponentials We represent a real valued sinusoid as the real part of a complex exponential Complex exponentials provide the link between time functions and phasors Complex exponentials make solving for AC steady state an algebraic problem EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 16 8 Complex Exponentials A complex number A z can be represented as A z z ej z cos j z sin A complex exponential is ej t cos t j sin t What do you get when you multiply A and ej t and find the real part EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 17 Complex Exponentials Aej t z ej ej t z ej t z ej t z cos t j z sin t Re Aej t z cos t EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 18 9 Sinusoids Complex Exponentials and Phasors Sinusoid z cos t Complex exponential Aej t z ej t Phasor V z EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 19 Phasor Relationships for Circuit Elements Phasors allow us to express currentvoltage relationships for inductors and capacitors much like we express the current voltage relationship for a resistor A complex exponential is the mathematical tool needed to obtain this relationship EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 20 10 I V Relationship for a Capacitor i t v t C i t C dv t dt Suppose that v t is a sinusoid v t VM ej t Find i t EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 21 Computing the Current dv t dVM e j t j i t C C dt dt i t j CVM e j t j j Cv t EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 22 11 Phasor Relationship Represent v t and i t as phasors V VM I j C V The derivative in the relationship between v t and i t becomes a multiplication by j in the relationship between V and I EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 23 Example v t 120V cos 377t 30 C 2 F What is V What is I What is i t EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 24 12 I V Relationship for an Inductor i t v t L v t L di t dt V j L I EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 25 Example i t 1 A cos 2 9 15 107t 30 L 1 H What is I What is V What is v t EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 26 13 Resistor I V relationship vR iRR VR IRR where R is the resistance in ohms VR phasor voltage IR phasor current boldface indicates complex quantity Capacitor I V relationship iC CdvC dt Phasor current IC phasor voltage VC capacitive impedance ZC IC VC ZC where ZC 1 j C j 1 1 2 and boldface indicates complex quantity Inductor I V relationship vL LdiL dt Phasor voltage VL phasor current IL inductive impedance ZL VL ILZL where ZL j L j 1 1 2 and boldface indicates complex quantity EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 27 Phasor Diagrams A phasor diagram is just a graph of several phasors on the complex plane using real and imaginary axes A phasor diagram helps to visualize the relationships between currents and voltages EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 28 14 Impedance AC steady state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm s law V IZ Z is called impedance EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 29 Some Thoughts on Impedance Impedance depends on the frequency Impedance is often a complex number Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 30 15 Example Single Loop Circuit 10V 0 20k VC 1 F 377 Find VC How do we find VC First compute impedances for resistor and capacitor ZR 20k 20k 0 ZC 1 j 377 1 F 2 65k 90 EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 31 Impedance Example 20k 0 10V 0 VC 2 65k 90 Now use the voltage divider to find VC 2 65k 90 VC 10V 0 2 65 k 90 20 k 0 VC 1 31V 82 4 EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 32 16 What happens when changes 10V 0 20k VC 1 F 10 Find VC EE40 Summer 2005 Lecture 6 Instructor Octavian Florescu 33 Low Pass Filter A Single Node pair Circuit 5mA 0 0 1 F 1k …
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