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Pusan National University power PNU 세계로 미래로 Electric Circuits Fall, 2014 Chapter 12 Phasors and Impedance 세계로 미래로 power PNU 12.1 Introduction 12.1 Introduction This chapter will cover alternating current. A discussion of complex numbers is included prior to introducing phasors. Applications of phasors and frequency domain analysis for circuits including resistors, capacitors, and inductors will be covered. The concept of impedance and admittance is also introduced. Advanced Broadcasting & Communications Lab. 1 세계로 미래로 power PNU 12.2 Phasors and Complex Number 12.2 Phasors and Complex Number Complex Numbers A powerful method for representing sinusoids is the phasor. But in order to understand how they work, we need to cover some complex numbers first. A complex number z can be represented in rectangular form as: z x jy It can also be written in polar or exponential form as: z r re j Advanced Broadcasting & Communications Lab. 2 세계로 미래로 power PNU 12.2 Phasors and Complex Number Complex Numbers II The different forms can be interconverted. Starting with rectangular form, one can go to polar: r x2 y 2 tan 1 y x Likewise, from polar to rectangular form goes as follows: x r cos y r sin Advanced Broadcasting & Communications Lab. 3 세계로 미래로 power PNU 12.2 Phasors and Complex Number Complex Numbers III The following mathematical operations are important Addition Subtraction Multiplication z1 z2 x1 x2 j y1 y2 z1 z2 x1 x2 j y1 y2 z1 z2 r1r2 1 2 Division z1 r1 1 2 z2 r2 Reciprocal 1 1 z r Square Root z r / 2 Complex Conjugate z* x jy r re j Advanced Broadcasting & Communications Lab. 4 세계로 미래로 power PNU 12.2 Phasors and Complex Number Phasors The idea of a phasor representation is based on Euler’s identity: e j cos j sin From this we can represent a sinusoid as the real component of a vector in the complex plane. The length of the vector is the amplitude of the sinusoid. The vector,V, in polar form, is at an angle with respect to the positive real axis. Advanced Broadcasting & Communications Lab. 5 세계로 미래로 power PNU 12.2 Phasors and Complex Number Phasors II Phasors are typically represented at t=0. As such, the transformation between time domain to phasor domain is: v t Vm cos t V Vm (Time-domain representation) (Phasor-domain representation) They can be graphically represented as shown here. Advanced Broadcasting & Communications Lab. 6 세계로 미래로 power PNU 12.2 Phasors and Complex Number Phasor Diagram A convenient way to compare sinusoids of the same frequency is to use a phasor diagram. Here the magnitude is represented as the length of the arrow and the phase is represented by the angle. Advanced Broadcasting & Communications Lab. 7 세계로 미래로 power PNU 12.2 Phasors and Complex Number Example and Problem Example 12.2 Example 12.3 Advanced Broadcasting & Communications Lab. 8 세계로 미래로 power PNU 12.3 Phasor Relationships for Circuit Elements 12.3 Phasor Relationships for Circuit Elements Here is a handy table for transforming various time domain sinusoids into phasor domain: Advanced Broadcasting & Communications Lab. 9 세계로 미래로 power PNU 12.3 Phasor Relationships for Circuit Elements Sinusoids-Phasor Transformation Note that the frequency of the phasor is not explicitly shown in the phasor diagram For this reason phasor domain is also known as frequency domain. Applying a derivative to a phasor yields: dv dt jV (Phasor domain) (Time domain) Applying an integral to a phasor yeilds: vdt (Time domain) Advanced Broadcasting & Communications Lab. V j (Phasor domain) 10 세계로 미래로 power PNU 12.3 Phasor Relationships for Circuit Elements Phasor Relationships for Resistors Each circuit element has a relationship between its current and voltage. These can be mapped into phasor relationships very simply for resistors capacitors and inductor. For the resistor, the voltage and current are related via Ohm’s law. As such, the voltage and current are in phase with each other. Advanced Broadcasting & Communications Lab. 11 세계로 미래로 power PNU 12.3 Phasor Relationships for Circuit Elements Phasor Relationships for Inductors Inductors on the other hand have a phase shift between the voltage and current. In this case, the voltage leads the current by 90°. Or one says the current lags the voltage, which is the standard convention. This is represented on the phasor diagram by a positive phase angle between the voltage and current. Advanced Broadcasting & Communications Lab. 12 세계로 미래로 power PNU 12.3 Phasor Relationships for Circuit Elements Voltage current relationships Advanced Broadcasting & Communications Lab. 13 세계로 미래로 power PNU 12.3 Phasor Relationships for Circuit Elements Example and Problem Example 12.6 v(t ) 12 cos(60t 45) Advanced Broadcasting & Communications Lab. 14 세계로 미래로 power PNU 12.4 Impedance and Admittance 12.4 Impedance and Admittance It is possible to expand Ohm’s law to capacitors and inductors. In time domain, this would be tricky as the ratios of voltage and current and always changing. But in frequency domain it is straightforward The impedance of a circuit element is the ratio of the phasor voltage to the phasor current. Z V I or V ZI Admittance is simply the inverse of impedance. Advanced Broadcasting & Communications Lab. 15 세계로 미래로 power PNU 12.4 Impedance and Admittance Impedance and Admittance II It is important to realize that in frequency domain, the values obtained for impedance are only valid at that frequency. Changing to a new frequency will require recalculating the values. The impedance of capacitors and inductors are shown here: Advanced Broadcasting & Communications Lab. 16 세계로 미래로 power PNU 12.4 Impedance and Admittance Impedance and Admittance III As a complex quantity, the impedance may be expressed in rectangular form. The separation of the real and imaginary components is useful. The real part is the resistance. The imaginary component is called the reactance, X. When it is positive, we say the impedance is inductive, and capacitive when it is negative. Advanced Broadcasting & Communications Lab. 17 세계로 미래로 power PNU 12.4 Impedance and Admittance Impedance and Admittance IV Admittance, being the reciprocal of the impedance, is also a complex number. It is measured in units of Siemens The real part of the admittance is called the conductance, G The imaginary part is called the susceptance, B These are all expressed in Siemens or (mhos) The impedance and admittance components can be related to each other: G R R2 X 2 Advanced Broadcasting & Communications Lab. B X R2 X 2 18 세계로 미래로 power PNU 12.4 Impedance and Admittance Impedance and Admittance V Advanced Broadcasting & Communications Lab. 19 세계로 미래로 power PNU 12.4 Impedance and Admittance Kirchoff’s Laws in Frequency Domain A powerful aspect of phasors is that Kirchoff’s laws apply to them as well. This means that a circuit transformed to frequency domain can be evaluated by the same methodology developed for KVL and KCL. One consequence is that there will likely be complex values. Advanced Broadcasting & Communications Lab. 20 세계로 미래로 power PNU 12.4 Impedance and Admittance Kirchoff’s Laws in Frequency Domain A powerful aspect of phasors is that Kirchoff’s laws apply to them as well. This means that a circuit transformed to frequency domain can be evaluated by the same methodology developed for KVL and KCL. One consequence is that there will likely be complex values. Advanced Broadcasting & Communications Lab. 21 세계로 미래로 power PNU 12.4 Impedance and Admittance Example and Problem Example 12.7 Advanced Broadcasting & Communications Lab. 22 세계로 미래로 power PNU 12.5 Impedance Combinations 12.5 Impedance Combinations Once in frequency domain, the impedance elements are generalized. Combinations will follow the rules for resistors: Advanced Broadcasting & Communications Lab. 23 세계로 미래로 power PNU 12.5 Impedance Combinations Seires Combinations Series combinations will result in a sum of the impedance elements: Z eq Z1 Z 2 Z3 ZN Here then two elements in series can act like a voltage divider Z1 V1 V Z1 Z 2 Z2 V2 V Z1 Z 2 Advanced Broadcasting & Communications Lab. 24 세계로 미래로 power PNU 12.5 Impedance Combinations Parallel Combinations Likewise, elements combined in parallel will combine in the same fashion as resistors in parallel: 1 1 1 1 Zeq Z1 Z 2 Z3 Advanced Broadcasting & Communications Lab. 1 ZN 25 세계로 미래로 power PNU 12.5 Impedance Combinations Delta-Wye transformation Zb Zc Z1 Z a Zb Zc Z1Z 2 Z 2 Z 3 Z 3 Z1 Za Z1 Zc Za Z2 Z a Zb Zc Z1Z 2 Z 2 Z 3 Z 3 Z1 Zb Z2 Z a Zb Z3 Z a Zb Zc Z1Z 2 Z 2 Z 3 Z 3 Z1 Zc Z3 Advanced Broadcasting & Communications Lab. 26 세계로 미래로 power PNU 12.5 Impedance Combinations Example and Problem Example 12.8 Example 12.9 Advanced Broadcasting & Communications Lab. 27 세계로 미래로 power PNU Homework Homework Read Text Chapter 13. Thevenin and Norton Equivalent Circuits pp. 341-362 Prepare Presentation Solve Problems 12.7, 12.20, 12.24, 12.35, 12.47 Computer Problem 12.53 Advanced Broadcasting & Communications Lab. 28