1 ECE 3144 Lecture 31 Dr. Rose Q. Hu Electrical and Computer Engineering Department Mississippi State University

2 Reminder from Lecture 30 Complex forcing function V M e j(  t+  ) results in complex response I M e j(  t+  ) Phasor notation: Phasor relationahships for R, L, C –For a resistor R: V=RI –For a capacitor C, I=j  CV –For an inductor L, V=j  LI

3 Impedance The current-voltage relationships for the three passive elements, R, C, and L, have been investigated. We find that the phasor voltage/phasor current ratios are simple quantities that dependent on element values and frequencies. We treat these ratios in the same manner that we treat resistances, with exception that they are complex quantities. The two-terminal input impedance Z, is defined as the ratio of the phasor voltage V to the phasor current I: ac circuit Notice that the passive sign convention still applies to the ac circuit defined by phasor voltage, phasor current and impedance.

4 Impedance Since Z is the ratio of V to I, the units of Z are ohms. An impedance in an ac circuit is analogous to the resistance in a dc circuit. Impedance Z can also be expressed as Z(  )=R(  )+jX(  ), where R(  ) is the real, or resistive, component and X(  ) is the imaginary, or reactive, component. We simply refer R as the resistance and X as the reactance. Both R and X are real functions of  => Z(  ) is frequency dependent. Z is a complex number. BUT it is not a phasor. Phasors only denote sinusoidal functions. We know Passive element impedance: => and Or => R=Zcos  z and X= Zsin  z Passive elementImpedance RZ=R L Z=j  L=  L  90 o C Z=1/j  C=j/-  C= 1/  C  -90 o

5 Series and Parallel impedance combinations KCL and KVL are both valid in the frequency domain. We can follow what was done for the resistors, to show that impedance can be combined in the same way. If Z 1, Z 2, …., Z n are connected in series, the equivalent impedance Z s is Z s = Z 1 +Z 2 +…+Z n If Z 1, Z 2, …, Z n are connected in parallel, the equivalent impedance Z p is given by

6 Admittance Admittance Y of a circuit element is defined as the ratio of phasor current to phasor voltage. We can see that admittance is the reciprocal of impedance Z. Admittance is analogous to conductance in resistive dc circuits. The units of Y are also siements. Y is also a complex number, which can be expressed as G and B are called conductance and susceptance, respectively. Based on the relationship between Y and Z, we have The admittance of the individual circuit elements are => and or and Passive elementImpedance RY=1/R L Y=1/j  L=-j/  L C Y=j  C

7 Series and parallel admittance combinations Once again, KCL and KVL are valid in the frequency domain. We can use the same approach outlined in Chapter 2 for conductance in resistive circuits to derive the following rules: –If Y 1, Y 2, …, Y n are connected in series, the equivalent admittance Y s is –If Y 1, Y 2, …, Y n are connected in parallel, the equivalent admittance Y p is Y p = Y 1 +Y 2 +…+Y n

8 Phasor diagrams The phasor diagram is a name given to sketch in the complex plane showing the relationships of the phasor voltages and phasor current for a specific circuit. It provides a graphical method for solving certain problems. Since phasor voltages and currents are complex numbers, they can be identified as points in the complex domain. Now we use complex domain to identify complex numbers, to do addition and subtractions, all graphically.

9 Examples Examples will be provided during the class.

10 Homework for lecture 31 Problems 7.8, 7.12, 7.16,7.18, 7.23 Due April 8