1. Applied Circuit Analysis Chapter 11 AC Voltage and Current Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. Overview 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. 2

3. Alternating Current Alternating Current, or AC, is the dominant form of electrical power that is delivered to homes and industry. In the late 1800’s there was a battle between proponents of DC and AC. AC won out due to its efficiency for long distance transmission. AC is a sinusoidal current, meaning the current reverses at regular times and has alternating positive and negative values. 3

4. Generating AC In the DC circuits the power source we used was a battery. But batteries cannot provide the very large amount of power required by homes and industry. Electrical power can also be created by rotating a coil of wire in a static magnetic field. This rotation causes the output voltage to alternate between positive and negative. 4

5. Generating AC II The output voltage can be expressed mathematically: Where V m is the maximum voltage and  is the angle (in radians) of rotation of the coil. 5

6. 6 Alternating Voltage

7. SINUSOIDAL ac VOLTAGE CHARACTERISTICS/DEFINITIONS FIG. 13.2 Various sources of ac power: (a) generating plant; (b) portable ac generator; (c) wind- power station; (d) solar panel; (e) function generator

8. Sinusoidal Voltage If the angle  is replaced with an expression indicating time (  t), we get: Where –V m is the amplitude of the sinusoid –  is the angular frequency in radians per second –t is the time in seconds 8

9. Sinusoids Sinusoids are interesting to us because there are a number of natural phenomenon that are sinusoidal in nature. It is also a very easy signal to generate and transmit. Also, through Fourier analysis, any practical periodic function can be made by adding sinusoids. Lastly, they are very easy to handle mathematically. 9

10. Sinusoids II A sinusoidal forcing function produces both a transient and a steady state response. When the transient has died out, we say the circuit is in sinusoidal steady state. A sinusoidal voltage may be represented as: The function repeats itself every T seconds. This is called the period 10

11. Sinusoids III The period is inversely related to another important characteristic, the frequency The units of this is cycles per second, or Hertz (Hz) It is often useful to refer to frequency in angular terms: Here the angular frequency is in radians per second 11

12. Radians Although degrees are more commonly used in engineering, radians are the more natural unit for sinusoids. Thus it is important to establish the relationship between degrees and radians. A 360-degree revolution is equal to 2  radians. Or: 12

13. 13 EXAMPLE

14. Phase More generally, we need to account for relative timing of one wave versus another. This can be done by including a phase shift,  : Consider the two sinusoids: 14

15. 15 v2 leads v1 by θ or v1 lags v2 by θ

16. 16 EXAMPLE

17. 17 EXAMPLE

18. Phase II If two sinusoids are in phase, then this means that the reach their maximum and minimum at the same time. Sinusoids may be expressed as sine or cosine. The conversion between them is: 18

19. Phase Relations Now let us compare two sinusoids operating at the same frequency: The starting point of v 2 occurs first in time. We thus say that v 2 leads v 1 by , or v 1 lags v 2. 19

20. Phase Relations II We can make this comparison because the two sinusoids are at the same frequency. The difference angle must be less than 180 degrees to be called leading or lagging. Sinusoids may also be expressed as cosines. This may require transforming one waveform to allow proper comparison. We can use trigonometric identities to do so: 20

21. Trig Identities 21

22. 22 EXAMPLE SOLUTIONS