1. AC SINUSOIDS Lecture 6 (I)

2. SCOPE Explain the difference between AC and DC Express angular measure in both degrees and radians. Compute the peak, peak-peak, and instantaneous values of a waveform. Define and solve for the RMS value Define cycle, period, and frequency Given the analytical expression, sketch and explain the graph of a sinusoid. Determine the relative phase of a sinusoidal waveform.

3. OBJECTIVES (cont) Determine the total voltages and currents that have DC and AC components. Apply Ohm’s Law, KCL, and KVL to analyze a simple AC circuit. Write the time domain equation for any sinusoidal waveform with a DC component.

4. SINE WAVES Voltage can be produced such that, over time, it follows the shape of a sine wave The magnitude of the voltage continually changes. Polarity may or may not change. When it does not change, the current does not change direction. When polarity does change, the current changes direction. When graphing a sinusoidal voltage, the polarity changes only when the magnitude alternates between “+” and “-” values.

5. AC SINEWAVE 1 cycle Voltage is positive Voltage is negative Polarity change t voltage + - 0 + - 0 Voltage is Polarity change t voltage + - 0 + - 0

6. OTHER ACs SINE WAVE TRIANGLE WAVE SQUARE WAVE

7. HOW IS A SINE WAVE GENERATED ? Electromagnetic Induction. (Ship AC generators produce sine wave voltages through electromagnetic induction):  magnetic field  conductor  relative motion between the two. Electronic Signal Generators Function Generators: multi-waveforms.

8. GENERATING AC VOLTAGES One way to generate ac voltage is to rotate a coil of wire at constant angular velocity in a fixed magnetic field

9. FARADAY’S LAW “ Voltage is induced in a circuit whenever the flux linking (i.e. passing through) the circuit is changing.. and that the magnitude of the voltage is proportional to the rate of change of the flux linkages”

10. DC vs AC DC Source: voltage POLARITY of the source and current DIRECTION do not change over time.

11. AC SOURCE AC source: Voltage polarity changes therefore the current changes direction.

12. PERIOD AND FREQUENCY Period: Time to complete one complete cycle Symbol: T Frequency: Number of cycles in one second Symbol: f Measured in hertz (Hz) t V

13. FREQUENCY Definition: the number of cycles per second of a waveform Denoted by the lower case letter f Its unit is the hertz (Hz)

14. 1 cycle 1 second f=1 Hz Ex.

15. f=2 Hz Ex. 1 cycle 1 second 1 cycle

16. 60 cycles 1 cycle 1 second Ex.

17. PERIOD Definition: the duration of one cycle. It is the inverse of frequency. Denoted by the upper case letter T Measured in second, s

18. The period of a waveform can be measured between any two corresponding point. Often it is measured between zero points because they are easy to establish on an oscilloscope trace

19. T (between peaks) T (between zero points) T (Any two identical points) t

20. Ex. Figure shows an oscilloscope trace of a square wave. Each horizontal division represents 50 μs. Determine the frequency.

21. Solution Since the wave repeats itself every 200 μs, its period (T) is 200 μs and,

22. Ex. Determine the period and frequency of the waveform of the figure above. T 2 = 10 ms T 1 = 8 ms

23. Solution Time interval T 1 does not represent a period as it is not measured between corresponding points. Interval T 2, however, is. Thus, T = 10 ms and,

24. PEAK VALUES (V P, I P ) Max Voltage (Current) Symbol V M ( I M ) The maximum value of V (I) measured from the point of inflection (“baseline or DC offset”) From the graph: V M - V DC Also called “Amplitude”

25. baseline V M or Amplitude V DC t V

26. PEAK TO PEAK VALUES (V PP, I PP ) Peak to Peak Voltage (Current) Symbol V PP ( I PP ) The difference between the maximum value of V (I) and the minimum value of V (I) From the graph: V MAX – V MIN Equals twice peak value V PP = 2V P

27. V PP V MIN V MAX t V

28. ROOT-MEAN-SQUARE (V RMS, I RMS ) Named for the mathematical process by which the value is calculated. “Effective Voltage (V EFF )”

29. COMPATIBILITY OF VALUES When Peak voltages are used as source values, current calculations will also be in Peak values. Likewise, an RMS source produces answers in RMS. When solving a problem make sure all values are expressed ONE way (peak, peak to peak, or RMS)! VMVMVMVM V rms V pp

30. VOLTAGE & CURRENT VALUES Ohm’s Law still applies: V=IR If current changes with time and R is a constant, voltage will also change with time Voltage will be proportional to current

31. VOLTAGE & CURRENT VALUES A graph of current and voltage in a resistor produces identical waveforms: Peak at the same time Cross the same baseline, at the same time Differ only in amplitude: I P is 1/R of V P

32. INSTANTANEOUS VALUES Instantaneous Values ( v, i ) value of voltage and current at any: instant in time or at at any angle Mathematically expressed 2 ways:

33. ANGULAR DOMAIN We can identify points on the sine wave in terms of an angular measurement (degrees or radians). 1 rotation = 360°=2 radians The instantaneous value of the sine wave can be related to the angular rotation of the generator, (1 rotation = 360°=2 radians)

34. Sine Wave Angles: Degrees & Radians 2 radians = 360 o 1 radian = 57.3 o

35. TIME DOMAIN Because the time to complete a cycle is frequency dependent, we can also identify points on the sine wave in terms of time. To convert between the time domain and angular domain remember:

36. PHASE ANGLE Symbol is  (theta). It is expressed as an angle Phase angle specifies the lateral shift in the position of a sine wave from a reference wave. Examine the same event, on each wave: Two events occurring at the same angle or at the same time are in phase. Events occurring at different angles or at different times are out of phase.

37. PHASE ANGLE (angular domain) Wave A is the reference wave: Wave B is 90° out of phase.

38. PHASE ANGLE (Time domain) Wave A is the reference wave. Compare the positive peak events: Wave A peaks at 30ms; Wave B at 60ms T=120ms  /360º = t/T = (60ms-30ms)/120ms.  = 90º

39. LEADING & LAGGING Since wave B peaked after the reference wave peaked, we say it LAGS the reference wave by 90º ;  = - 90º If wave B was the reference, wave A would peak before the reference wave (B). We would say it LEADS the reference wave;  = + 90º Note: Because it is the reference wave,  for ANY reference wave is 0 º

40. Ex: Compute the phase angle if: V 1 (t) is the reference wave V 2 (t) is the reference wave t = 1 ms/div

41. Ex: V 2 is the reference. Write the equations.

42. SUPERIMPOSED DC & AC A circuit can have both a DC voltage source and an AC We say that the “AC rides on the DC” The graph of the voltage is displaced vertically from 0, to the DC voltage level. Algebraically:

43. REVIEW QUIZ The difference between DC and AC ? 3 items required for electromagnetic induction. Frequency is equal to ? Name 3 different Sine wave values. How many radians in 360 degrees ? If the peak value is 170 V, the RMS value =? What type of shift does a phase angle represent?