Sinusoidal waveform characteristics Phase relationships Average and RMS values

Transcription

1 Chapter 13 Sinusoidal waveform characteristics Phase relationships Average and RMS values 2 1

2 Voltage, Current DC Capacitor just stored energy Voltages were simple numbers E, I readings were constant, no matter when they were taken. 3 New York City Utility Lines,

3 Alternating current will be carried primarily in the outer portions of the conductor Based on material, signal frequency Factors into high power transmission systems Ex- Hollow tubes in 50 kw radio transmitters Ex- HV transmission line bundles Current density distribution in a conductor carrying alternating current

4 The path traced by a quantity, such as voltage, plotted as a function of some variable, such as time, temperature, etc FIG Alternating waveforms. FIG Alternating waveforms. 7 FIG Important parameters for a sinusoidal voltage. 8 4

5 Instantaneous value (e 1, e 2 ) Value at any given moment Peak amplitude (E m ) Value from average to highest value Peak value Max instantaneous value compared to zero With no DC offset, same as peak amplitude 9 Peak-to-peak value (E p-p ) Full voltage between positive and negative peaks E = 2A Periodic waveform Waveform that repeats itself after the same time interval Period (T ) Time of a periodic waveform 10 5

6 Portion of the waveform contained in one period of time FIG Defining the cycle and period of a sinusoidal waveform. 11 Cycles completed per second (Hz) FIG Demonstrating the effect of a changing frequency on the period of a sinusoidal waveform. 12 6

7 FIG Example FIG Areas of application for specific frequency bands. 14 7

8 Polarity determined by relation to axis Above - positive FIG (a) Sinusoidal ac voltage sources; (b) sinusoidal current sources. 15 Closest to AC behavior Unaffected by RLC circuits AC generation results in sine wave Shape unaffected by R, L, or C components 16 8

9 FIG Defining the radian. 17 FIG There are 2π radians in one full circle of

10 FIG Plotting a sine wave versus (a) degrees and (b) radians. 19 FIG Generating a sinusoidal waveform through the vertical projection of a rotating vector

11 ω angular velocity α angle t time T Period of waveform f - frequency α = ωt ω = 2πf ω = Shorter waveform period, higher angular velocity 21 FIG Demonstrating the effect of ψ on the frequency and period

12 The basic mathematical format for the sinusoidal waveform is: 24 12

13 FIG Basic sinusoidal function. 25 A sin ωt or A sin α FIG Example

14 Write sinusoidal expression for the following a) Vp = 6V, f=500 hz b) v = 4.5 V, α= 25, t= 1mS c) Ip = 2.3 ma, ω= A sin(ωt ± θ) FIG Defining the phase shift for a sinusoidal function that crosses the horizontal axis with a positive slope before

15 Negative shift Positive Shift 29 FIG Example 13.12(a): i leads y by

16 FIG Example 13.12(e): y and i are in phase. 31 A. V=10sin(wt+30), I=5sin(wt+70) B. I=15sin(wt+60), V=10sin(wt-20) C. V=120sin(377t+120), I=5sin(377t-20) D. V=25sin(wt+180), I=15sin(wt-180) 32 16

17 Determine values based on image on screen (Divisions)x(Sensitivity) FIG Example Using screenshot, find E I Phase shift between E and I FIG Finding the phase angle between waveforms using a dual-trace oscilloscope

18 Capacitive circuit Current leads Voltage Inductive circuit Voltage leads Current FIG Example 13.12(b): i leads y by 80. FIG Example 13.12(c): i leads y by

19 FIG Effect of distance (length) on average value. FIG Effect of depressions (negative excursions) on average value. 37 FIG Example

20 Contains both AC and DC information If DC offset is negative, first value will be negative 1.5 V (DC) V (AC) sin (7500t) 39 Area for sine must use integration Area per section = 2A G = ( ) ( ) = 0.637A 40 20

21 G = 2A ( ) 2A ( ) 2π FIG Example FIG Example The equivalent dc value of a sinusoidal current or voltage is of its peak value Idc = Iac(rms) = 0.707Im Any time power is delivered to a resistive load, energy is delivered no matter what the polarity 42 21

22 FIG Example FIG Example

23 V = Vdc + Vac(rms) FIG Generation and display of a waveform having a dc and an ac component. 45 Find the RMS value for the following a) 15 ma sin (377t) b) 14 V sin (100t) c) 1.5 V+2.5 sin (7500t) Convert RMS to sinusoidal expression d) 25 Vrms 46 23

24 Q. 18 Draw the waveform, labeling start, halfway, and end Q. 38 Part C, find values for both E and I 47 24