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ECE 1100: Introduction to Electrical and Computer Engineering

ECE 1100: Introduction to Electrical and Computer Engineering. Spring 2011. Wanda Wosik Associate Professor, ECE Dept. Notes 20. Power in AC Circuits and RMS. Notes prepared by Dr. Jackson. +. v ( t ). -. AC Power. V p = peak voltage. R [ ]. f = frequency [Hz].

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ECE 1100: Introduction to Electrical and Computer Engineering

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  1. ECE 1100: Introduction toElectrical and Computer Engineering Spring 2011 Wanda Wosik Associate Professor, ECE Dept. Notes 20 Power in AC Circuits and RMS Notes prepared by Dr. Jackson

  2. + v (t) - AC Power Vp = peak voltage R [] f = frequency [Hz] Note: The phase of the voltage wave is assumed to be zero here for convenience. Goal: Find the average power absorbedby resistor:

  3. cos2 (t) Tp = T / 2 = 0.5 / f [s] AC Power (cont.) cos (t) T = 1/f[s]

  4. AC Power (cont.) Note: We obtain the same result if we integrate over Tp or T.

  5. AC Power (cont.) Consider the integral that needs to be evaluated:

  6. AC Power (cont.) “The average value of cos2 is 1/2.”

  7. AC Power (cont.) Ic = T/2 so Hence

  8. + v (t) R [] - Summary

  9. Effective Voltage Veff Define: Note: Veffis used the same way we use V in a DC power calculation. Then we have:

  10. + V R [] v (t) R [] - Effective Voltage Veff DC AC same formula

  11. Example In the U. S., 60 Hz line voltage has an effective voltage of 120 [V]. Describe the voltage waveform mathematically. Veff = 120 [V] so

  12. + R = 144 [] - Example 60 Hz line voltage is connected to a 144 [] resistor. Determine the average power being absorbed. 120 [V] (eff)

  13. RMS (Root Mean Square) This is a general way to calculate the effective voltage for any periodic waveform (not necessarily sinusoidal). tp v(t) t digital pulse waveform T Duty cycle: D = tp / T

  14. RMS (cont.) By definition, Also, Hence,

  15. RMS (cont.) Hence

  16. RMS (cont.) Define VRMS is the root (square root) of the mean (average) of the square of the voltage waveform Comparing with the formula for Veff , we see that Veff = VRMS

  17. RMS (cont.) For sinusoidal (AC) signals, For other periodic signals, there will be a different relationship between VRMS and Vp. (See the example at the end of these notes.)

  18. + R [] - v (t) i (t) RMS Current The concept of effective (RMS) current works the same as for voltage. Define:

  19. + R [] - v (t) i (t) RMS Current (cont.) RMS current can be easily related to RMS voltage. where

  20. + R = 144 [] - IRMS Example 60 Hz line voltage is connected to a 144 [] resistor. Determine the RMS current and the average power absorbed (using the current formula). 120 [V] (RMS)

  21. + + VRMS R [] - - IRMS RMS Voltage and Current Power can also be expressed in terms of both RMS voltage and current.

  22. + R = 144 [] - IRMS 120 [V] (RMS) Example 60 Hz line voltage is connected to a 144 [] resistor. Determine the average power (using the voltage-current formula).

  23. + + VRMS R [] - - IRMS Summary of AC Power

  24. Vp v (t) T t Example (non-sinusoidal) Find the RMS voltage of a sawtooth waveform:

  25. Example (cont.) Hence

  26. v (t) + R [] - Vp v (t) t T Example (sawtooth wave) Vp= 10 [V] Given: R = 100 [] Find the average power absorbed by the resistor.

  27. Vp= 10 [V] v (t) v (t) + 100 [] - t T Example (cont.) (for sawtooth)

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