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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D1 ECE 3183 – EE Systems Chapter 5 – Part D AC Power, Power Factor

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D2 EFFECTIVE OR RMS VALUES The effective value is the equivalent DC value that supplies the same average power

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D3 EFFECTIVE OR RMS VALUES

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D4 EFFECTIVE OR RMS VALUES Definition is valid for ANY periodic signal with period T

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D5 EFFECTIVE OR RMS VALUES If the current is sinusoidal the average power is

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D6 LEARNING EXAMPLE Compute the rms value of the voltage waveform

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D7 LEARNING EXAMPLE Compute the rms value of the voltage waveform

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D8 LEARNING EXAMPLE Compute the rms value of the voltage waveform and use it to determine the average power supplied to the resistor

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D9 LEARNING EXAMPLE Compute the rms value of the voltage waveform and use it to determine the average power supplied to the resistor

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D10 LEARNING EXTENSION Compute rms value of the voltage waveform

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D11 LEARNING EXTENSION Compute the rms value for the current waveforms and use them to determine average power supplied to the resistor

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D12 LEARNING EXTENSION Compute the rms value for the current waveforms and use them to determine average power supplied to the resistor

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D13 STEADY-STATE POWER ANALYSIS LEARNING GOALS Instantaneous Power For the special case of steady state sinusoidal signals Average Power Power absorbed or supplied during one cycle Maximum Average Power Transfer When the circuit is in sinusoidal steady state

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D14 STEADY-STATE POWER ANALYSIS LEARNING GOALS Effective or RMS Values For the case of sinusoidal signals Power Factor A measure of the angle between current and voltage phasors Power Factor Correction How to improve power transfer to a load by “aligning” phasors

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D15 STEADY-STATE POWER ANALYSIS LEARNING GOALS Complex Power Measure of power using phasors Single Phase Three-Wire Circuits Typical distribution method for households and small loads

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D16 INSTANTANEOUS POWER FIND i(t) and p(t) We know that: where yielding:

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D17 INSTANTANEOUS POWER LEARNING EXAMPLE

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D18 INSTANTANEOUS POWER

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D19 AVERAGE POWER For sinusoidal (and other periodic signals) we compute averages over one period DOES IT MATTER WHO LEADS?

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D20 AVERAGE POWER If voltage and current are in phase If voltage and current are in quadrature Purely resistive Purely inductive or capacitive (cos(0)=1) (cos(90)=0)

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D21 AVERAGE POWER LEARNING EXAMPLE Find the average power absorbed by the impedances

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D22 AVERAGE POWER LEARNING EXAMPLE Find the average power absorbed by the impedances Since an inductor does not absorb power, one can use voltages and currents for the resistive part only.

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D23 LEARNING EXAMPLE Determine the average power absorbed by each resistor, the total average power absorbed and the average power supplied by the source

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D24 LEARNING EXTENSION Find the AVERAGE power absorbed by each PASSIVE component and the total power supplied by the source Power supplied by source Method 1. Method 2:

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D25 LEARNING EXAMPLE Determine average power absorbed or supplied by each element To determine power absorbed/supplied by sources we need the currents I1, I3 Passive sign convention

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D26 LEARNING EXTENSION Determine average power absorbed/supplied by each element

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D27 LEARNING EXTENSION Determine average power absorbed/supplied by each element

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D28 LEARNING EXTENSION Determine average power absorbed/supplied by each element

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D29 LEARNING EXTENSION Determine average power absorbed/supplied by each element

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D30 THE POWER FACTOR

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D31 THE POWER FACTOR

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D32 LEARNING EXAMPLE Find the power supplied by the power company. Determine how it changes if the power factor is changed to 0.9 Power company

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D33 LEARNING EXAMPLE Find the power supplied by the power company. Determine how it changes if the power factor is changed to 0.9 Power company Current lags the voltage To solve for z, take the arccos of p.f. z = cos -1 (0.707) = 45°. Which one to choose? Since the load is lagging (inductive), we know 0 < z < 90° So pick z = + 45°.

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D34 LEARNING EXAMPLE Find the power supplied by the power company. Determine how it changes if the power factor is changed to 0.9 Power company Current lags the voltage If pf=0.9 Losses can be reduced by 2kW!

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D35 LEARNING EXAMPLE Find the power supplied by the power company. Determine how it changes if the power factor is changed to 0.9 Power company Current lags the voltage

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D36 LEARNING EXTENSION Determine the power savings if the power factor can be increased to 0.94

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D37 COMPLEX POWER

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D38 COMPLEX POWER P=Active PowerQ=Reactive Power

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ECE 3183 – Chapter 5 – Part D CHAPTER 5 - D39 COMPLEX POWER Let a load be Z L = R + jX. Then Q is: Positive if the load is inductive (X>0) Negative if the load is capacitive (X<0) Zero if the load is resistive (X=0) inductive capacitive

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