Dr. Sahar Abd El Moneim Moussa 1

Pharos UniversityEE-385

Electrical Power & Machines“Electrical Engineering Dept”

Prepared By:Dr. Sahar Abd El Moneim Moussa

Dr. Sahar Abd El Moneim Moussa 2

Single Phase System

Dr. Sahar Abd El Moneim Moussa 3

Review on AC CircuitBasic Principles:Sinusoidal voltage source: it is a source that

produces voltage that varies with time as sine wave

Waveform:

Where: T: periodic time: the time of one complete cycle.f: number of cycles per second = 1/T Hzω: angular frequency of the sine wave = 2πf rad/sec

T

Dr. Sahar Abd El Moneim Moussa 4

Equation: V(instantaneous)= Vm sin ωt

Where: Vm

: The maximum voltage value & it is knows as the amplitude

Vrms : Root mean square of the voltage =

Symbol:

~ V(t)= Vm sin ωt

Dr. Sahar Abd El Moneim Moussa 5

Resistive Circuit Circuit diagram:

Equation: V= Vm sin ωt , I= Im sin ωt Waveform: “in terms of the time domain”

Phasor diagram:

∴ The Resistive current is in phase with the voltage

VmIm

V(t)= Vm sin ωt

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Capacitive CircuitCircuit diagram:Equation: V= Vm sin ωt , I= Im sin (ωt+90) Waveform:

Phasor Diagram:

Where θ is the angle between the voltage and the current ( cos θ is called the power factor )

∴ The Capacitive current leads the voltage by 90o

Vm

Im

Xc = 1/ωc Ω

θ

Dr. Sahar Abd El Moneim Moussa

7

Inductive Circuit Circuit diagram:

Equation: V= Vm sin ωt , I= Im sin (ωt-90) Waveform:

Phasor Diagram:

Where θ is the angle between the voltage and the current ( cos θ is called the power factor )

∴ The Inductive current lags its voltage by 90o

XL = ωL Ω

Vm

Im

θ

Dr. Sahar Abd El Moneim Moussa

Dr. Sahar Abd El Moneim Moussa 8

Power in a single-phase system

Active PowerP

Watt

Reactive Power

QVAR

Apparent Power

SVA

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Power in a Single-Phase SystemActive Power, P, Watt:

The electric energy per second consumed by the resistive loads in the circuit which can be withdrawn outside the circuit as thermal, mechanical, light or another forms of energy

P= Vrms Irms Cos θ watt

Reactive Power, Q, VAR:This type of power is the source of the energy stored in the circuit & used to maintain the magnetic field in case of the inductor OR the electric field in case of the capacitor

P= Vrms Irms Sin θ VAR

This type of energy cannot be withdrawn outside the circuit

10Dr. Sahar Abd El Moneim Moussa

Apparent Power, S, VA:The resultant of the active & reactive power

S= Vrms I* rms

S= Vrms Irms ∠ -θ VA

Q

P

S

θ

V

Load

Q

P

P

Source

Q

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NOTE: Circuit diagram: Equations:

v(t)= Vm sinωt

iR (t)= IRm sin ωt

iL (t)= ILm sin (ωt-90)

iC (t)= ICm sin (ωt+90)

Phasor diagram of the current:

V

I

IR

IC

IL

IL - IC

Itotal

Ѳ

IR

IC

IL

Itotal

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12

Phasor diagram of the power:

V

I

P=S Cos θ = I2 R

Q=S Sin θ = I2 XS=Vrms Irms

Ѳ

Dr. Sahar Abd El Moneim Moussa

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Therefore, the power triangle will be

ѲQ

S

P