AC Signals and Power

CHUI Chee Kong, PhD

Control & Mechatronics Group

Mechanical Engineering, NUS

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What you need to know

Identify the frequency, angular frequency, peak

value, root-mean-square (rms) value, and phase

of a sinusoidal signal

Determine rms value of periodic current or

voltage

Solve steady-state AC circuits using phasors

and complex impedances

Compute power for steady-state AC circuits

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Contents

1. Sinusoidal Signals

2. Phasors

3. Complex Impedances

4. Circuit Analysis with Phasors and

Complex Impedances

5. Power in AC Circuits

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1. Sinusoidal Signals

A sinusoidal voltage is given by

v(t) =Vm cos(ωt +θ )

where Vm is the maximum value of the voltage, ω is the

angular frequency in rad/s, and θ is the phase angle, in

radian.

Sinusoidal signals are periodic, repeating after each

period T (s): ωT = 2π.

The frequency of the signal is f=1/T Hz

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A sinusoidal voltage waveform given by v(t) = Vm cos (ωt + θ). Note: Assuming that θ is in

degrees, we have tmax = –θ/360 × T. For the waveform shown, θ is –45°.

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Root-Mean-Square (RMS) Values

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Root-Mean-Square (RMS) Values (continue)

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Example: Power Delivered to a Resistance

by a Sinusoidal Source

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2. Phasors

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Example: Using Phasors to Add Sinusoids

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Example: Using Phasors to Add Sinusoids

(continue)

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Phasors as Rotating Vectors

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A sinusoid can be represented as

the real part of a vector rotating

counterclockwise in the complex

plane.

Phase Relationships

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Since the vectors rotate

counterclockwise, V1 leads V2 by

60°(or, equivalently, V2 lags V1 by 60°)

v1(t) leads v2(t) by 60°= The peaks of v1(t) occur 60°before the peaks of v2(t).

3. Complex Impedances

Impedance: the ratio of the phasor of the sinusoidal

voltage to the phasor of the sinusoidal current

Complex numbers for inductance and capacitance

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Inductance

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Inductance (continue)

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Inductance (continue)

Current lags voltage by 90°in a pure inductance.

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Capacitance

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Capacitance (continue)

Current leads voltage by 90°in a pure capacitance.

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4. Circuit Analysis with Phasors

and Complex Impedances

In a circuit with

sinusoidal sources,

inductors, capacitors

and resistors, the

sources can be

replaced by their

phasors, and the

inductance and

capacitance can be

replaced by their

impedance, follow by

the application of

Kirchoff’s laws

(KVL,KCL).

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Example: Steady-State AC Analysis of a

Series Circuit

Find the steady-state current for the following circuit. Also, find the phasor voltage

across each element and construct a phasor diagram.

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Example: Steady-State AC Analysis of a

Series Circuit (continue)

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Example: Steady-State AC Analysis of a

Series Circuit (continue)below.

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100 + j100

= 141.4 [cos(45o)

+ j sin(45o)]

Example: Steady-State AC Analysis of a

Series Circuit (continue)

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below.

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I = 0.707 L-15o

VR = 70.7 L-15o

VL= 106.1 L75o

VC = 35.4 L-105o

VS = 100 L30o

5. Power in AC Circuits

Instantaneous and average power

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Single-Phase Power

Three-Phase Power

Three-phase power is an arrangement

in which three sinusoidal voltages are

generated out of phase with each other.

The most common case is the balanced

voltage case with 3 voltages of equal

amplitude and frequency but with phase

120 degrees apart.

Most of the AC power is generated and

distributed as three-phase power

because

Efficient – less wiring needed

Total power delivery is constant

Three-phase motor has non-zero starting

torque.

Voltage-vs-time plot

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Balanced Voltage Case

The voltages are balanced!

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Three Phase Source – Wye connected

(Y-connected)

Phase sequence can be important because

the direction of rotation of a three-phase

motor is opposite for the two phase

sequences. Hence, direction of the

motor can be reversed by interchanging the b

and c connection.30

Wye-Wye Connection

A three-phase source

is connected to a

balanced three-phase

load (i.e. the 3 loads

are equal).

Each source is referred

to as phase. The term

phase also refers to

the load.

Examples, phase A of

source is van(t); phase

A of load is the

impedance between A

and N.

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Wye-Wye Connection (continue)

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Note: IL = VY/Z.

Six wires are needed to connect three single-phase sources to three

loads. In a three-phase system, the same power transfer can be

accomplished with three wires. (Four if the neutral wire is used, for the

three-phase connection to achieve the same power transfer.)

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Advantage 1: wiring for connecting the sources to the loads is less

expensive for three-phase AC compared to that of single-phase AC.

Power in Three-Phase Circuit

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Advantage 2: the total power is constant (as a

function of time) rather than pulsating for

three-phase AC compared to that of single-

phase AC.

Line-to-Line Voltages

Phasor

diagram with

all phasors

from origin

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van = VY L0o

Example: Analysis of a Wye-Wye System

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below.

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To find the line-to-line phasors:

The power delivered to the load is:

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The reactive power is given by:

The phasor diagram is shown below. Note

that a different scale has been chosen for

the currents than for the voltages.

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Reactive power is the

power that flow back and

forth between the sources

and energy-storage

elements contained in a

three-phase or single-

phase load.

Example: Analysis of a Wye-Wye system

with 0.2 H inductance and 50 ohms

resistance

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A balance positive-sequence wye-connected 60 Hz three-phase source has

line-to-line voltage of VL = 1000 V. This source is connected to a balanced

wye-connected load. Each phase of the load consists of a 0.2-H inductance in

series with a 100-ohms resistance. Find the line-to-neural voltages, the line

currents and the power delivered to the load. Assume that the phase of Van is

zero.

Delta-Connected Sources and Loads

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Delta-connected three-

phase source

Loads can be either wye connected or delta connected

For a given wye-

connected source, there is

an equivalent delta-

connected source (or vice

versa).