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Lecture 4

- AC Fundamentals

Alternating Current

- Voltages of ac sources alternate in polarity and

vary in magnitude. - These voltages produce currents which vary in

magnitude and alternate in direction. - A sinusoidal ac waveform starts at zero,

increases to a positive maximum, decreases to

zero, changes polarity, increases to a negative

maximum, then returns to zero.

Sinusoidal waveforms

Generating AC Voltages

Generating AC Voltages

Coil voltage vs angular position

Current direction

Voltage and Current Conventions for AC

- Assign a reference polarity for the source.
- When the voltage e has a positive value, its

actual polarity is the same as the reference

polarity. - When e is negative, its actual polarity is

opposite that of the reference polarity. - When i has a positive value, its actual direction

is the same as the reference arrow. - If i is negative, its actual direction is

opposite that of the reference.

References for voltage and current

Frequency

- The number of cycles per second of a waveform is

called its frequency. - Frequency is denoted f.
- The unit of frequency is the hertz.
- 1 Hz 1 cycle per second

Period

- The period of a waveform is the duration of one

cycle. - It is measured in units of time.
- It is the inverse of frequency.
- T 1/f
- For 50 Hz T0.02 s

Amplitude and Peak-to-Peak Value

- The amplitude of a sine wave is the distance from

its average to its peak. - We use Em for amplitude.
- Peak-to-peak voltage is measured between the

minimum and maximum peaks. - We use Ep-p or Vp-p.

Peak Value

- The peak value of a voltage or current is its

maximum value with respect to zero. - If a sine wave rides on top of a dc value, the

peak is the sum of the dc voltage and the ac

waveform amplitude.

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The Basic Sine Wave Equation

- The voltage produced by a generator is
- e Emsin ?.
- Em is the maximum voltage and ? is the

instantaneous angular position of the rotating

coil of the generator. - The voltage at any point on the sine wave may be

found by multiplying Em times the sine of angle

at that point.

Angular Velocity

- The rate at which the generator coil rotates is

called its angular velocity, ?. - The units for ? are revolutions/second,

degrees/sec, or radians/sec.

Radian Measure

- ? is usually expressed in radians.
- 2? radians 360
- To convert from degrees to radians, multiply by

?/180. - To convert from radians to degrees, multiply by

180/?.

Relationship between ?,T, and f

- One cycle of a sine wave may be represented by ?

2? rads or t T s.

Voltages and Currents as Functions of Time

- Since ?? ?t, the equation e Emsin ? becomes e

Emsin ?t. - Also v Vmsin ?t and i Imsin ?t.
- These equations may be used to compute voltages

and currents at any instant of time.

Voltages and Currents with Phase Shifts

- If a sine wave does not pass through zero at
- t 0, it has a phase shift.
- For a waveform shifted left,
- v Vmsin(?t ?).
- For a waveform shifted right,
- v Vmsin(?t - ?).

Phase shifts

Phasors

- A phasor is a rotating line whose projection on a

vertical axis can be used to represent

sinusoidally varying quantities. - A sinusoidal waveform can be created by plotting

the vertical projection of a phasor that rotates

in the counterclockwise direction at a constant

angular velocity ?. - Phasors apply only to sinusoidal waveforms.

Rotating phasor

Shifted Sine Waves

- Phasors may be used to represent shifted

waveforms. - The angle ? is the position of the phasor at t

0 seconds.

Phase Difference

- Phase difference is the angular displacement

between waveforms at the same frequency. - If the angular displacement is 0, the waveforms

are in phase otherwise they are out of phase. - If v1 5 sin(100t) and v2 3 sin(100t - 30),

v1 leads v2 by 30.

Phase differences

Average Value

- To find an average value of a waveform, divide

the area under the waveform by the length of its

base. - Areas above the axis are positive, areas below

the axis are negative. - Average values are also called dc values because

dc meters indicate average values rather than

instantaneous values.

Sine Wave Averages

- The average value of a sine wave over a complete

cycle is zero. - The average over a half cycle is not zero.
- The full-wave average is 2/?0.637 times the

maximum value. - The half-wave average is 1/? 0.318 times the

maximum value.

Effective Values

- An effective value is an equivalent dc value.
- It tells how many volts or amps of dc that an ac

waveform is equal to in terms of its ability to

produce the same average power. - In Australia, house voltage is 240 V(ac). This

means that the voltage is capable of producing

the same average power as 240 V(dc).

Effective Values

- To determine the effective power, we set

Power(dc) Power(ac). - Pdc pac
- I2R i2R where i Imsin ?t
- By applying a trigonometric identity, we are able

to solve for I in terms of Im. - Ieff Im/ ?2 0.707Im
- Veff 0.707Vm
- The effective value is also known as the RMS

value.

- R,L, and C Elements and the Impedance Concept

Introduction

- To analyze ac circuits in the time domain is not

very practical. - It is more practical to represent voltages and

currents as phasors, circuit elements as

impedances, and use complex algebra to analyze. - With this approach, ac circuits can be handled

much like dc circuits - the relationships and

laws still apply.

Complex Number Review

- A complex number is in the form a jb, where j

- a is the real part and b is the imaginary part of

the complex number. - This called the rectangular form.
- A complex number may be represented graphically

with a being the horizontal component and b being

the vertical component.

Conversion between Rectangular and Polar Forms

- If C a jb in rectangular form, then C C??,

where

Complex Number Review

- j 2 -1
- j 3 -j
- j 4 1
- 1/j -j
- To add complex numbers, add the real parts and

imaginary parts separately. - Subtraction is done similarly.

Review of Complex Numbers

- To multiply or divide complex numbers, it is best

to convert to polar form first. - (A??)(B??) (AB)?(? ?)
- (A??)/(B??) (A/B)?(? - ?)
- (1/C??) (1/C)?-?
- The complex conjugate of a jb is a - jb.

Voltages and Currents as Complex Numbers

- AC voltages and currents can be represented as

phasors. - Since phasors have magnitude and angle, they can

be viewed as complex numbers. - A voltage given as 100 sin(314t 30) can be

written as 100?30.

Voltages and Currents as Complex Numbers

- We can represent a source by its phasor

equivalent from the start. - The phasor representation contains all the

information we need except for the angular

velocity. - By doing this, we have transformed from the time

domain to the phasor domain. - KVL and KCL apply in both time domain and phasor

domain.

Sinusoidal source complex number

Summing AC Voltages and Currents

- To add or subtract waveforms in time domain is

very tedious. - This can be done easier by converting to phasors

and adding as complex numbers. - Once the waveforms are added, the corresponding

time equation and companion waveform can be

determined.

Summing waveforms point by point

Summing phasors

Important Notes

- Until now, we have used peak values when writing

voltages and current in phasor form. It is more

common that they be written as RMS values. - To add or subtract sinusoidal voltages or

currents, convert to phasor form, add or

subtract, then convert back to sinusoidal form. - Quantities expressed as phasors are said to be in

phasor domain or frequency domain.

R,L, and C Circuits with Sinusoidal Excitation

- R, L, and C circuit elements each have quite

different electrical properties. - These differences result in different

voltage-current relationships. - When a circuit is connected to a sinusoidal

source, all currents and voltages in the circuit

will be sinusoidal. - These sine waves will have the same frequency as

the source and will differ from it only in terms

of their magnitudes and angles.

Resistance and Sinusoidal AC

- In a purely resistive circuit, Ohms Law applies

the current is proportional to the voltage. - Current variations follow voltage variations,

each reaching their peak values at the same time. - The voltage and current of a resistor are in

phase.

Resistance

- For a resistor, the voltage and current are in

phase. - If the voltage has a phase angle, the current has

the same angle. - The impedance of a resistor is equal to R?0.

Inductive Circuit

- The voltage of an inductor is proportional to the

rate of change of the current. - Because the voltage is greatest when the rate of

change (or the slope) of the current is greatest,

the voltage and current are not in phase. - The voltage phasor leads the current by 90 for

an inductor.

Inductive Reactance

- Inductive reactance, XL, represents the

opposition that inductance presents to current

for the sinusoidal ac case. - XL is frequency-dependent.
- XL V/I and has units of ohms.
- XL ?L 2?fL

Inductance

- For an inductor, voltage leads current by 90.
- If the voltage has an angle of 0, the current

has an angle of -90. - The impedance of an inductor is XL?90.

Inductance V and I

Inductance V and I

Capacitive Circuits

- In a capacitive circuit, the current is

proportional to the rate of change of the

voltage. - The current will be greatest when the rate of

change of the voltage is greatest, so the voltage

and current are out of phase. - For a capacitor, the current leads the voltage by

90.

Capacitive Reactance

- Capacitive reactance, XC, represents the

opposition that capacitance presents to current

for the sinusoidal case. - XC is frequency-dependent. As the frequency

increases, XC decreases. - XC V/I and has units of ohms.

Capacitance

- For a capacitor, the current leads the voltage by

90. - If the voltage has an angle of 0, the current

has an angle of 90. - The impedance of a capacitor is given as XC?-90.

Capacitance V and I

Impedance

- The opposition that a circuit element presents to

current is the impedance, Z. - Z V/I, is in units of ohms
- Z in phasor form is Z?? where ? is the phase

difference between the voltage and current.

Impedance

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