AC Fundamentals

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Lecture 4

• AC Fundamentals

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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.

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Sinusoidal waveforms
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Generating AC Voltages
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Generating AC Voltages
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Coil voltage vs angular position
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Current direction
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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.

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References for voltage and current
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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

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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

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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.

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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.

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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.

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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/?.

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Relationship between ?,T, and f

• One cycle of a sine wave may be represented by ?
  2? rads or t T s.

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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.

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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 - ?).

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Phase shifts
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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.

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Rotating phasor
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Shifted Sine Waves

• Phasors may be used to represent shifted
  waveforms.
• The angle ? is the position of the phasor at t
  0 seconds.

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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.

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Phase differences
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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.

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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.

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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).

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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.

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

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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.

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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.

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Conversion between Rectangular and Polar Forms

• If C a jb in rectangular form, then C C??,
  where

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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.

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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.

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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.

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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.

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Sinusoidal source complex number
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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.

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Summing waveforms point by point
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Summing phasors
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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.

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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.

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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.

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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.

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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.

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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

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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.

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Inductance V and I
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Inductance V and I
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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.

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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.

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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.

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Capacitance V and I
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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.

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Impedance