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Chapter 13



Sinusoidal waveform characteristics
Phase relationships
Average and RMS values
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
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
Voltage, Current – DC
Capacitor just stored energy
Voltages were simple numbers
E, I readings were constant, no matter when
they were taken.
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New York City
Utility Lines,
1890
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
Alternating current will be
carried primarily in the outer
portions of the conductor
◦ Based on material, signal
frequency

Factors into high power
transmission systems
◦ Ex- Hollow tubes in 50 kW radio
transmitters
◦ Ex- HV transmission line bundles
Current density distribution
in a conductor carrying
alternating current
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13.1 – 13.4
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
The path traced by a quantity, such as
voltage, plotted as a function of some
variable, such as time, temperature, etc…
FIG. 13.1 Alternating
waveforms.
FIG. 13.1 Alternating
waveforms.
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FIG. 13.3 Important parameters for a sinusoidal
voltage.
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Instantaneous value (e1, e2)
◦ Value at any given moment

Peak amplitude (Em)
◦ Value from average to highest value

Peak value
◦ Max instantaneous value compared to zero
◦ With no DC offset, same as peak amplitude
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Peak-to-peak value (Ep-p)
◦ Full voltage between positive and negative peaks
◦
=2

Periodic waveform
◦ Waveform that repeats itself after the same time
interval

Period (T )
◦ Time of a periodic waveform
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Portion of the waveform contained in one
period of time
FIG. 13.4 Defining the cycle and period of a
sinusoidal waveform.
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Cycles completed per second (Hz)
FIG. 13.5 Demonstrating the effect of a changing frequency on
the period of a sinusoidal waveform.
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FIG. 13.7 Example 13.1
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FIG. 13.8 Areas of
application for specific
frequency bands.
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Polarity determined by relation to axis
◦ Above - positive
FIG. 13.11 (a) Sinusoidal ac voltage sources;
(b) sinusoidal current sources.




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Closest to AC behavior
Unaffected by RLC
circuits
AC generation results in
sine wave
Shape unaffected by R, L,
or C components
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FIG. 13.13 Defining the
radian.
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FIG. 13.14 There are 2π radians in one full
circle of 360°.
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FIG. 13.15 Plotting a sine wave versus (a) degrees and
(b) radians.
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FIG. 13.16 Generating a sinusoidal
waveform through the vertical projection
of a rotating vector.
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



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ω – angular
velocity
α – angle
t – time
T – Period of
waveform
f - frequency

α = ωt
=2

ω=

Shorter waveform period, higher angular velocity
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FIG. 13.17 Demonstrating the effect
of ψ on the frequency and period.
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13.5 – 13.6

The basic mathematical format for the
sinusoidal waveform is:
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FIG. 13.18 Basic sinusoidal
function.
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sin
or
sin
FIG. 13.19 Example
13.9.
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
Write sinusoidal expression for the following
a) Vp = 6V, f=500 hz
b) v = 4.5 V, α= 25, t= 1mS
c) Ip = 2.3 mA, ω=1500
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sin(
27
± )
FIG. 13.27 Defining the phase shift for a
sinusoidal function that crosses the
horizontal axis with a positive slope before
0°.
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Negative shift

Positive Shift
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FIG. 13.31 Example 13.12(a): i leads
y by 40°.
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FIG. 13.35 Example 13.12(e): y and i are in
phase.
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A.
V=10sin(wt+30), I=5sin(wt+70)
B.
I=15sin(wt+60), V=10sin(wt-20)
C.
V=120sin(377t+120), I=5sin(377t-20)
D.
V=25sin(wt+180), I=15sin(wt-180)
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
Determine values based
on image on screen
◦ (Divisions)x(Sensitivity)
FIG. 13.38 Example 13.13.
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Using screenshot, find
◦ E
◦ I
◦ Phase shift between E and I
FIG. 13.39 Finding the phase
angle between waveforms using a
dual-trace oscilloscope.
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Capacitive circuit
◦ Current leads Voltage

Inductive circuit
◦ Voltage leads Current
FIG. 13.32 Example 13.12(b): i
leads y by 80°.
FIG. 13.33 Example 13.12(c): i
leads y by 110°.
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13.7 -13.8
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FIG. 13.41 Effect of distance
(length) on average value.
FIG. 13.42 Effect of depressions
(negative excursions) on
average value.
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FIG. 13.44 Example
13.14.
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
Contains both AC and DC information
◦ If DC offset is negative, first value will be negative
1.5 V(DC) + 2.5 V(AC) sin (7500t)
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Area for sine must use integration
=2
=
(
)
(
)
= 0.637
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=
FIG. 13.54 Example
13.17.
2
(
)
−2
2
(
FIG. 13.55 Example
13.18.
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
)
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The equivalent dc value of a sinusoidal
current or voltage is 0.707 of its peak value
Idc = Iac(rms) = 0.707Im
Any time power is delivered to a resistive load, energy is
delivered no matter what the polarity
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FIG. 13.60 Example
13.20.
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FIG. 13.61 Example
13.21.
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=
+
(
)
FIG. 13.68 Generation and display of a waveform having a
dc and an ac component.
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Find the RMS value for the following
a) 15 mA sin (377t)
b) 14 V sin (100t)
c) 1.5 V+2.5 sin (7500t)

Convert RMS to sinusoidal expression
d) 25 Vrms
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
Q. 18
◦ Draw the waveform, labeling start, halfway, and end

Q. 38
◦ Part C, find values for both E and I
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