SINUSOIDAL
WAVEFORMS
The sinusoidal waveform is the only waveform
whose shape is not affected by the response
characteristics of R, L, and C elements.
Enzo Paterno
1
CIRCUIT ELEMENTS
R[Ω]
Ω: Ohms
Resistance:
Georg Simon Ohm
Capacitance:
C[F]
+
F: Farads
Michael Faraday
L[H]
Inductance:
L: Henries
Joseph Henry
Enzo Paterno
2
SINUSOIDAL FUNCTION
ω = 2πf ; 2π rad = 360°; π = 3.1415 ; 1 rad = 57.29°
Sine wave
A
A
θ
0
T =
t
1
f
2πt
v(t ) = A sin(ωt + θ ) = A sin( 2πft + θ ) = A sin(
+θ )
T
• A = peak amplitude [volts]
•
•
•
•
•
f = frequency [cycles/s, hertz]
ω = angular frequency [rad/s]
T = period [sec]
θ = phase [radians or degrees]
t = time [sec]
ac source  Alternating Current
Creates an alternating (+, -, …)
varying voltage
Enzo Paterno
3
SINE WAVE GEOMETRY
http://www.ies.co.jp/math/java/samples/graphSinX.html
JSP-Applet: Sine Wave
Enzo Paterno
4
SINE WAVE – Degrees Vs Radians
90°
0°
0
r
r
r
π
π
2
180°
270°
3π
2
360°
2π
degrees
radians
A radian is defined by a quadrant of a circle where the distance
subtended on the circumference equals the radius of the circle
C = 2πr = 360°
360° = 2π radians → π ≅ 3.14159
1 rad ≅ 57.3°
Enzo Paterno
5
SINE WAVE – Degrees Vs Radians
90°
0°
0
π
π
2
180°
270°
3π
2
360°
2π
degrees
radians
 π 
Radians = 
 x degrees
 180° 
 180° 
Degrees = 
 x radians
 π 
Enzo Paterno
6
SINE WAVE – Angular Velocity
The velocity with which the radius vector rotates about the
center, called the angular velocity, can be determined by:
distance [ radians ]
Angular Velocity =
time [s]
α 2π
1
ω= =
= 2π f → f ∝ ω , T ω
t
T
∝
α =ωt
α ≈ angle
Note : 2πft
Enzo Paterno
=α
7
General Format for I & V SINUSOIDALS
A
0°
A sin α → A ≈ Peak amplitude, α ≈ unit of measure
for the horizontal axis
α
2π
A sin ω t → ω =
→ ω=
→ ω = 2πf
t
T
i
i (α ) = I p sin α
i (t ) = I p sin ω t → ac current
α = sin -1
Ip
v(α ) = V p sin α
v(t ) = V p sin ω t
v
→ ac voltage α = sin
Vp
Enzo Paterno
-1
8
RELATIVE PHASES OF SINUSOIDALS
A
t
0°
@ θ = 0°  f(t) = 0
A sin ω t → phase = 0° (no vertical axis shift)
A sin (ω t ± θ ) → phase = θ ° (θ ° shift from vertical axis)
Shift Right
Asin (ω t + θ )
+v
Asin (ω t − θ )
Shift Left
θ
θ
0°
α
0°
α
-v
Shift wave left
Enzo Paterno
Shift wave right
9
RELATIVE PHASES OF SINUSOIDALS
va (t ) = 10 sin(ω t + 45°)
vb (t ) = 10 sin(ω t + 105°)
10
va and vb have same frequency
but different phases:
vb va
8
6
4
105° - 45° = 60°
2
vb
60 º
1
0
va
t
2
-2
 vb leads va by 60º (leads in time)
-6
lags
leads
-4
-8
 va lags vb by 60º (lags in time)
-10
0
2
Enzo Paterno
4
60º
6
8
10
12
14
16
18
10
20
RELATIVE PHASES OF SINUSOIDALS
cos ωt = sin(ωt + 90 )

+cos leads +sin by 90º
+sin lags +cos by 90º
- sine leads +cos by 90º
- cos lags +sin by 90º
+cos
-sin
CCW:
+ angles
+sin
-cos
CW:
- angles
sin ωt = cos(ωt − 90 )

− sin ωt = cos(ωt + 90 )

− cos ωt = sin(ωt − 90 )
sin( −ωt ) = − sin ωt  Odd Function

cos(−ωt ) = cos ωt 
Even Function
− sin ωt = sin(ωt ± 180 )

ω
ω
t
t
− cos
=
±
cos(
180
)
Enzo Paterno

11
RELATIVE PHASES OF SINUSOIDALS
90 º
t
0°
cos ωt = sin(ωt + 90 )
+cos leads +sin by 90º
+cos
90°
0°
t
-sin
+sin
-cos
Enzo Paterno
CCW:
+ angles
CW:
- angles
12
Oscilloscope Frequency Measurements
A
T
 H sensitivity 
T = no. of H div 
;
div


Enzo Paterno
1
f =
T
t
13
Oscilloscope Phase Measurements
Note: 2 possible answers
i.e. θº or 360º – θº
1
2
Ɵ
Enzo Paterno
e leads i by 144º
14
Oscilloscope Phase Measurements
Phase and Time Measurements Continued
Phase and time measurements are
related as shown by the equation:
θ
t
=
360° T
The time from the start of a waveform
to a given phase angle can be found
using:
θ
t =T
360°
Enzo Paterno
15
Oscilloscope Phase Measurements
Enzo Paterno
16
RELATIVE PHASES OF SINUSOIDALS
Example:
v(t ) = 10 sin(ωt + 30°)
i (t ) = 5 sin(ωt + 70°)
Note that v(t) and i(t) are both sin, thus:
 i leads v by 40º
 v lags i by 40º
40°
30°
70°
Example:
v(t ) = 10 sin(ωt − 20°)
i (t ) = 15 sin(ωt + 60°)
Note that v(t) and i(t) are both sin, thus:
 i leads v by 80º
 v lags i by 80º
80°
-20°
Enzo Paterno
60°
17
PHASOR REPRESENTATION
Whenever an arbitrary phase angle, θ, is associated with a sine or cosine
function of amplitude A, we can sketch this function using a phasor diagram.
We sketch a phasor of length A at an angle θ degrees with respect to the
appropriate axis.
f (t ) = A sin(ωt + θ )
f (t ) = A∠θ
A
θ
We only sketch the
magnitude and phase.
We have in effect “frozen”
the wave in time.
Phasor
Diagram
If we desire to plot sine and cosine functions onto the same phasor diagram,
we either convert the sine functions to its cosine function equivalent or
convert the cosine functions to its sine function equivalent.
Enzo Paterno
18
RELATIVE PHASES OF SINUSOIDALS
Relative to cos
Example:
v(t ) = 3 sin(ωt − 10°)
10º
2
i (t ) = 2 cos(ωt + 10°)
Relative to sin
100º
i (t ) = 2 cos(ωt + 10°) = 2 sin(ωt + 100°)
OR
v(t ) = 3 sin(ωt − 10°) = 3 cos(ωt − 100°)
 i leads v by 110º
 v lags i by 110º
100º
3
Enzo Paterno
10º
19
RELATIVE PHASES OF SINUSOIDALS
Example:
v(t ) = 2 sin(ωt + 10°)
i (t ) = − sin(ωt + 30°)
Although, one can use
± 180°, use -180° because
the resulting angle gives:
|θ| ≤ 180°
30º 1
-150º
i (t ) = − sin(ωt + 30°) = sin(ωt − 150°)
 v leads i by 160º
 i lags v by 160º
Enzo Paterno
20
PHASOR REPRESENTATION
Example:
Express e = 5 cos (ωt + 30º) in terms of the sine function.
We know:
Thus:
cos ωt = sin(ωt + 90)
5 cos(ωt + 30°) = 5 sin(ωt + 120°)
We know:
+cos
Relative to cos
e
30º
5
-sin
+sin
Relative to sin
120º
-cos
Enzo Paterno
21
PHASOR REPRESENTATION
Example:
Determine the phase of e2 with respect to e1 when
e1 = 5 sin (ωt - 30º) and e2 = -10 sin (ωt - 40º)
We know:
Thus:
− sin ωt = sin(ωt ± 180)
e2 = −10 sin(ωt − 40°) = 10 sin(ωt + 140°)
We know:
+cos
e2
Δθ = 170º
10
-sin
+sin
-cos
40º
140º
5
30º
e1
Enzo Paterno
22
ADDITION OF SINUSOIDS
WITH SAME FREQUENCY
Sinusoidal form
f (t ) = A1 sin(ωt + θ1 ) + A2 sin(ωt + θ 2 ) +  + An sin(ωt + θ n )
f1 = f2 means same velocity
Convert to Phasor form
f (t ) = A1∠θ1 + A2 ∠θ 2 +  + An ∠θ n
Convert to Rectangular form
f (t ) = ( A1 cos θ1 + A2 cos θ 2 +  + An cos θ n ) +
j
A sin θ
A
θ
A cos θ
j ( A1 sin θ1 + A2 sin θ 2 +  + An sin θ n )
Convert to Sinusoidal form
f (t ) = A sin(ωt + θ )
Enzo Paterno
23
ADDITION OF SINUSOIDS
Example:
Determine e(t) = e1(t) + e2(t) + e3(t) for
e1 = 10 sin (500t + 30º), e2 = -20 cos (500t - 45º), e3 = 30 cos (500t + 30º)
Sinusoidal form
e1 = 10 sin (500t + 30º), e2 = 20 sin (500t - 135º), e3 = 30 sin (500t + 120º)
Convert to Phasor form
Convert to
We know:
Rectangular
e1 (t ) = 10∠30° = 8.66 + j 5
+cos
form
e2 (t ) = 20∠ − 135° = −14.14 − j14.14
-sin
+sin
-cos
e3 (t ) = 30∠120° = −15 + j 25.98
e(t ) = 26.51∠ + 140.58°
e(t ) = 26.51sin(500t + 140.58°)
e(t ) = 26.51 cos(500t + 50.58°)
Enzo Paterno
Convert to
Sinusoidal
form
24
AVERAGE VALUE – dc VALUE
What do we mean by the average value?
For example we want the average height of the mound of sand:
Height
Mound
of sand
Compact Sand
in the Sandbox
d
Average Height
Compacted sand
d
If d is increased:
Height
Mound
of sand
Average Height decreases
Compacted sand
d
d
Enzo Paterno
25
AVERAGE VALUE – dc VALUE
What do we mean by the average value?
If a depression exists:
Height
Average Height decreases
even further
Compacted
Mound
of sand
sand
d
d
Average Area = Average Height x d
If we knew the mathematical equation for the curve representing the original
mound of sand:
Average Area =
Area under the curve
Total distance
Enzo Paterno
26
AVERAGE VALUE – dc VALUE
The average value (i.e. dc value) of a periodic function f(t) is defined as;
Area under the curve for 1 cycle
Average value =
Period
T
and is given by:
1
Fdc = ∫ f (t ) dt
T0
If f(t) is defined differently over various regions of the cycle, then the
Average value will be several integrals, one for each region
Remark:
Alegbraic sum of areas for 1 cycle
Average value =
Period
A true dc voltmeter or ammeter reads the average value of a periodic
Waveform when set to dc measurement.
Enzo Paterno
27
AVERAGE VALUE – dc VALUE
Example 30:
Find the average value of the current waveform:
I(t) [A]
8
5
7
2
4 5
8 10
t
-4
T=8
Fdc = [( 5 x 2 ) + ( 8 x 1) + ( -4 x 1)] / 8 = 1.75 A
Enzo Paterno
28
AVERAGE VALUE – dc VALUE
Example 31:
Find the average value of the sine wave:
1
Fdc =
2π
sin t
1
0
π
t
2π
2π
1
2π
∫0 sin tdt = − 2π cos t 0
1
(cos 2π − cos 0)
Fdc = −
2π
1
1
(0)
(1 − 1) = −
Fdc = −
2π
2π
Fdc = 0
Enzo Paterno
29
EFFECTIVE / RMS VALUES
Question: What is the equivalent dc (average) power to an ac
power for a particular load R when an ac source is applied?
This equivalent dc power is called the effective power or the rms
power (Root Mean Square).
vac = Vm sin ωt → iac = Im sin ωt
If the ac source is a
sinusoidal signal:
1

Pac = (iac ) R = ( I m sin ωt ) R = I m  (1 − cos 2ωt )  R
2

1
I 2 mR I 2 mR
2
sin
ω
t
=
(1 − cos 2ωt )
−
cos 2ωt
Pac =
2
2
2
2
2
2
2
0 (since average power)
I 2 mR I 2 mR
I 2 mR
Pav ( ac ) =
−
cos 2ωt =
2
2
2
Enzo Paterno
30
EFFECTIVE / RMS VALUES
We want:
Pav ( ac ) = Pdc
I 2 mR
= I 2 dcR
2
∴ Im = 2 Idc
Im
∴ Idc =
= 0.707 Im = Ieff = I rms
2
Another word, the equivalent dc value of a sinusoidal current or voltage is
0.707 of its maximum value. This equivalent dc value is called the effective
value of the sinusoidal quantity.
Enzo Paterno
31
EFFECTIVE / RMS VALUES
The effective value of any quantity plotted as a function of time, f(t), can be
found using the formula:
T
Frms =
RMS
1) Square the function
2) Find the Mean (average)
3) Take the square root
∫
f 2 (t )dt
0
T
The average power due
to a periodic (ac) source is:
2
Pav = I rms
R
2
rms
E
Pav =
R
Remark:
A true rms voltmeter or ammeter reads the effective value of a periodic
waveform when set to ac measurement.
Enzo Paterno
32
EFFECTIVE / RMS VALUES
Example:
Find the rms value of the current waveform:
I(t) [A]
8
5
7
2
4 5
8 10
t
-4
T=8
Irms =
(25 x 2) + (64 x1) + (16 x1)
= 4.03 A
8
Enzo Paterno
33