Schedule…
Date
Day
8 Oct
Wed
9 Oct
Thu
10 Oct
Fri
11 Oct
Sat
12 Oct
Sun
13 Oct
Mon
Class
No.
11
Title
Dynamic Circuits
Chapters
Lab
Due date
Exam
4.2 – 4.4
Recitation
12
HW
Due date
HW 5
Exam 1 Review
LAB 4
14 Oct
Tue
15 Oct
Wed
ECEN 301
EXAM 1
13
AC Circuit Analysis
4.5
Discussion #11 – Dynamic Circuits
1
Change
1 Corinthians 15:51-57
51 Behold, I shew you a mystery; We shall not all sleep, but we shall all be
changed,
52 In a moment, in the twinkling of an eye, at the last trump: for the trumpet
shall sound, and the dead shall be raised incorruptible, and we shall be
changed.
53 For this corruptible must put on incorruption, and this mortal must put on
immortality.
54 So when this corruptible shall have put on incorruption, and this mortal
shall have put on immortality, then shall be brought to pass the saying that is
written, Death is swallowed up in victory.
55 O death, where is thy sting? O grave, where is thy victory?
56 The sting of death is sin; and the strength of sin is the law.
57 But thanks be to God, which giveth us the victory through our Lord Jesus
Christ.
ECEN 301
Discussion #11 – Dynamic Circuits
2
Lecture 11 – Dynamic Circuits
Time-Dependent Sources
ECEN 301
Discussion #11 – Dynamic Circuits
3
Time Dependent Sources
Periodic signals: repeating patterns that appear
frequently in practical applications
A periodic signal x(t) satisfies the equation:
x(t )
x(t nT )
1.0
2.00
time
ECEN 301
1.5
x(t)
x(t)
x(t)
1.5
0.0
0.00
0.0
0.00
n 1, 2, 3, ...
2.00
4.00
0.0
0.00
2.00
4.00
-1.5
-1.5
time
Discussion #11 – Dynamic Circuits
time
4
Time Dependent Sources
Sinusoidal signal: a periodic waveform satisfying the
following equation:
x(t )
A cos( t
1.5
A – amplitude
ω – radian frequency
φ – phase
T
A
x(t)
φ/ω
0.0
0.00
2.00
)
4.00
-A
-1.5
time
ECEN 301
Discussion #11 – Dynamic Circuits
5
Sinusoidal Sources
Helpful identities:
f
T
1
Hz (cycles / s )
T
2 f rad / s
2
s
t
2
rad
T
t
360
deg
T
ECEN 301
sin( t ) cos
t
cos( t ) sin
t
sin( t
cos( t
2
cos t 90 
sin
t 90 
2
) cos( ) sin( t ) sin( ) cos( t )
) cos( ) cos( t )  sin( ) sin( t )
Discussion #11 – Dynamic Circuits
6
Sinusoidal Sources
Why sinusoidal sources?
• Sinusoidal AC is the fundamental current type supplied to homes throughout the
world by way of power grids
• Current war:
• late 1880’s AC (Westinghouse and Tesla) competed with DC (Edison) for
the electric power grid standard
• Low frequency AC (50 - 60Hz) can be more dangerous than DC
• Alternating fluctuations can cause the heart to lose coordination
(death)
• High frequency DC can be more dangerous than AC
• causes muscles to lock in position – preventing victim from releasing
conductor
• DC has serious limitations
• DC cannot be transmitted over long distances (greater than 1 mile)
without serious power losses
• DC cannot be easily changed to higher or lower voltages
ECEN 301
Discussion #11 – Dynamic Circuits
7
Measuring Signal Strength
Methods of quantifying the strength of timevarying electric signals:
Average (DC) value
• Mean voltage (or current) over a period of time
Root-mean-square (RMS) value
• Takes into account the fluctuations of the signal about its
average value
ECEN 301
Discussion #11 – Dynamic Circuits
8
Measuring Signal Strength
Time – averaged signal strength: integrate signal
x(t) over a period (T) of time
x(t )
ECEN 301
1
T
T
0
x( )d
Discussion #11 – Dynamic Circuits
9
Measuring Signal Strength
Example1: compute the average value of the signal –
x(t) = 10cos(100t)
ECEN 301
Discussion #11 – Dynamic Circuits
10
Measuring Signal Strength
Example1: compute the average value of the signal –
x(t) = 10cos(100t)
T
2
2
100
ECEN 301
Discussion #11 – Dynamic Circuits
11
Measuring Signal Strength
Example1: compute the average value of the signal –
x(t) = 10cos(100t)
T
2
x(t )
2
100
1 T
x( )d
0
T
100 2 /100
10 cos(100 t )dt
0
2
10
sin( 2 ) sin( 0)
2
0
NB: in general, for any sinusoidal signal
A cos( t
ECEN 301
)
0
Discussion #11 – Dynamic Circuits
12
Measuring Signal Strength
Root–mean–square (RMS): since a zero average
signal strength is not useful, often the RMS value is
used instead
The RMS value of a signal x(t) is defined as:
x(t ) rms
1
T
T
0
x( ) 2 d
NB: often ~
x (t ) notation is
used instead of x(t ) rms
NB: the rms value is simply the square root of the average
(mean) after being squared – hence: root – mean – square
ECEN 301
Discussion #11 – Dynamic Circuits
13
Measuring Signal Strength
Example2: Compute the rms value of the sinusoidal
current i(t) = I cos(ωt)
ECEN 301
Discussion #11 – Dynamic Circuits
14
Measuring Signal Strength
Example2: Compute the rms value of the sinusoidal
current i(t) = I cos(ωt)
1
irms
T
T
0
i 2 ( )d
2 /
2
2
cos (t )
cos( 2t ) 1
2
0
2 /
2
0
1 2
I
2
Integrating a sinusoidal waveform
over 2 periods equals zero
ECEN 301
I 2 cos 2 (
1
2
I2
2 /
2
0
)d
1
cos( 2
2
) d
I2
cos( 2
2
)d
1 2
I 0
2
I
2
Discussion #11 – Dynamic Circuits
15
Measuring Signal Strength
Example2: Compute the rms value of the sinusoidal
current i(t) = I cos(ωt)
irms
1
T
T
0
i 2 ( )d
2 /
2
0
2 /
2
The RMS value of any sinusoid
signal is always equal to 0.707
times the peak value (regardless of
amplitude or frequency)
ECEN 301
1 2
I
2
1 2
I
2
I
2
0
I 2 cos 2 (2
1
2
I2
2 /
2
0
)d
1
cos( 2
2
) d
I2
cos( 2
2
)d
0
Discussion #11 – Dynamic Circuits
16
Network Analysis with Capacitors
and Inductors (Dynamic Circuits)
Differential Equations
ECEN 301
Discussion #11 – Dynamic Circuits
17
Dynamic Circuit Network Analysis
Kirchoff’s law’s (KCL and KVL) still apply, but they
will now produce differential equations.
+ R–
vs(t)
+
~
–
ECEN 301
iR
iC
+
C
–
Discussion #11 – Dynamic Circuits
18
Dynamic Circuit Network Analysis
Kirchoff’s law’s (KCL and KVL) still apply, but they
will now produce differential equations.
KVL :
vS (t ) vR (t ) vC (t ) 0
+ R–
vs(t)
+
~
–
iR
iC
+
C
–
1 t
vS (t ) RiC (t )
iC ( )d
0
C
Differenti ate both sides :
diC (t )
dt
ECEN 301
Discussion #11 – Dynamic Circuits
1
iC (t )
RC
1 dvS (t )
R dt
19
Dynamic Circuit Network Analysis
Kirchoff’s law’s (KCL and KVL) still apply, but they
will now produce differential equations.
KCL :
iR iC
+ R–
vs(t)
+
~
–
ECEN 301
iR
iC
+
C
–
dvC (t )
vR (t )
C
R
dt
[vs (t ) vC (t )]
dvC (t )
C
R
dt
dvC (t ) 1
1
vC (t )
vS (t )
dt
RC
RC
Discussion #11 – Dynamic Circuits
20
Sinusoidal Source Responses
Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
dvC (t )
dt
+ R–
vs(t)
+
~
–
iR
iC
+
C
–
1
vC (t )
RC
The solution to this diff EQ will be a sinusoid:
vC (t )
A sin( t ) B cos( t )
C cos( t
ECEN 301
1
V cos( t )
RC
Discussion #11 – Dynamic Circuits
)
21
Sinusoidal Source Responses
Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
Substitute the solution form into the diff EQ:
dvC (t )
dt
+ R–
vs(t)
+
~
–
ECEN 301
iR
iC
+
C
–
1
vC (t )
RC
Substitute vC(t)
A sin (ωωt B cos (ωωt
d [ A sin ( t) B cos ( t )]
dt
1
[ A sin ( t ) B cos ( t )]
RC
Discussion #11 – Dynamic Circuits
1
V cos( t )
RC
1
V cos ( t )
RC
22
Sinusoidal Source Responses
Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
1
1
[ A sin ( t ) B cos ( t )]
V cos ( t )
RC
RC
B
V
sin t A
cos t 0
RC RC
A cos ( t) B sin ( t )
A
RC
B
+ R–
vs(t)
+
~
–
iR
iC
+
C
–
For this equation to hold, both the sin(ωt) and
cos(ωt) coefficients must be zero
A
RC
ECEN 301
B
0
Discussion #11 – Dynamic Circuits
A
B
RC
V
RC
0
23
Sinusoidal Source Responses
Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
A
RC
B
0
B
RC
A
V
RC
0
+ R–
vs(t)
+
~
–
ECEN 301
iR
iC
+
C
–
Solving these equations for A and B gives:
V RC
A
2
1
( RC ) 2
Discussion #11 – Dynamic Circuits
B
1
V
2
( RC ) 2
24
Sinusoidal Source Responses
Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
+ R–
vs(t)
+
~
–
iR
iC
+
C
–
Writing the solution for vC(t):
V RC
vC (t )
sin( t )
2
2
1
( RC )
1
V
cos( t )
2
2
( RC )
NB: This is the solution for a single-order
diff EQ (i.e. with only one capacitor)
ECEN 301
Discussion #11 – Dynamic Circuits
25
Sinusoidal Source Responses
vc(t) has the same frequency, but different amplitude
and different phase than vs(t)
vs(t)
vc(t)
What happens when
R or C is small?
vC (t )
ECEN 301
V RC
sin( t )
2
2
1
( RC )
1
V
cos( t )
2
2
( RC )
Discussion #11 – Dynamic Circuits
26
Sinusoidal Source Responses
In a circuit with an AC source: all branch voltages and
currents are also sinusoids with the same frequency as the
source. The amplitudes of the branch voltages and currents
are scaled versions of the source amplitude (i.e. not as large
as the source) and the branch voltages and currents may be
shifted in phase with respect to the source.
+ R–
vs(t)
+
~
–
ECEN 301
iR
iC
+
C
–
3 parameters that uniquely identify a sinusoid:
• frequency
• amplitude
• phase
Discussion #11 – Dynamic Circuits
27