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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 . - PowerPoint PPT PresentationTRANSCRIPT
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ECEN 301 Discussion #11 – Dynamic Circuits 1
Change1 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.
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ECEN 301 Discussion #11 – Dynamic Circuits 2
Lecture 11 – Dynamic Circuits
Time-Dependent Sources
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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:
...,3,2,1)()( nnTtxtx
0.0
1.0
0.00 2.00
time
x(t)
-1.5
0.0
1.5
0.00 2.00 4.00
time
x(t)
-1.5
0.0
1.5
0.00 2.00 4.00
timex(t)
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ECEN 301 Discussion #11 – Dynamic Circuits 4
Time Dependent Sources• Sinusoidal signal: a periodic waveform satisfying the
following equation:
)cos()( tAtx
A – amplitude ω – radian frequencyφ – phase
-1.5
0.0
1.5
0.00 2.00 4.00
time
x(t)
A
-A
T
φ/ω
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ECEN 301 Discussion #11 – Dynamic Circuits 5
Sinusoidal Sources• Helpful identities:
deg360
2
2/2
)/(1
Tt
radT
t
sT
sradf
scyclesHzT
f
)sin()sin()cos()cos()cos()cos()sin()sin()cos()sin(
90sin2
sin)cos(
90cos2
cos)sin(
tttttt
ttt
ttt
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ECEN 301 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 voltage 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
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ECEN 301 Discussion #11 – Dynamic Circuits 7
Measuring Signal Strength• Methods of quantifying the strength of time-
varying 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
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ECEN 301 Discussion #11 – Dynamic Circuits 8
Measuring Signal Strength• Time – averaged signal: integrate signal x(t) over a
period (T) of time (the average or mean value)
T
dxT
tx0
)(1)(
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ECEN 301 Discussion #11 – Dynamic Circuits 9
Measuring Signal Strength• Example 1: compute the average value of the signal
– x(t) = 10cos(100t)
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ECEN 301 Discussion #11 – Dynamic Circuits 10
Measuring Signal Strength• Example 1: compute the average value of the signal
– x(t) = 10cos(100t)
1002
2
T
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ECEN 301 Discussion #11 – Dynamic Circuits 11
Measuring Signal Strength• Example 1: compute the average value of the signal –
x(t) = 10cos(100t)
0
)0sin()2sin(210
)100cos(102100
)(1)(
100/2
0
0
dtt
dxT
txT
1002
2
T
NB: in general, for any sinusoidal signal
0)cos( tA
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ECEN 301 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:
T
rms dxT
tx0
2)(1)(
NB: the rms value is simply the square root of the average (mean) after being squared – hence: root – mean – square
NB: often notation is used instead of
)(~ txrmstx )(
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ECEN 301 Discussion #11 – Dynamic Circuits 13
Measuring Signal Strength• Example 2: Compute the rms value of the sinusoidal
current i(t) = I cos(ωt)
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ECEN 301 Discussion #11 – Dynamic Circuits 14
Measuring Signal Strength• Example 2: Compute the rms value of the sinusoidal
current i(t) = I cos(ωt)
2
021
)2cos(222
1
)2cos(21
21
2
)(cos2
)(1
2
/2
0
22
/2
0
2
/2
0
22
0
2
I
I
dII
dI
dI
diT
iT
rms
Integrating a sinusoidal waveform over 2 periods equals zero
21)2cos()(cos2
tt
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ECEN 301 Discussion #11 – Dynamic Circuits 15
Measuring Signal Strength• Example 2: Compute the rms value of the sinusoidal
current i(t) = I cos(ωt)
2
021
)2cos(222
1
)2cos(21
21
2
)2(cos2
)(1
2
/2
0
22
/2
0
2
/2
0
22
0
2
I
I
dII
dI
dI
diT
iT
rms
The RMS value of any sinusoid signal is always equal to 0.707=1/sqrt(2) times the peak value (regardless of amplitude or frequency)
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ECEN 301 Discussion #11 – Dynamic Circuits 16
Network Analysis with Capacitors and Inductors (Dynamic Circuits)
Differential Equations
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ECEN 301 Discussion #11 – Dynamic Circuits 17
Dynamic Circuit Network AnalysisKirchoff’s law’s (KCL and KVL) still apply, but they
now produce differential equations.
+ R –
iR +C–
iCvs(t)+–~
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ECEN 301 Discussion #11 – Dynamic Circuits 18
Dynamic Circuit Network AnalysisKirchoff’s law’s (KCL and KVL) still apply, but they
now produce differential equations.
+ R –
iR +C–
iCvs(t)+–~
dttdv
Rti
RCdttdi
diC
tRitv
tvtvtv
SC
C
t
CCS
CRS
)(1)(1)(:sidesboth ateDifferenti
0)(1)()(
0)()()(:KVL
t
C diC
tv )(1)(Recall:
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ECEN 301 Discussion #11 – Dynamic Circuits 19
Dynamic Circuit Network AnalysisKirchoff’s law’s (KCL and KVL) still apply, but they
now produce differential equations.
+ R –
iR +C–
iCvs(t)+–~
)(1)(1)(
)()]()([
)()(
:KCL
tvRC
tvRCdt
tdvdt
tdvCR
tvtvdt
tdvCR
tvii
SCC
CCs
CR
CR
Recall:dt
tdvCti )()(
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ECEN 301 Discussion #11 – Dynamic Circuits 20
Sinusoidal Source Responses• Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
)cos()cos()sin()(
tCtBtAtvC
+ R –
iR +C–
iCvs(t)+–~
The solution to this diff EQ is a sinusoid:
)cos(1)(1)( tVRC
tvRCdt
tdvC
C
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ECEN 301 Discussion #11 – Dynamic Circuits 21
Sinusoidal Source Responses• Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
+ R –
iR +C–
iCvs(t)+–~
Substitute the solution form into the diff EQ:
)cos1)]cos)sin[1
)]cossin[
)cos)sin
)cos(1)(1)(
t(VRC
t(Bt(ARC
dtt(Bt)(Ad
t(Bt(A(t)v
tVRC
tvRCdt
tdv
C
CC
Substitute
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ECEN 301 Discussion #11 – Dynamic Circuits 22
Sinusoidal Source Responses• Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
0cossin
)cos1)]cos)sin[1)sincos
tRCV
RCBAtB
RCA
t(VRC
t(Bt(ARC
t(Bt)(A
+ R –
iR +C–
iCvs(t)+–~
For this equation to hold, both the sin(ωt) and cos(ωt) coefficients must be zero
0 BRCA 0
RCV
RCBA
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ECEN 301 Discussion #11 – Dynamic Circuits 23
Sinusoidal Source Responses• Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
0 BRCA
+ R –
iR +C–
iCvs(t)+–~
Solving these equations for A and B gives:
0RCV
RCBA
22 )(1 RCRCVA
22 )(1 RC
VB
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ECEN 301 Discussion #11 – Dynamic Circuits 24
Sinusoidal Source Responses• Consider the AC source producing the voltage:
vs(t) = Vcos(ωt)
+ R –
iR +C–
iCvs(t)+–~
Writing the solution for vC(t):
)cos()(1
)sin()(1
)( 2222 tRC
VtRC
RCVtvC
NB: This is the solution for a single-order diff EQ (i.e. with only one capacitor)
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ECEN 301 Discussion #11 – Dynamic Circuits 25
Sinusoidal Source Responsesvc(t) has the same frequency, but different amplitude
and different phase than vs(t)
)cos()(1
)sin()(1
)( 2222 tRC
VtRC
RCVtvC
vs(t)
vc(t)
What happens when R or C is small?
vc lags vs
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ECEN 301 Discussion #11 – Dynamic Circuits 26
Sinusoidal Source ResponsesIn 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 –
iR +C–
iCvs(t)+–~
3 parameters that uniquely identify a sinusoid:• frequency• amplitude (magnitude)• phase