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1/89
Revision Lecture 4: Transients &
Lines
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 1 / 9
7/25/2019 90400 Transients
2/89
Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
7/25/2019 90400 Transients
3/89
Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
1st order circuits only: one capacitor/inductor
7/25/2019 90400 Transients
4/89
Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
1st order circuits only: one capacitor/inductor
All voltage/current waveforms are: Steady State + Transient
Steady States:find with nodal analysis or transfer function
Note:Steady Stateis not the same asDC Level
Need steady states beforeandafter the sudden change
7/25/2019 90400 Transients
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Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
1st order circuits only: one capacitor/inductor
All voltage/current waveforms are: Steady State + Transient
Steady States:find with nodal analysis or transfer function
Note:Steady Stateis not the same asDC Level
Need steady states beforeandafter the sudden change
Transient: Always a negative exponential: Ae
t
7/25/2019 90400 Transients
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Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
1st order circuits only: one capacitor/inductor
All voltage/current waveforms are: Steady State + Transient
Steady States:find with nodal analysis or transfer function
Note:Steady Stateis not the same asDC Level
Need steady states beforeandafter the sudden change
Transient: Always a negative exponential: Ae
t
Time Constant: =RCor LR whereR is the Thveninresistance at the terminals ofCorL
7/25/2019 90400 Transients
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Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
1st order circuits only: one capacitor/inductor
All voltage/current waveforms are: Steady State + Transient
Steady States:find with nodal analysis or transfer function
Note:Steady Stateis not the same asDC Level
Need steady states beforeandafter the sudden change
Transient: Always a negative exponential: Ae
t
Time Constant: =RCor LR whereR is the Thveninresistance at the terminals ofCorL
Find transient amplitude,A, from continuity sinceVC or
IL cannot change instantly.
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Transients: Basic Ideas
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 2 / 9
Transients happen in response to asudden change
Input voltage/current abruptly changes its magnitude,
frequency or phase
A switch alters the circuit
1st order circuits only: one capacitor/inductor
All voltage/current waveforms are: Steady State + Transient
Steady States:find with nodal analysis or transfer function
Note:Steady Stateis not the same asDC Level
Need steady states beforeandafter the sudden change
Transient: Always a negative exponential: Ae
t
Time Constant: =RCor LR whereR is the Thveninresistance at the terminals ofCorL
Find transient amplitude,A, from continuity sinceVC or
IL cannot change instantly. andA can also be found from the transfer function.
7/25/2019 90400 Transients
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Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
10/89
Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
11/89
Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
12/89
7/25/2019 90400 Transients
13/89
Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
14/89
Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
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Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
16/89
Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
17/89
Steady States
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 3 / 9
Asteady-stateoutput assumes the input frequency, phase and amplitude
are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t
7/25/2019 90400 Transients
18/89
Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
7/25/2019 90400 Transients
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
7/25/2019 90400 Transients
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R
7/25/2019 90400 Transients
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=
RTh
C= 2
RC
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=
RTh
C= 2
RC
Method 2: Transfer function
7/25/2019 90400 Transients
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=
RTh
C= 2
RC
Method 2: Transfer function
(a)Calculate transfer function using nodal analysis
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=RTh
C= 2RC
Method 2: Transfer function
(a)Calculate transfer function using nodal analysis
KCL @ V:
VX
4R + V
8R +jCV + VY
2R = 0KCL @ Y: YV2R +
YX6R = 0
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=RTh
C= 2RC
Method 2: Transfer function
(a)Calculate transfer function using nodal analysis
KCL @ V:
VX
4R + V
8R +jCV + VY
2R = 0KCL @ Y: YV2R +
YX6R = 0
EliminateV to get transfer Function: YX = 8jRC+1332jRC+16
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=RTh
C= 2RC
Method 2: Transfer function
(a)Calculate transfer function using nodal analysis
KCL @ V:
VX
4R + V
8R +jCV + VY
2R = 0KCL @ Y: YV2R +
YX6R = 0
EliminateV to get transfer Function: YX = 8jRC+1332jRC+16
(b)Time Constant = 1
Denominator corner frequency
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Determining Time Constant
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 4 / 9
Method 1: Thvenin
(a)Remove the capacitor/inductor
(b)Set all sources to zero (including the input
voltage source). Leave output unconnected.
(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:
RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor
LRTh
=RTh
C= 2RC
Method 2: Transfer function
(a)Calculate transfer function using nodal analysis
KCL @ V:
VX
4R + V
8R +jCV + VY
2R = 0KCL @ Y: YV2R +
YX6R = 0
EliminateV to get transfer Function: YX = 8jRC+1332jRC+16
(b)Time Constant = 1
Denominator corner frequencyd = 1632RC =
1d
= 2RC
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Determining Transient Amplitude
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
30/89
Determining Transient Amplitude
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
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Determining Transient Amplitude
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
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Determining Transient Amplitude
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuity
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
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Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
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Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mA
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
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Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
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Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current source
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
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Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
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Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
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39/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
40/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
Input step,x=x(0+) x(0) = +4-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
41/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
Input step,x=x(0+) x(0) = +4y(0+) =y(0) +H(j)x
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
42/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
Input step,x=x(0+) x(0) = +4y(0+) =y(0) +H(j)x
= 1 + 1 4 = 5
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
ySS
7/25/2019 90400 Transients
43/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
Input step,x=x(0+) x(0) = +4y(0+) =y(0) +H(j)x
= 1 + 1 4 = 5
(ii)A =y(0+) ySS(0+) = 5 3 = 2
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
yTr
ySS
7/25/2019 90400 Transients
44/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
Input step,x=x(0+) x(0) = +4y(0+) =y(0) +H(j)x
= 1 + 1 4 = 5
(ii)A =y(0+) ySS(0+) = 5 3 = 2
(iii)y(t) =ySS(t) +Aet/
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
yTr
ySS
7/25/2019 90400 Transients
45/89
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 5 / 9
After an input change att = 0,y(t) =ySS(t) +Ae t .
y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA
(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5
(i)Version 2: Transfer function
H(j) = YX (j) = R+jL2R+jL
Input step,x=x(0+) x(0) = +4y(0+) =y(0) +H(j)x
= 1 + 1 4 = 5
(ii)A =y(0+) ySS(0+) = 5 3 = 2
(iii)y(t) =ySS(t) +Aet/
= 3 + 2e
t/2
-5 0 5 10 15 20
0
2
4
6
t (s)
-5 0 5 10 15 20
0
2
4
6
t (s)
yTr
ySS
y
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
0 2 4 6 8 10Time (ns)
f(t-0/u)
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u)
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
Knowingfx(t)forx=x0 fixes it for allotherx.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
Knowingfx(t)forx=x0 fixes it for allotherx.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Backward wave: gx(t)is the same but travelling : gx(t) =g0
t+ xu
.
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
Knowingfx(t)forx=x0 fixes it for allotherx.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Backward wave: gx(t)is the same but travelling : gx(t) =g0
t+ xu
.
Voltage and current are: vx =fx+gx andix = fxgx
Z0whereix is
positive in the+xdirection () andZ0 =
L0C0
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Transmission Lines Basics
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 6 / 9
Transmission Line: constantL0 andC0 : inductance/capacitance permetre.
Forward wavetravels along the line: fx(t) =f0
t xu
.
Velocityu =
1L0C0
12c= 15cm/ns
fx(t)equalsf0(t)but delayed by xu .
Knowingfx(t)forx=x0 fixes it for allotherx.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Backward wave: gx(t)is the same but travelling : gx(t) =g0
t+ xu
.
Voltage and current are: vx =fx+gx andix = fxgx
Z0whereix is
positive in the+xdirection () andZ0 =
L0C0
Waveforms offx andgx are determined by the connections at both ends.
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t)
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
R fl i
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
L g0(t +
2Lu )
R fl i
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
L g0(t +
2Lu )
0 f0(t +
2Lu )
R fl ti
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
L g0(t +
2Lu )
0 f0(t +
2Lu )
L g0(t +
4Lu )
R fl ti
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
L g0(t +
2Lu )
0 f0(t +
2Lu )
L g0(t +
4Lu )
0
R fl ti
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
L g0(t +
2Lu )
0 f0(t +
2Lu )
L g0(t +
4Lu )
0
Infinite sum:
f0(t) =0vS(t)+0L0vS(t2Lu )+. . .
Reflections
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Reflections
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 7 / 9
vx =fx+gxix =
fxgxZ0
Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0
fL(t).
Reflection coefficient:L= gL(t)fL(t)
= RLZ0RL+Z0L [1, +1]and increases withRL
Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x
Atx = 0:f0(t) = Z0RS+Z0
vS(t) +RSZ0RS+Z0
g0(t) =0vS(t) +0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t) 0 f0(t)
L g0(t +
2Lu )
0 f0(t +
2Lu )
L g0(t +
4Lu )
0
Infinite sum:
f0(t) =0vS(t)+0L0vS(t2Lu )+. . .=
i=00iL
i0vSt
2Liu
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:
2. Time delays are just phase shifts:
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
kis thewavenumber: radians per metre (c.f. in rad per second)
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
kis thewavenumber: radians per metre (c.f. in rad per second)
As before: Vx =Fx+Gx andIx = FxGx
Z0
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
kis thewavenumber: radians per metre (c.f. in rad per second)
As before: Vx =Fx+Gx andIx = FxGx
Z0
As before:
GL =LFLF0 =0VS+0G0
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
kis thewavenumber: radians per metre (c.f. in rad per second)
As before: Vx =Fx+Gx andIx = FxGx
Z0
As before:
GL =LFLF0 =0VS+0G0
ButG0 =F0Le2jkL : roundtrip delay of 2Lu + reflection atx =L.
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
kis thewavenumber: radians per metre (c.f. in rad per second)
As before: Vx =Fx+Gx andIx = FxGx
Z0
As before:
GL =LFLF0 =0VS+0G0
ButG0 =F0Le2jkL : roundtrip delay of 2Lu + reflection atx =L.
Substituting forG0 in source end equation: F0 =0VS+0F0Le2jkL
Sinewaves and Phasors
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Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej
2. Time delays are just phase shifts:
fx(t) =A cos
t xu
+ Fx =Ae
j(ux)=F0ejkx
kis thewavenumber: radians per metre (c.f. in rad per second)
As before: Vx =Fx+Gx andIx = FxGx
Z0
As before:
GL =LFLF0 =0VS+0G0
ButG0 =F0Le2jkL : roundtrip delay of 2Lu + reflection atx =L.
Substituting forG0 in source end equation: F0 =0VS+0F0Le2jkL
F0= 0
10Lexp(2jkL)VSso no infinite sums needed
Standing Waves
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Standing Waves
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
Standing Waves
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Standing Waves
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Standing Waves
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Standing Waves
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx)
Standing Waves
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Standing Waves
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Standing Waves
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Standing Waves
Revision Lecture 4:Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx
Standing Waves
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Sta d g a es
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx=F0ejkx
1 +Le
2jk(Lx)
Standing Waves
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g
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx=F0ejkx
1 +Le
2jk(Lx)
Voltage Magnitude : |Vx| = |F0|1 +Le2jk(Lx)
Standing Waves
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g
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx=F0ejkx
1 +Le
2jk(Lx)
Voltage Magnitude : |Vx| = |F0|1 +Le2jk(Lx)
: depends onx
Standing Waves
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g
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx=F0ejkx
1 +Le
2jk(Lx)
Voltage Magnitude : |Vx| = |F0|1 +Le2jk(Lx)
: depends onx
IfL 0,max magnitudeis(1 +L) |F0| whenevere2jk(Lx)
= +1
Standing Waves
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g
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of xu
Fx =F0ejkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx=F0ejkx
1 +Le
2jk(Lx)
Voltage Magnitude : |Vx| = |F0|1 +Le2jk(Lx)
: depends onx
IfL 0,max magnitudeis(1 +L) |F0| whenevere2jk(Lx)
= +1 x=Lorx =L k orx =L
2k or. . .
Standing Waves
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g
Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 9 / 9
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of x
u Fx =F0e
jkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu
Voltage atx: Vx =Fx+Gx=F0ejkx
1 +Le
2jk(Lx)
Voltage Magnitude : |Vx| = |F0|1 +Le2jk(Lx)
: depends onx
IfL 0,max magnitudeis(1 +L) |F0| whenevere2jk(Lx)
= +1 x=Lorx =L k orx =L
2k or. . .
Min magnitudeis(1 L) |F0| whenevere2jk(Lx) = 1
Standing Waves
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Revision Lecture 4:
Transients & Lines
Transients: Basic Ideas
Steady States
Determining Time
Constant
Determining Transient
Amplitude
Transmission Lines Basics
Reflections
Sinewaves and Phasors
Standing Waves
Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add
but where they areanti-phasetheir amplitudes subtract.
At any pointx,
delay of x
u Fx =F0e
jkx
Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2LxuVoltage atx: Vx =Fx+Gx=F0e
jkx
1 +Le2jk(Lx)
Voltage Magnitude : |Vx| = |F0|
1 +Le2jk(Lx): depends onx
IfL 0,max magnitudeis(1 +L) |F0| whenevere2jk(Lx)
= +1 x=Lorx =L k orx =L
2k or. . .
Min magnitudeis(1 L) |F0| whenevere2jk(Lx) = 1
x=L 2k orx =L 32k orx =L
52k or. . .
Recommended