revision lecture 4: transients & lines · e1.1 analysis of circuits (2014-4649) revision...
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Revision Lecture 4: Transients &Lines
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 1 / 9
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden 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 State is not the same as DC Level• Need steady states before and after the sudden change
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden 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 State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden 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 State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
• Time Constant: τ = RC or LR where R is the Thévenin
resistance at the terminals of C or L
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden 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 State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
• Time Constant: τ = RC or LR where R is the Thévenin
resistance at the terminals of C or L• Find transient amplitude, A, from continuity since VC orIL cannot change instantly.
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 a sudden 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 State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
• Time Constant: τ = RC or LR where R is the Thévenin
resistance at the terminals of C or L• Find transient amplitude, A, from continuity since VC orIL cannot change instantly.
• τ and A can also be found from the transfer function.
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
set ω = 0: YX (0) = 1
2
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
set ω = 0: YX (0) = 1
2ySS(0−) = 1, ySS(0+) = 3
-5 0 5 10 15 200
2
4
6
t (µs)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
set ω = 0: YX (0) = 1
2ySS(0−) = 1, ySS(0+) = 3
-5 0 5 10 15 200
2
4
6
t (µs)
Sinusoidal input⇒ Sinusoidal steady state⇒ use phasors.Then convert phasors to time waveforms to calculate the actual outputvoltages ySS(0−) and ySS(0+) at t = 0.
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
→ Eliminate V to get transfer Function: YX = 8jωRC+13
32jωRC+16
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
→ Eliminate V to get transfer Function: YX = 8jωRC+13
32jωRC+16
(b) Time Constant = 1Denominator corner frequency
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
→ Eliminate V to get transfer Function: YX = 8jωRC+13
32jωRC+16
(b) Time Constant = 1Denominator corner frequency
ωd = 1632RC ⇒ τ = 1
ωd= 2RC
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuity
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mA
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current source
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer function
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
= 1 + 1× 4 = 5
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
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 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
yTr
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
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) +Ae−t/τ
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
yTr
ySS
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
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) +Ae−t/τ
= 3 + 2e−t/2µ
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
yTr
ySS
y
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
0 2 4 6 8 10Time (ns)
f(t-0/u)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
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
)
.
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
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 and ix = fx−gxZ0
where ix is
positive in the +x direction (→) and Z0 =√
L0
C0
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
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 and ix = fx−gxZ0
where ix is
positive in the +x direction (→) and Z0 =√
L0
C0
Waveforms of fx and gx are determined by the connections at both ends.
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
×ρ0
−→ · · ·
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
×ρ0
−→ · · ·
Infinite sum:f0(t) = τ0vS(t)+τ0ρLρ0vS(t−
2Lu )+. . .
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
×ρ0
−→ · · ·
Infinite sum:f0(t) = τ0vS(t)+τ0ρLρ0vS(t−
2Lu )+. . .=
∑
∞
i=0 τ0ρiLρ
i0vS
(
t− 2Liu
)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t:
2. Time delays are just phase shifts:
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
But G0 = F0ρLe−2jkL : roundtrip delay of 2L
u + reflection at x = L.
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
But G0 = F0ρLe−2jkL : roundtrip delay of 2L
u + reflection at x = L.Substituting for G0 in source end equation: F0 = τ0VS + ρ0F0ρLe
−2jkL
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• 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 eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
But G0 = F0ρLe−2jkL : roundtrip delay of 2L
u + reflection at x = L.Substituting for G0 in source end equation: F0 = τ0VS + ρ0F0ρLe
−2jkL
⇒ F0 = τ01−ρ0ρL exp(−2jkL)VS so no infinite sums needed ,
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1⇒ x = L or x = L− π
k or x = L− 2πk or . . .
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1⇒ x = L or x = L− π
k or x = L− 2πk or . . .
Min magnitude is (1− ρL) |F0| whenever e−2jk(L−x) = −1
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1⇒ x = L or x = L− π
k or x = L− 2πk or . . .
Min magnitude is (1− ρL) |F0| whenever e−2jk(L−x) = −1⇒ x = L− π
2k or x = L− 3π2k or x = L− 5π
2k or . . .
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