18: Phasors and TransmissionLines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 1 / 7
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux= F0e
−jkx
where the wavenumber is k , ωu
.
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux= F0e
−jkx
where the wavenumber is k , ωu
.Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux= F0e
−jkx
where the wavenumber is k , ωu
.Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0e+jkx.
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux= F0e
−jkx
where the wavenumber is k , ωu
.Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0e+jkx.
Everything is time-invariant: phasors do not depend on t.
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux= F0e
−jkx
where the wavenumber is k , ωu
.Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0e+jkx.
Everything is time-invariant: phasors do not depend on t.
Nice things about sine waves:(1) a time delay is just a phase shift
Phasors and transmision lines
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 2 / 7
For a transmission line: v(t, x) = f(
t− xu
)
+ g(
t+ xu
)
and
i(t, x) = Z−10
(
f(t− xu)− g(t+ x
u))
We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.
Then fx(t) = f(t− xu) = A cos
(
ω(
t− xu
)
+ φ)
⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω
ux= F0e
−jkx
where the wavenumber is k , ωu
.Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0e+jkx.
Everything is time-invariant: phasors do not depend on t.
Nice things about sine waves:(1) a time delay is just a phase shift(2) sum of delayed sine waves is another sine wave
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
|Fx| ≡ |F |indep of x
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
|Fx| ≡ |F |indep of x
fy(t) = fx
(
t− (y−x)u
)
Fy = Fxe−jk(y−x)
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
|Fx| ≡ |F |indep of x
fy(t) = fx
(
t− (y−x)u
)
Fy = Fxe−jk(y−x) Delayed by y−x
u
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
|Fx| ≡ |F |indep of x
fy(t) = fx
(
t− (y−x)u
)
Fy = Fxe−jk(y−x) Delayed by y−x
u
gy(t) = gx
(
t+ (y−x)u
)
Gy = Gxe+jk(y−x) Advanced by y−x
u
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
|Fx| ≡ |F |indep of x
fy(t) = fx
(
t− (y−x)u
)
Fy = Fxe−jk(y−x) Delayed by y−x
u
gy(t) = gx
(
t+ (y−x)u
)
Gy = Gxe+jk(y−x) Advanced by y−x
u
vx(t) = fx(t) + gx(t) Vx = Fx +Gx
Phasor Relationships
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 3 / 7
Time Domain Phasor Notes
f(t) = A cos (ωt+ φ) F = Aejφ F indep of t
fx(t) = f(
t− xu
)
= A cos(
ωt+ φ− ωux)
Fx = Aej(φ−ωux)
= Fe−jkx
|Fx| ≡ |F |indep of x
fy(t) = fx
(
t− (y−x)u
)
Fy = Fxe−jk(y−x) Delayed by y−x
u
gy(t) = gx
(
t+ (y−x)u
)
Gy = Gxe+jk(y−x) Advanced by y−x
u
vx(t) = fx(t) + gx(t) Vx = Fx +Gx
ix(t) =fx(t)−gx(t)
Z0Ix = Fx−Gx
Z0
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x)
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Note that∣
∣
∣
Gx
Fx
∣
∣
∣≡ |ρL| has the same value for all x.
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Note that∣
∣
∣
Gx
Fx
∣
∣
∣≡ |ρL| has the same value for all x.
Vx = Fx +Gx
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Note that∣
∣
∣
Gx
Fx
∣
∣
∣≡ |ρL| has the same value for all x.
Vx = Fx +Gx = Fx
(
1 + ρLe−2jk(L−x)
)
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Note that∣
∣
∣
Gx
Fx
∣
∣
∣≡ |ρL| has the same value for all x.
Vx = Fx +Gx = Fx
(
1 + ρLe−2jk(L−x)
)
Ix = Z−10 (Fx −Gx)
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Note that∣
∣
∣
Gx
Fx
∣
∣
∣≡ |ρL| has the same value for all x.
Vx = Fx +Gx = Fx
(
1 + ρLe−2jk(L−x)
)
Ix = Z−10 (Fx −Gx) = Z−1
0 Fx
(
1− ρLe−2jk(L−x)
)
Phasor Reflection
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 4 / 7
Phasors obey Ohm’s law: VL
IL= RL = FL+GL
Z−10 (FL−GL)
So GL = ρLFL where ρL = RL−Z0
RL+Z0
At any x, Gx
Fx= GLe−jk(L−x)
FLe+jk(L−x) = ρLe−2jk(L−x)
Ohm’s law at the load determines the ratio Gx
Fxeverywhere on the line.
Note that∣
∣
∣
Gx
Fx
∣
∣
∣≡ |ρL| has the same value for all x.
Vx = Fx +Gx = Fx
(
1 + ρLe−2jk(L−x)
)
Ix = Z−10 (Fx −Gx) = Z−1
0 Fx
(
1− ρLe−2jk(L−x)
)
The exponent −2jk (L− x) is the phase delay from travelling from x to L
and back again (hence the factor 2).
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Max amplitude equals 1 + |ρL| at values of x where Fx and Gx are in phase. Thisoccurs every λ
2 away from L
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Max amplitude equals 1 + |ρL| at values of x where Fx and Gx are in phase. Thisoccurs every λ
2 away from L where λ is the wavelength, λ = 2πk
= uf
.
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Max amplitude equals 1 + |ρL| at values of x where Fx and Gx are in phase. Thisoccurs every λ
2 away from L where λ is the wavelength, λ = 2πk
= uf
.
Min amplitude equals 1− |ρL| at values of x where Fx and Gx are out of phase.
Standing Waves
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 5 / 7
Forward wave phasor: Fx = Fe−jkx
Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx
Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(
1 + ρLe−2jk(L−x)
)
Line Voltage Amplitude: |Vx| = |F |∣
∣1 + ρLe−2jk(L−x)
∣
∣ varies with x but not t
Max amplitude equals 1 + |ρL| at values of x where Fx and Gx are in phase. Thisoccurs every λ
2 away from L where λ is the wavelength, λ = 2πk
= uf
.
Min amplitude equals 1− |ρL| at values of x where Fx and Gx are out of phase.
Standing waves arise whenever a periodic wave meets its reflection: e.g. ponds,musical instruments, microwave ovens.
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f
(
t− xu
)
→ Fx = F0e−jkx
◦ gx(t) = g(
t+ xu
)
→ Gx = G0e+jkx
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f
(
t− xu
)
→ Fx = F0e−jkx
◦ gx(t) = g(
t+ xu
)
→ Gx = G0e+jkx
⊲ k = ωu
is the wavenumber in “radians per metre”
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f
(
t− xu
)
→ Fx = F0e−jkx
◦ gx(t) = g(
t+ xu
)
→ Gx = G0e+jkx
⊲ k = ωu
is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fy = Fxe−jk(y−x)
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f
(
t− xu
)
→ Fx = F0e−jkx
◦ gx(t) = g(
t+ xu
)
→ Gx = G0e+jkx
⊲ k = ωu
is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fy = Fxe−jk(y−x)
• When a periodic wave meets its reflection you get a standing wave:
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f
(
t− xu
)
→ Fx = F0e−jkx
◦ gx(t) = g(
t+ xu
)
→ Gx = G0e+jkx
⊲ k = ωu
is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fy = Fxe−jk(y−x)
• When a periodic wave meets its reflection you get a standing wave:
◦ Oscillation amplitude varies with x: ∝∣
∣1 + ρLe−2jk(L−x)
∣
∣
Summary
18: Phasors andTransmission Lines• Phasors and transmisionlines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 6 / 7
• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f
(
t− xu
)
→ Fx = F0e−jkx
◦ gx(t) = g(
t+ xu
)
→ Gx = G0e+jkx
⊲ k = ωu
is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fy = Fxe−jk(y−x)
• When a periodic wave meets its reflection you get a standing wave:
◦ Oscillation amplitude varies with x: ∝∣
∣1 + ρLe−2jk(L−x)
∣
∣
◦ Max amplitude of (1 + |ρL|) occurs every λ2
Merry Xmas
E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 7 / 7