18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
18: Phasors and Transmission
Lines
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 1 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
For a transmission line:
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2016-8284)
x
u
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
For a transmission line:
x
u
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
For a transmission line:
x
u
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
For a transmission line:
x
u
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
E1.1 Analysis of Circuits (2016-8284)
x
u)
= A cos ω t −
x
u
+φ
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
x
u
+φ
j (− ω
u x+φ)
⇒ Fx = Ae
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
E1.1 Analysis of Circuits (2016-8284)
x
u
+φ
jφ −j ω
ux
= Ae e
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
jφ −j ω
ux
= Ae e
x
u
+φ
= F0 e−jkx
where the wavenumber is k , ω
u.
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
jφ −j ω
ux
= Ae e
x
u
+φ
= F0 e−jkx
where the wavenumber is k , ω
u.
Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
jφ −j ω
ux
= Ae e
x
u
+φ
= F0 e−jkx
where the wavenumber is k , ω
u.
Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0 e+jkx .
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
jφ −j ω
ux
= Ae e
x
u
+φ
= F0 e−jkx
where the wavenumber is k , ω
u.
Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0 e+jkx .
Everything is time-invariant: phasors do not depend on t.
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
jφ −j ω
ux
= Ae e
x
u
+φ
= F0 e−jkx
where the wavenumber is k , ω
u.
Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0 e+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
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasors and transmision lines
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
x
u
For a transmission line:
x
u
v(t, x) = f t −
+g t+
and
−1
x
x
i(t, x) = Z0 f (t − u ) − g(t + u )
We can use phasors to eliminate t from the equations if f () and g() are
sinusoidal with the same ω : f0 (t) = A cos (ωt + φ) ⇒ F0 = Aejφ .
Then fx (t) = f (t −
x
u)
= A cos ω t −
j (− ω
u x+φ)
⇒ Fx = Ae
jφ −j ω
ux
= Ae e
x
u
+φ
= F0 e−jkx
where the wavenumber is k , ω
u.
Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).
Similarly Gx = G0 e+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
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 2 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
|Fx | ≡ |F |
indep of x
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
(x−y)
fx (t) = fy t − u
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
|Fx | ≡ |F |
indep of x
Fx = Fy e−jk(x−y)
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
(x−y)
fx (t) = fy t − u
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
Fx = Fy e−jk(x−y)
|Fx | ≡ |F |
indep of x
Delayed by
x−y
u
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
(x−y)
fx (t) = fy t − u
(x−y)
gx (t) = gy t + u
F = Aejφ
F indep of t
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
Fx = Fy e−jk(x−y)
Gx = Gy e+jk(x−y)
|Fx | ≡ |F |
indep of x
x−y
u
x−y
by u
Delayed by
Advanced
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
(x−y)
fx (t) = fy t − u
(x−y)
gx (t) = gy t + u
F = Aejφ
F indep of t
Gx = Gy e+jk(x−y)
vx (t) = fx (t) + gx (t)
Vx = Fx + G x
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
Fx = Fy e−jk(x−y)
|Fx | ≡ |F |
indep of x
x−y
u
x−y
by u
Delayed by
Advanced
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Relationships
Time Domain
Phasor
Notes
f (t) = A cos (ωt + φ)
x
fx (t) = f t − u = A cos ωt + φ − ωu x
(x−y)
fx (t) = fy t − u
(x−y)
gx (t) = gy t + u
F = Aejφ
F indep of t
Gx = Gy e+jk(x−y)
vx (t) = fx (t) + gx (t)
Vx = Fx + G x
ix (t) =
fx (t)−gx (t)
Z0
E1.1 Analysis of Circuits (2016-8284)
j (φ− ω
u x)
Fx = Ae
= F e−jkx
Fx = Fy e−jk(x−y)
Ix =
|Fx | ≡ |F |
indep of x
x−y
u
x−y
by u
Delayed by
Advanced
Fx −Gx
Z0
Phasors and Transmission Lines: 18 – 3 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL
L
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
E1.1 Analysis of Circuits (2016-8284)
FL +GL
−1
Z0 (FL −GL )
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
E1.1 Analysis of Circuits (2016-8284)
FL +GL
−1
Z0 (FL −GL )
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
E1.1 Analysis of Circuits (2016-8284)
Gx
Fx
=
FL +GL
−1
Z0 (FL −GL )
GL e−jk(L−x)
FL e+jk(L−x)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
E1.1 Analysis of Circuits (2016-8284)
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
x
Note that G
Fx ≡ |ρL | has the same value for all x.
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
x
Note that G
Fx ≡ |ρL | has the same value for all x.
Vx = Fx + G x
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
x
Note that G
Fx ≡ |ρL | has the same value for all x.
−2jk(L−x)
Vx = F x + G x = F x 1 + ρL e
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
x
Note that G
Fx ≡ |ρL | has the same value for all x.
−2jk(L−x)
Vx = F x + G x = F x 1 + ρL e
Ix = Z0−1 (Fx − Gx )
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
x
Note that G
Fx ≡ |ρL | has the same value for all x.
−2jk(L−x)
Vx = F x + G x = F x 1 + ρL e
Ix =
Z0−1
E1.1 Analysis of Circuits (2016-8284)
(Fx − Gx ) =
Z0−1 Fx
1 − ρL e
−2jk(L−x)
Phasors and Transmission Lines: 18 – 4 / 7
Phasor Reflection
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
Phasors obey Ohm’s law: VI L = RL =
L
L −Z0
So GL = ρL FL where ρL = R
RL +Z0
At any x,
Gx
Fx
=
GL e−jk(L−x)
FL e+jk(L−x)
FL +GL
−1
Z0 (FL −GL )
= ρL e−2jk(L−x)
x
Ohm’s law at the load determines the ratio G
F everywhere on the line.
x
x
Note that G
Fx ≡ |ρL | has the same value for all x.
−2jk(L−x)
Vx = F x + G x = F x 1 + ρL e
Ix =
Z0−1
(Fx − Gx ) =
Z0−1 Fx
1 − ρL e
−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).
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 4 / 7
Standing Waves
t=0.0 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
f = 300 MHz
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.0 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
Forward wave phasor: Fx = F e−jkx
E1.1 Analysis of Circuits (2016-8284)
30
40
50
x (cm)
60
70
80
90
f = 300 MHz
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.0 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
Forward wave phasor: Fx = F e−jkx
Backward wave phasor: Gx = ρL Fx e−2jk(L−x)
E1.1 Analysis of Circuits (2016-8284)
70
80
90
f = 300 MHz
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.0 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.2 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.4 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.6 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.8 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=1.0 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=1.2 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=1.4 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
−jkx
E1.1 Analysis of Circuits (2016-8284)
1 + ρL e
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
Volts ÷ |F|
1
f
g
0
-1
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.0 ns
t=0.0 ns
1
f
Volts ÷ |F|
Volts ÷ |F|
1
g
0
-1
0
|v|
v
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.2 ns
t=0.2 ns
1
f
Volts ÷ |F|
Volts ÷ |F|
1
g
0
-1
0
v
|v|
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.4 ns
t=0.4 ns
1
f
Volts ÷ |F|
Volts ÷ |F|
1
g
0
-1
0
v
|v|
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.6 ns
t=0.6 ns
f
g
0
-1
0
|v|
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=0.8 ns
t=0.8 ns
f
g
0
-1
0
|v|
1
Volts ÷ |F|
Volts ÷ |F|
1
v f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=1.0 ns
t=1.0 ns
f
g
0
-1
0
|v|
1
Volts ÷ |F|
Volts ÷ |F|
1
f
v
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=1.2 ns
t=1.2 ns
f
g
0
-1
0
|v|
1
Volts ÷ |F|
Volts ÷ |F|
1
f
v
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
t=1.4 ns
t=1.4 ns
f
g
0
-1
0
|v|
1
Volts ÷ |F|
Volts ÷ |F|
1
f v
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
f
g
0
-1
0
λ = u/f =50 cm
|v|
t=1.6 ns
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
f
g
0
-1
0
λ = u/f =50 cm
|v|
t=1.6 ns
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
E1.1 Analysis of Circuits (2016-8284)
−2jk(L−x)
varies with x
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
f
g
0
-1
0
λ = u/f =50 cm
|v|
t=1.6 ns
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
−2jk(L−x)
varies with x
Max amplitude equals 1 + |ρL | at values of x where Fx and Gx are in phase. This
occurs every λ
2 away from L
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
f
g
0
-1
0
λ = u/f =50 cm
|v|
t=1.6 ns
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
−2jk(L−x)
varies with x
Max amplitude equals 1 + |ρL | at values of x where Fx and Gx are in phase. This
2π
u
wavelength,
λ
=
occurs every λ
away
from
L
where
λ
is
the
=
2
k
f.
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
f
g
0
-1
0
λ = u/f =50 cm
|v|
t=1.6 ns
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
−2jk(L−x)
varies with x
Max amplitude equals 1 + |ρL | at values of x where Fx and Gx are in phase. This
2π
u
wavelength,
λ
=
occurs every λ
away
from
L
where
λ
is
the
=
2
k
f.
Min amplitude equals 1 − |ρL | at values of x where Fx and Gx are out of phase.
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Standing Waves
λ = u/f =50 cm
t=1.6 ns
f
g
0
-1
0
λ = u/f =50 cm
|v|
t=1.6 ns
v
1
Volts ÷ |F|
Volts ÷ |F|
1
f
g
0
-1
10
20
30
40
50
x (cm)
60
70
80
90
0
10
20
30
40
50
x (cm)
60
70
80
90
Forward wave phasor: Fx = F e−jkx
f = 300 MHz
Backward wave phasor: Gx = ρL Fx e−2jk(L−x) = ρL F e−2jkL e+jkx
Line Voltage phasor: Vx = Fx + Gx = F e
1 + ρL
e
Line Voltage Amplitude: |Vx | = |F | 1 + ρL e−2jk(L−x) −jkx
−2jk(L−x)
varies with x
Max amplitude equals 1 + |ρL | at values of x where Fx and Gx are in phase. This
2π
u
wavelength,
λ
=
occurs every λ
away
from
L
where
λ
is
the
=
2
k
f.
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.
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 5 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 6 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
x
◦ fx (t) = f t − u
→ Fx = F0 e−jkx
x
→ Gx = G0 e+jkx
◦ gx (t) = g t + u
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 6 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
x
◦ fx (t) = f t − u
→ Fx = F0 e−jkx
x
→ Gx = G0 e+jkx
◦ gx (t) = g t + u
⊲ k = ωu is the wavenumber in “radians per metre”
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 6 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
x
◦ fx (t) = f t − u
→ Fx = F0 e−jkx
x
→ Gx = G0 e+jkx
◦ gx (t) = g t + u
⊲ k = ωu is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fx = Fy e−jk(x−y)
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 6 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
x
◦ fx (t) = f t − u
→ Fx = F0 e−jkx
x
→ Gx = G0 e+jkx
◦ gx (t) = g t + u
⊲ k = ωu is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fx = Fy e−jk(x−y)
• When a periodic wave meets its reflection you get a standing wave:
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 6 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
x
◦ fx (t) = f t − u
→ Fx = F0 e−jkx
x
→ Gx = G0 e+jkx
◦ gx (t) = g t + u
⊲ k = ωu is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fx = Fy e−jk(x−y)
• When a periodic wave meets its reflection you get a standing wave:
−2jk(L−x) ◦ Oscillation amplitude varies with x: ∝ 1 + ρL e
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 6 / 7
Summary
18: Phasors and
Transmission Lines
• Phasors and transmision
lines
• Phasor Relationships
• Phasor Reflection
• Standing Waves
• Summary
• Merry Xmas
• Use phasors if forward and backward waves are sinusoidal with the
same ω .
x
◦ fx (t) = f t − u
→ Fx = F0 e−jkx
x
→ Gx = G0 e+jkx
◦ gx (t) = g t + u
⊲ k = ωu is the wavenumber in “radians per metre”
• Time delays ≃ phase shifts: Fx = Fy e−jk(x−y)
• When a periodic wave meets its reflection you get a standing wave:
−2jk(L−x) ◦ Oscillation amplitude varies with x: ∝ 1 + ρL e
◦ Max amplitude of (1 + |ρL |) occurs every
E1.1 Analysis of Circuits (2016-8284)
λ
2
Phasors and Transmission Lines: 18 – 6 / 7
Merry Xmas
E1.1 Analysis of Circuits (2016-8284)
Phasors and Transmission Lines: 18 – 7 / 7