17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
17: Transmission Lines
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 1 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
This is not true.
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
This is not true.
Characteristics
• Summary
If fact signals travel at around half the speed of light (c = 30 cm/ns).
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
This is not true.
Characteristics
• Summary
If fact signals travel at around half the speed of light (c = 30 cm/ns).
Reason: all wires have capacitance to ground and to neighbouring
conductors and also self-inductance. It takes time to change the current
through an inductor or voltage across a capacitor.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
This is not true.
Characteristics
• Summary
If fact signals travel at around half the speed of light (c = 30 cm/ns).
Reason: all wires have capacitance to ground and to neighbouring
conductors and also self-inductance. It takes time to change the current
through an inductor or voltage across a capacitor.
A transmission line is a wire with a uniform goemetry along its length: the
capacitance and inductance of any segment is proportional to its length.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
This is not true.
Characteristics
• Summary
If fact signals travel at around half the speed of light (c = 30 cm/ns).
Reason: all wires have capacitance to ground and to neighbouring
conductors and also self-inductance. It takes time to change the current
through an inductor or voltage across a capacitor.
A transmission line is a wire with a uniform goemetry along its length: the
capacitance and inductance of any segment is proportional to its length.
We represent as a large number of small inductors and capacitors spaced
along the line.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Lines
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Previously assume that any change in v0 (t) appears instantly at vL (t).
This is not true.
Characteristics
• Summary
If fact signals travel at around half the speed of light (c = 30 cm/ns).
Reason: all wires have capacitance to ground and to neighbouring
conductors and also self-inductance. It takes time to change the current
through an inductor or voltage across a capacitor.
A transmission line is a wire with a uniform goemetry along its length: the
capacitance and inductance of any segment is proportional to its length.
We represent as a large number of small inductors and capacitors spaced
along the line.
The signal speed along a transmisison line is predictable.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 2 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 3 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Small δx ⇒ ignore 2nd order derivatives:
∂v(x,t)
∂t
=
∂v(x+δx,t)
∂t
,
∂v
∂t .
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 3 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Small δx ⇒ ignore 2nd order derivatives:
∂v(x,t)
∂t
=
∂v(x+δx,t)
∂t
,
∂v
∂t .
Basic Equations
KVL: v(x, t) = V2 + v(x + δx, t) + V1
KCL: i(x, t) = iC + i(x + δx, t)
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 3 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Small δx ⇒ ignore 2nd order derivatives:
∂v(x,t)
∂t
=
∂v(x+δx,t)
∂t
,
∂v
∂t .
Basic Equations
KVL: v(x, t) = V2 + v(x + δx, t) + V1
KCL: i(x, t) = iC + i(x + δx, t)
∂i
Capacitor equation: C ∂v
∂t = iC = i(x, t) − i(x + δx, t) = − ∂x δx
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 3 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Small δx ⇒ ignore 2nd order derivatives:
∂v(x,t)
∂t
=
∂v(x+δx,t)
∂t
,
∂v
∂t .
Basic Equations
KVL: v(x, t) = V2 + v(x + δx, t) + V1
KCL: i(x, t) = iC + i(x + δx, t)
∂i
Capacitor equation: C ∂v
∂t = iC = i(x, t) − i(x + δx, t) = − ∂x δx
Inductor equation (L1 and L2 have the same current):
∂v
∂i
= V1 + V2 = v(x, t) − v(x + δx, t) = − ∂x
δx
(L1 + L2 ) ∂t
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 3 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Small δx ⇒ ignore 2nd order derivatives:
∂v(x,t)
∂t
=
∂v(x+δx,t)
∂t
,
∂v
∂t .
Basic Equations
KVL: v(x, t) = V2 + v(x + δx, t) + V1
KCL: i(x, t) = iC + i(x + δx, t)
∂i
Capacitor equation: C ∂v
∂t = iC = i(x, t) − i(x + δx, t) = − ∂x δx
Inductor equation (L1 and L2 have the same current):
∂v
∂i
= V1 + V2 = v(x, t) − v(x + δx, t) = − ∂x
δx
(L1 + L2 ) ∂t
Transmission Line Equations
∂i
C0 ∂v
=
−
∂t
∂x
∂v
∂i
L0 ∂t = − ∂x
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 3 / 13
Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
A short section of line δx long:
v(x, t) and i(x, t) depend on both
position and time.
Small δx ⇒ ignore 2nd order derivatives:
∂v(x,t)
∂t
=
∂v(x+δx,t)
∂t
,
∂v
∂t .
Basic Equations
KVL: v(x, t) = V2 + v(x + δx, t) + V1
KCL: i(x, t) = iC + i(x + δx, t)
∂i
Capacitor equation: C ∂v
∂t = iC = i(x, t) − i(x + δx, t) = − ∂x δx
Inductor equation (L1 and L2 have the same current):
∂v
∂i
= V1 + V2 = v(x, t) − v(x + δx, t) = − ∂x
δx
(L1 + L2 ) ∂t
Transmission Line Equations
∂i
C0 ∂v
=
−
∂t
∂x
∂v
∂i
L0 ∂t = − ∂x
E1.1 Analysis of Circuits (2016-8284)
C
is the capacitance per unit length
where C0 = δx
+L2
(Farads/m) and L0 = L1δx
is the total
inductance per unit length (Henries/m).
Transmission Lines: 17 – 3 / 13
Solution to Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Transmission Line Equations:
∂i
C0 ∂v
∂t = − ∂x
∂i
∂v
L0 ∂t
= − ∂x
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
Transmission Line Equations:
General solution:
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
∂i
C0 ∂v
∂t = − ∂x
∂i
∂v
L0 ∂t
= − ∂x
v(t, x) = f (t − ux ) + g(t + ux )
i(t, x) =
where u =
x
x
)−g(t+ u
)
f (t− u
Z0
q
1
L0 C0
and Z0 =
q
L0
C0
.
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
Transmission Line Equations:
General solution:
• Forward Wave
• Forward + Backward
Characteristics
∂i
∂v
L0 ∂t
= − ∂x
v(t, x) = f (t − ux ) + g(t + ux )
i(t, x) =
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
∂i
C0 ∂v
∂t = − ∂x
where u =
x
x
)−g(t+ u
)
f (t− u
Z0
q
1
L0 C0
and Z0 =
q
L0
C0
.
u is the propagation velocity and Z0 is the characteristic impedance.
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
Transmission Line Equations:
General solution:
• Forward Wave
• Forward + Backward
Characteristics
∂i
∂v
L0 ∂t
= − ∂x
v(t, x) = f (t − ux ) + g(t + ux )
i(t, x) =
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
∂i
C0 ∂v
∂t = − ∂x
where u =
x
x
)−g(t+ u
)
f (t− u
Z0
q
1
L0 C0
and Z0 =
q
L0
C0
.
u is the propagation velocity and Z0 is the characteristic impedance.
• Summary
f () and g() can be any differentiable functions.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
Transmission Line Equations:
General solution:
• Forward Wave
• Forward + Backward
Characteristics
∂i
∂v
L0 ∂t
= − ∂x
v(t, x) = f (t − ux ) + g(t + ux )
i(t, x) =
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
∂i
C0 ∂v
∂t = − ∂x
where u =
x
x
)−g(t+ u
)
f (t− u
Z0
q
1
L0 C0
and Z0 =
q
L0
C0
.
u is the propagation velocity and Z0 is the characteristic impedance.
• Summary
f () and g() can be any differentiable functions.
Verify by substitution:
∂i
− ∂x
=−
E1.1 Analysis of Circuits (2016-8284)
x
x
)−g ′ (t+ u
)
−f ′ (t− u
Z0
×
1
u
Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
General solution:
• Forward Wave
• Forward + Backward
Characteristics
∂i
∂v
L0 ∂t
= − ∂x
v(t, x) = f (t − ux ) + g(t + ux )
x
x
)−g(t+ u
)
f (t− u
Z0
i(t, x) =
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
∂i
C0 ∂v
∂t = − ∂x
Transmission Line Equations:
where u =
q
1
L0 C0
and Z0 =
q
L0
C0
.
u is the propagation velocity and Z0 is the characteristic impedance.
• Summary
f () and g() can be any differentiable functions.
Verify by substitution:
∂i
− ∂x
=−
x
x
)−g ′ (t+ u
)
−f ′ (t− u
Z0
′
= C0 f (t −
E1.1 Analysis of Circuits (2016-8284)
x
u)
′
1
u
x
u)
×
+ g (t +
= C0 ∂v
∂t
Transmission Lines: 17 – 4 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
Waves
x
u
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
f(t-0/u)
0
2
4
6
8
Time (ns)
10
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
f(t-0/u)
0
f(t-45/u)
2
4
6
8
Time (ns)
10
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
f (t −
E1.1 Analysis of Circuits (2016-8284)
45
u )
f(t-0/u)
0
f(t-45/u)
2
4
6
8
Time (ns)
10
is exactly the same as f (t) but delayed by 45
u = 3 ns.
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
f(t-0/u)
0
f(t-45/u)
2
4
f(t-90/u)
6
8
Time (ns)
10
45
f (t − 45
)
is
exactly
the
same
as
f
(t)
but
delayed
by
u
u = 3 ns.
• At x = 90 cm [N], vR (t) = f (t − 90
u ); now delayed by 6 ns.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
f(t-0/u)
0
f(t-45/u)
2
4
f(t-90/u)
6
8
Time (ns)
10
45
f (t − 45
)
is
exactly
the
same
as
f
(t)
but
delayed
by
u
u = 3 ns.
• At x = 90 cm [N], vR (t) = f (t − 90
u ); now delayed by 6 ns.
Waveform at x = 0 completely determines the waveform everywhere else.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
f(t-0/u)
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
0
f(t-45/u)
2
4
f(t-90/u)
6
8
Time (ns)
10
45
f (t − 45
)
is
exactly
the
same
as
f
(t)
but
delayed
by
u
u = 3 ns.
• At x = 90 cm [N], vR (t) = f (t − 90
u ); now delayed by 6 ns.
Waveform at x = 0 completely determines the waveform everywhere else.
Snapshot at t0 = 4 ns:
the waveform has just
arrived at the point
x = ut0 = 60 cm.
E1.1 Analysis of Circuits (2016-8284)
t = 4 ns
0
f(4-x/u)
20
40
60
80
Position (cm)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
f(t-0/u)
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
0
f(t-45/u)
2
4
f(t-90/u)
6
8
Time (ns)
10
45
f (t − 45
)
is
exactly
the
same
as
f
(t)
but
delayed
by
u
u = 3 ns.
• At x = 90 cm [N], vR (t) = f (t − 90
u ); now delayed by 6 ns.
Waveform at x = 0 completely determines the waveform everywhere else.
Snapshot at t0 = 4 ns:
the waveform has just
arrived at the point
x = ut0 = 60 cm.
E1.1 Analysis of Circuits (2016-8284)
t = 4 ns
0
f(4-x/u)
20
40
60
80
Position (cm)
Transmission Lines: 17 – 5 / 13
Forward Wave
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Suppose:
u = 15 cm/ns
and g(t) ≡ 0
⇒ v(x, t) = f t −
• At x = 0 cm [N],
vS (t) = f (t − u0 )
x
u
f(t-0/u)
• At x = 45 cm [N],
v(45, t) = f (t − 45
u )
0
f(t-45/u)
2
4
f(t-90/u)
6
8
Time (ns)
10
45
f (t − 45
)
is
exactly
the
same
as
f
(t)
but
delayed
by
u
u = 3 ns.
• At x = 90 cm [N], vR (t) = f (t − 90
u ); now delayed by 6 ns.
Waveform at x = 0 completely determines the waveform everywhere else.
Snapshot at t0 = 4 ns:
the waveform has just
arrived at the point
x = ut0 = 60 cm.
t = 4 ns
0
f(4-x/u)
20
40
60
80
Position (cm)
f (t − ux ) is a wave travelling forward (i.e. towards +x) along the line.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 5 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
At x = 0 cm [N],
vS (t) = f (t) + g(t)
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
At x = 0 cm [N],
vS (t) = f (t) + g(t)
At x = 90 cm [N], g starts at t = 1 and f starts at t = 6.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
At x = 0 cm [N],
vS (t) = f (t) + g(t)
At x = 45 cm [N], g is only 1 ns behind f and they add together.
At x = 90 cm [N], g starts at t = 1 and f starts at t = 6.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
At x = 0 cm [N],
vS (t) = f (t) + g(t)
At x = 45 cm [N], g is only 1 ns behind f and they add together.
At x = 90 cm [N], g starts at t = 1 and f starts at t = 6.
A vertical line on the diagram
gives a snapshot of the entire
line at a time instant t.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
At x = 0 cm [N],
vS (t) = f (t) + g(t)
At x = 45 cm [N], g is only 1 ns behind f and they add together.
At x = 90 cm [N], g starts at t = 1 and f starts at t = 6.
A vertical line on the diagram
gives a snapshot of the entire
line at a time instant t.
f and g first meet at t = 3.5
and x = 52.5.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Forward + Backward Waves
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
x
) is a wave travelling backwards, i.e. in the −x direction.
Similarly g(t + u
v(x, t) =
f (t − ux ) + g(t + ux )
At x = 0 cm [N],
vS (t) = f (t) + g(t)
At x = 45 cm [N], g is only 1 ns behind f and they add together.
At x = 90 cm [N], g starts at t = 1 and f starts at t = 6.
A vertical line on the diagram
gives a snapshot of the entire
line at a time instant t.
f and g first meet at t = 3.5
and x = 52.5.
Magically, f and g pass
through each other entirely
unaltered.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 6 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
x
u
to be the forward and
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
to be the forward and
i is always
• Forward Wave
• Forward + Backward
measured in the
+ve x direction.
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
x
u
Then
vx (t) = fx (t) + gx (t)
and
ix (t) = Z0−1 (fx (t) − gx (t)).
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
x
u
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
x
u
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
Power Flow
The power transferred into the shaded region across the boundary at x is
Px (t) = vx (t)ix (t)
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
x
u
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
Power Flow
The power transferred into the shaded region across the boundary at x is
Px (t) = vx (t)ix (t) = Z0−1 (fx (t) + gx (t)) (fx (t) − gx (t))
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
x
u
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
Power Flow
The power transferred into the shaded region across the boundary at x is
Px (t) = vx (t)ix (t) = Z0−1 (fx (t) + gx (t)) (fx (t) − gx (t))
2
fx2 (t)
gx
(t)
= Z0 − Z0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
x
u
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
Power Flow
The power transferred into the shaded region across the boundary at x is
Px (t) = vx (t)ix (t) = Z0−1 (fx (t) + gx (t)) (fx (t) − gx (t))
2
fx2 (t)
gx
(t)
= Z0 − Z0
fx carries power into shaded area and gx carries power out independently.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
x
u
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
Power Flow
The power transferred into the shaded region across the boundary at x is
Px (t) = vx (t)ix (t) = Z0−1 (fx (t) + gx (t)) (fx (t) − gx (t))
2
fx2 (t)
gx
(t)
= Z0 − Z0
fx carries power into shaded area and gx carries power out independently.
Power travels in the same direction as the wave.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Power Flow
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
x
u
and gx (t) = g t +
Define fx (t) = f t −
backward waveforms at any point, x.
• Summary
to be the forward and
measured in the
+ve x direction.
Waves
Characteristics
x
u
i is always
• Forward Wave
• Forward + Backward
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Then vx (t) = fx (t) + gx (t) and ix (t) = Z0−1 (fx (t) − gx (t)).
Note: Knowing the waveform fx (t) or gx (t)at any position x, tells you it at
y−x
y−x
and gy (t) = gx t + u .
all other positions: fy (t) = fx t − u
Power Flow
The power transferred into the shaded region across the boundary at x is
Px (t) = vx (t)ix (t) = Z0−1 (fx (t) + gx (t)) (fx (t) − gx (t))
2
fx2 (t)
gx
(t)
= Z0 − Z0
fx carries power into shaded area and gx carries power out independently.
Power travels in the same direction as the wave.
The same power as would be absorbed by a [ficticious] resistor of value Z0 .
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 7 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 8 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Hence (fL (t) + gL (t)) = Z0−1 (fL (t) − gL (t)) RL
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 8 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Hence (fL (t) + gL (t)) = Z0−1 (fL (t) − gL (t)) RL
L −Z0
From this: gL (t) = R
R +Z × fL (t)
L
0
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 8 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Hence (fL (t) + gL (t)) = Z0−1 (fL (t) − gL (t)) RL
L −Z0
From this: gL (t) = R
R +Z × fL (t)
L
0
g (t)
L −Z0
We define the reflection coefficient: ρL = fL (t) = R
RL +Z0 = +0.5
L
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 8 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Hence (fL (t) + gL (t)) = Z0−1 (fL (t) − gL (t)) RL
L −Z0
From this: gL (t) = R
R +Z × fL (t)
L
0
g (t)
L −Z0
We define the reflection coefficient: ρL = fL (t) = R
RL +Z0 = +0.5
L
Substituting gL (t) = ρL fL (t) gives
vL (t) = (1 + ρL ) fL (t) and iL (t) = (1 − ρL ) Z0−1 fL (t)
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 8 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Hence (fL (t) + gL (t)) = Z0−1 (fL (t) − gL (t)) RL
L −Z0
From this: gL (t) = R
R +Z × fL (t)
L
0
g (t)
L −Z0
We define the reflection coefficient: ρL = fL (t) = R
RL +Z0 = +0.5
L
Substituting gL (t) = ρL fL (t) gives
vL (t) = (1 + ρL ) fL (t) and iL (t) = (1 − ρL ) Z0−1 fL (t)
v0(t)
0
2
4
6
At source end:
E1.1 Analysis of Circuits (2016-8284)
8
10
12
14
16
18
Time (ns)
g0 (t) = ρL f0 t −
2L
u
i.e. delayed by 2L
u = 12 ns.
Transmission Lines: 17 – 8 / 13
Reflections
vx = fx + gx
ix = Z0−1 (fx − gx )
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
From Ohm’s law at x = L, we have vL (t) = iL (t)RL
Hence (fL (t) + gL (t)) = Z0−1 (fL (t) − gL (t)) RL
L −Z0
From this: gL (t) = R
R +Z × fL (t)
L
0
g (t)
L −Z0
We define the reflection coefficient: ρL = fL (t) = R
RL +Z0 = +0.5
L
Substituting gL (t) = ρL fL (t) gives
vL (t) = (1 + ρL ) fL (t) and iL (t) = (1 − ρL ) Z0−1 fL (t)
v0(t)
0
2
i0(t)
4
6
8
10
12
14
16
18
Time (ns)
0
2L
u
2
4
6
8
10
12
14
16
18
Time (ns)
At source end: g0 (t) = ρL f0 t −
i.e. delayed by 2L
u = 12 ns.
Note that the reflected current has been multiplied by −ρ.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 8 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
ρ=
R−Z0
R+Z0
=
Equations
• Solution to Transmission
Line Equations
R
Z0
R
Z0
−1
1
+1
ρ
17: Transmission Lines
0
-1
0
• Forward Wave
• Forward + Backward
2
3
4
5
RZ-1
0
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
1
ρ depends on the ratio
R
Z0
ρ
3
+0.5
vL (t)
f (t)
R
Z0 .
iL (t)Z0
f (t)
Comment
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
ρ=
R−Z0
R+Z0
vL (t)
f (t)
=
R
Z0
R
Z0
−1
1
+1
ρ
17: Transmission Lines
0
=1+ρ
-1
0
• Forward Wave
• Forward + Backward
2
3
4
5
RZ-1
0
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
1
ρ depends on the ratio
R
Z0
ρ
vL (t)
f (t)
3
+0.5
1.5
R
Z0 .
iL (t)Z0
f (t)
Comment
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
1
+1
ρ
17: Transmission Lines
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0 .
R
Z0
ρ
vL (t)
f (t)
iL (t)Z0
f (t)
3
+0.5
1.5
0.5
Comment
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
1
+1
ρ
17: Transmission Lines
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0 .
R
Z0
ρ
vL (t)
f (t)
iL (t)Z0
f (t)
Comment
3
+0.5
1.5
0.5
R > Z0 ⇒ ρ > 0
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
1
+1
ρ
17: Transmission Lines
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0 .
R
Z0
ρ
vL (t)
f (t)
iL (t)Z0
f (t)
Comment
3
+0.5
1.5
0.5
R > Z0 ⇒ ρ > 0
1
3
−0.5
0.5
1.5
R < Z0 ⇒ ρ < 0
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
1
+1
ρ
17: Transmission Lines
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0 .
R
Z0
ρ
vL (t)
f (t)
iL (t)Z0
f (t)
3
1
+0.5
0
−0.5
1.5
1
0.5
0.5
1
1.5
Comment
Characteristics
• Summary
1
3
E1.1 Analysis of Circuits (2016-8284)
R > Z0 ⇒ ρ > 0
Matched: No reflection at all
R < Z0 ⇒ ρ < 0
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
+1
∞
3
1
1
3
E1.1 Analysis of Circuits (2016-8284)
ρ
+1
+0.5
0
−0.5
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0
1
ρ
17: Transmission Lines
R
Z0 .
vL (t)
f (t)
iL (t)Z0
f (t)
2
1.5
1
0.5
0
0.5
1
1.5
Comment
Open circuit: vL = 2f , iL ≡ 0
R > Z0 ⇒ ρ > 0
Matched: No reflection at all
R < Z0 ⇒ ρ < 0
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
+1
∞
3
1
1
3
0
E1.1 Analysis of Circuits (2016-8284)
ρ
+1
+0.5
0
−0.5
−1
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0
1
ρ
17: Transmission Lines
R
Z0 .
vL (t)
f (t)
iL (t)Z0
f (t)
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Comment
Open circuit: vL = 2f , iL ≡ 0
R > Z0 ⇒ ρ > 0
Matched: No reflection at all
R < Z0 ⇒ ρ < 0
Short circuit: vL ≡ 0, iL =
2f
Z0
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
R
Z0
R
Z0
−1
+1
∞
3
1
1
3
0
ρ
+1
+0.5
0
−0.5
−1
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0
1
ρ
17: Transmission Lines
R
Z0 .
vL (t)
f (t)
iL (t)Z0
f (t)
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Comment
Open circuit: vL = 2f , iL ≡ 0
R > Z0 ⇒ ρ > 0
Matched: No reflection at all
R < Z0 ⇒ ρ < 0
Short circuit: vL ≡ 0, iL =
2f
Z0
1+ρ
Note: Reverse mapping is R = vi L = 1−ρ × Z0
L
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 9 / 13
Reflection Coefficients
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
ρ=
R−Z0
R+Z0
=
vL (t)
f (t) = 1 + ρ
iL (t)Z0
f (t) = 1 −
−1
+1
∞
3
1
1
3
0
ρ
+1
+0.5
0
−0.5
−1
0
-1
0
ρ
1
2
3
4
5
RZ-1
0
ρ depends on the ratio
R
Z0
1
ρ
R
Z0
R
Z0
17: Transmission Lines
R
Z0 .
vL (t)
f (t)
iL (t)Z0
f (t)
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Comment
Open circuit: vL = 2f , iL ≡ 0
R > Z0 ⇒ ρ > 0
Matched: No reflection at all
R < Z0 ⇒ ρ < 0
Short circuit: vL ≡ 0, iL =
2f
Z0
1+ρ
Note: Reverse mapping is R = vi L = 1−ρ × Z0
L
Remember:
E1.1 Analysis of Circuits (2016-8284)
ρ ∈ {−1, +1} and increases with R.
Transmission Lines: 17 – 9 / 13
Driving a line
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
From Ohm’s law at x = 0, we have v0 (t) = vS (t) − i0 (t)RS where RS is
the Thévenin resistance of the voltage source.
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 10 / 13
Driving a line
17: Transmission Lines
• Transmission Lines
• Transmission Line
vx = fx + gx
x
ix = fxZ−g
0
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
From Ohm’s law at x = 0, we have v0 (t) = vS (t) − i0 (t)RS where RS is
the Thévenin resistance of the voltage source.
Substituting v0 (t) = f0 + g0 and i0 (t) =
• Summary
f0 (t) =
E1.1 Analysis of Circuits (2016-8284)
Z0
RS +Z0 vS (t)
+
f0 −g0
Z0
leads to:
RS −Z0
RS +Z0 g0 (t)
Transmission Lines: 17 – 10 / 13
Driving a line
17: Transmission Lines
• Transmission Lines
• Transmission Line
vx = fx + gx
x
ix = fxZ−g
0
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
From Ohm’s law at x = 0, we have v0 (t) = vS (t) − i0 (t)RS where RS is
the Thévenin resistance of the voltage source.
Substituting v0 (t) = f0 + g0 and i0 (t) =
• Summary
f0 (t) =
E1.1 Analysis of Circuits (2016-8284)
Z0
RS +Z0 vS (t)
+
f0 −g0
Z0
RS −Z0
RS +Z0 g0 (t) ,
leads to:
τ0 vS (t) + ρ0 g0 (t)
Transmission Lines: 17 – 10 / 13
Driving a line
17: Transmission Lines
• Transmission Lines
• Transmission Line
vx = fx + gx
x
ix = fxZ−g
0
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
From Ohm’s law at x = 0, we have v0 (t) = vS (t) − i0 (t)RS where RS is
the Thévenin resistance of the voltage source.
Substituting v0 (t) = f0 + g0 and i0 (t) =
• Summary
f0 (t) =
Z0
RS +Z0 vS (t)
+
f0 −g0
Z0
RS −Z0
RS +Z0 g0 (t) ,
leads to:
τ0 vS (t) + ρ0 g0 (t)
So f0 (t) is the superposition of two terms:
0
(1) Input vS (t) multiplied by τ0 = R Z+Z
which is the same as a
S
0
potential divider if you replace the line with a [ficticious] resistor Z0 .
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 10 / 13
Driving a line
17: Transmission Lines
• Transmission Lines
• Transmission Line
vx = fx + gx
x
ix = fxZ−g
0
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
From Ohm’s law at x = 0, we have v0 (t) = vS (t) − i0 (t)RS where RS is
the Thévenin resistance of the voltage source.
Substituting v0 (t) = f0 + g0 and i0 (t) =
• Summary
f0 (t) =
Z0
RS +Z0 vS (t)
+
f0 −g0
Z0
RS −Z0
RS +Z0 g0 (t) ,
leads to:
τ0 vS (t) + ρ0 g0 (t)
So f0 (t) is the superposition of two terms:
0
(1) Input vS (t) multiplied by τ0 = R Z+Z
which is the same as a
S
0
potential divider if you replace the line with a [ficticious] resistor Z0 .
(2) The incoming backward wave, g0 (t), multiplied by a reflection
S −Z0
coefficient: ρ0 = R
R +Z .
S
E1.1 Analysis of Circuits (2016-8284)
0
Transmission Lines: 17 – 10 / 13
Driving a line
17: Transmission Lines
• Transmission Lines
• Transmission Line
vx = fx + gx
x
ix = fxZ−g
0
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
From Ohm’s law at x = 0, we have v0 (t) = vS (t) − i0 (t)RS where RS is
the Thévenin resistance of the voltage source.
Substituting v0 (t) = f0 + g0 and i0 (t) =
• Summary
f0 (t) =
Z0
RS +Z0 vS (t)
+
f0 −g0
Z0
RS −Z0
RS +Z0 g0 (t) ,
leads to:
τ0 vS (t) + ρ0 g0 (t)
So f0 (t) is the superposition of two terms:
0
(1) Input vS (t) multiplied by τ0 = R Z+Z
which is the same as a
S
0
potential divider if you replace the line with a [ficticious] resistor Z0 .
(2) The incoming backward wave, g0 (t), multiplied by a reflection
S −Z0
coefficient: ρ0 = R
R +Z .
S
0
100
= 0.83
For RS = 20: τ0 = 20+100
E1.1 Analysis of Circuits (2016-8284)
and
ρ0 =
20−100
20+100
= −0.67.
Transmission Lines: 17 – 10 / 13
Multiple Reflections
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
ρ0 = − 23
ρL = 12
vx = fx + gx
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
f 0(t)
0
5
10
15
20
25
30
Time (ns)
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
Characteristics
• Summary
0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
Characteristics
• Summary
0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
Characteristics
• Summary
0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
Characteristics
• Summary
0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
Characteristics
• Summary
0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
t−
f 0(t)
0
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
E1.1 Analysis of Circuits (2016-8284)
5
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
E1.1 Analysis of Circuits (2016-8284)
t−
L
u
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
v0 (t) =
f0 (t) + gL t −
E1.1 Analysis of Circuits (2016-8284)
t−
L
u
L
u
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
E1.1 Analysis of Circuits (2016-8284)
L
u
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
E1.1 Analysis of Circuits (2016-8284)
L
u
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
E1.1 Analysis of Circuits (2016-8284)
L
u
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
vL (t) =
f0 t −
E1.1 Analysis of Circuits (2016-8284)
L
u
L
u
0
+ gL (t)
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
vL (t) =
f0 t −
E1.1 Analysis of Circuits (2016-8284)
L
u
0
vL(t)
L
u
+ gL (t)
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
vL (t) =
f0 t −
E1.1 Analysis of Circuits (2016-8284)
L
u
0
vL(t)
L
u
+ gL (t)
0
Transmission Lines: 17 – 11 / 13
Multiple Reflections
ρ0 = − 23
ρL = 12
vx = fx + gx
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Each extra bit of f0 is
delayed by 2L
u (=12 ns)
and multiplied by ρL ρ0 :
f0P
(t) =
∞
i i
i=0 τ0 ρL ρ0 vS
gL (t) = ρL f0 t −
t−
L
f 0(t)
0
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
5
10
15
20
25
30
Time (ns)
gL(t)
2Li
u
0
u
v0(t)
v0 (t) =
f0 (t) + gL t −
vL (t) =
f0 t −
E1.1 Analysis of Circuits (2016-8284)
L
u
0
vL(t)
L
u
+ gL (t)
0
Transmission Lines: 17 – 11 / 13
Transmission Line Characteristics
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
Integrated circuits & Printed circuit boards
High speed digital or high frequency analog
interconnections
Z0 ≈ 100 Ω, u ≈ 15 cm/ns.
Long Cables
Coaxial cable (“coax”): unaffacted by external fields;
use for antennae and instrumentation.
Z0 = 50 or 75 Ω, u ≈ 25 cm/ns.
Twisted Pairs: cheaper and thinner than coax and
resistant to magnetic fields; use for computer network
and telephone cabling. Z0 ≈ 100 Ω, u ≈ 19 cm/ns.
When do you have to bother?
Answer: long cables or high frequencies. You can completely ignore
u
transmission line effects if length ≪ frequency
= wavelength.
• Audio (< 20 kHz) never matters.
• Computers (1 GHz) usually matters.
• Radio/TV usually matters.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 12 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
• Voltage and current are: vx = fx + gx and ix =
fx −gx
Z0
Characteristics
• Summary
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
• Voltage and current are: vx = fx + gx and ix =
fx −gx
Z0
• Terminating line with R at x = L links the forward and backward waves:
0
◦ backward wave is gL = ρL fL where ρL = R−Z
R+Z0
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
• Voltage and current are: vx = fx + gx and ix =
fx −gx
Z0
• Terminating line with R at x = L links the forward and backward waves:
0
◦ backward wave is gL = ρL fL where ρL = R−Z
R+Z0
◦ the reflection coefficient, ρL ∈ {−1, +1} and increases with R
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
• Voltage and current are: vx = fx + gx and ix =
fx −gx
Z0
• Terminating line with R at x = L links the forward and backward waves:
0
◦ backward wave is gL = ρL fL where ρL = R−Z
R+Z0
◦ the reflection coefficient, ρL ∈ {−1, +1} and increases with R
◦ R = Z0 avoids reflections: matched termination.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
• Voltage and current are: vx = fx + gx and ix =
fx −gx
Z0
• Terminating line with R at x = L links the forward and backward waves:
0
◦ backward wave is gL = ρL fL where ρL = R−Z
R+Z0
◦ the reflection coefficient, ρL ∈ {−1, +1} and increases with R
◦ R = Z0 avoids reflections: matched termination.
◦ Reflections go on for ever unless one or both ends are matched.
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13
Summary
17: Transmission Lines
• Transmission Lines
• Transmission Line
Equations
• Solution to Transmission
Line Equations
• Forward Wave
• Forward + Backward
Waves
• Power Flow
• Reflections
• Reflection Coefficients
• Driving a line
• Multiple Reflections
• Transmission Line
Characteristics
• Summary
• Signals travel at around u ≈ 12 c = 15 cm/ns.
Only matters for high frequencies or long cables.
• Forward and backward waves travel along the line:
x
x
fx (t) = f0 t − u
and gx (t) = g0 t + u
◦ Knowing fx and gx at any single x position tells you everything
• Voltage and current are: vx = fx + gx and ix =
fx −gx
Z0
• Terminating line with R at x = L links the forward and backward waves:
0
◦ backward wave is gL = ρL fL where ρL = R−Z
R+Z0
◦ the reflection coefficient, ρL ∈ {−1, +1} and increases with R
◦ R = Z0 avoids reflections: matched termination.
◦ Reflections go on for ever unless one or both ends are matched.
◦ f is infinite sum of copies of the input signal delayed successively
by the round-trip delay, 2L
u , and multiplied by ρL ρ0 .
E1.1 Analysis of Circuits (2016-8284)
Transmission Lines: 17 – 13 / 13