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Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Transmission Line Theory
S. R. [email protected]
School of Electronics EngineeringVellore Institute of Technology
April 26, 2013
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
First of All, You Know How to Represent a WaveRight!
The wave shown in the above diagram can be represented as
F (x, t) = sin (βx− βvt) = sin (βx−ωt) (1)
where,ω = 2πf = βv. (2)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Now, What is Electrical Length?
The wave shown in the above diagram can be represented as
λ ⇔ 2π
l ⇔ ?? (3)
Electrical length θ is given as
θ =
(2π
λ
)l = βl. (4)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Remember these Equations from Wave Propagation inan Isotropic Medium?
• v = ωβ
• εs = ε(1− j σ
ωε
); tan θlt =
σωε
• γ2 = γ2x + γ2
y + γ2z = −ω2µε
• γ = α + jβ = ω
√µε2
[√1 +
(σ
ωε
)2 − 1]+ jω
√µε2
[√1 +
(σ
ωε
)2+ 1]
• ~P = ~E× ~H
• η =√
µεs
=√
jωµσ+jωε
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Reflection of Plane Wave at Normal IncidenceElectric fields on both sides are given as,
~Ei = Eie−γz,1zx,
~Et = Ete−γz,1zx, and
~Er = Ereγz,1zx.
Similarly, magnetic fields on both sides are given as,
~Hi = Hie−γz,1zy =
Ei
η1e−γz,1zy,
~Ht = Hte−γz,2zy =Et
η2e−γz,2zy, and
~Hr = Hreγz,1zx = − Er
η1eγz,1zy.
From the boundary conditions (at z = 0),
Ei + Er = Et, and
Ei
η1− Er
η1=
Et
η2. (5)
x
From (5),
Γ =Er
Ei=
η2 − η1
η2 + η1, (6)
τ =Et
Ei=
2η2
η2 + η1, and (7)
1 + Γ = τ. (8)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Let’s Just Use a Different NotationElectric fields on both sides are given as,
~Ei = V+1 e−γz,1zx,
~Et = V−2 e−γz,2zx, and
~Er = V−1 eγz,1zx.
Similarly, magnetic fields on both sides are given as,
~Hi = I+1 e−γz,1zx =V+
1
Z10
e−γz,1zy,
~Ht = I−2 e−γz,2zx =V−2Z2
0e−γz,2zy, and
~Hr = I−1 eγz,1zx = −V−1Z1
0eγz,1zy.
From the boundary conditions (at z = 0),
V+1 + V−1 = V−2 , and
V+1
Z10−V−1
Z10
=V−2Z2
0. (9)
x
From (9),
Γ =V−1V+
1=
Z20 − Z1
0
Z20 + Z1
0, (10)
τ =V−2V−1
=2Z2
0
Z20 + Z1
0, and (11)
1 + Γ = τ. (12)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Analogy
x
No Power Missmatch
here
+

+

LHS:
Vtot (z < 0) = V+1 e−γz,1z + V−1 eγz,1z
Itot (z < 0) = I+1 e−γz,1z + I−1 eγz,1z
=V+
1Z0,1
e−γz,1z− V−1Z0,1
eγz,1z
RHS:
Vtot (z > 0) = V−2 e−γz,2z + V+2 eγz,2z
Itot (z > 0) = I−2 e−γz,2 + I+2 eγz,2z
=V−2Z0,2
e−γz,2z− V+2
Z0,2eγz,2z
︸ ︷︷ ︸this term is zero here
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
TX Line Connected to a Load
+

V (z) = V+e−γz︸ ︷︷ ︸incident wave
+ V−eγz︸ ︷︷ ︸reflected wave
I (z) = I+e−γz︸ ︷︷ ︸incident wave
+ I−eγz︸ ︷︷ ︸reflected wave
=V+
Z0e−γz−V−
Z0eγz (13)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Reflection Coefficient (Γ/Γ0)  Definition
+

Γ (z) =voltage of the reflected wavevoltage of the incident wave
=V−eγz
V+e−γz
=
(V−
V+
)e2γz
= Γ0e2γz (14)
Γ0 = Γ (z = 0) =V−
V+= − I−
I+. (15)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
(Γ/Γ0)  Derivation
+

From (13)
V (z = 0)I (z = 0)
= ZL =V+ + V−(V+
Z0− V−
Z0
)= Z0
1 + V−V+
1− V−V+
= Z0
(1 + Γ0
1− Γ0
).
By rearranging the above equation, one gets
Γ0 =ZL − Z0
ZL + Z0⇒ Γ (z) =
(ZL − Z0
ZL + Z0
)e2γz (16)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Input Impedance (Zin)  Definition
+

V (z) = V+e−γz + V−eγz
I (z) = V+
Z0e−γz − V−
Z0eγz
Zin =V (z)I (z)
= Z0
(V+e−γz + V−eγz
V+e−γz −V−eγz
)= Z0
(1 + Γ0e2γz
1− Γ0e2γz
)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Zin  Lossless Case  Derivation
Zin = Z0
(1 + Γ0e2γz
1− Γ0e2γz
)
= Z0
1 +(
ZL−Z0ZL+Z0
)ej2βz
1−(
ZL−Z0ZL+Z0
)ej2βz
, since γ = jβ, for lossless case
= Z0
[ZL + Z0 + (ZL − Z0) ej2βz
ZL + Z0 − (ZL − Z0) ej2βz
]= Z0
[ZL + Z0 + ZLej2βz − Z0ej2βz
ZL + Z0 − ZLej2βz + Z0ej2βz
]
= Z0
[ZL(1 + ej2βz
)+ Z0
(1− ej2βz
)ZL (1− e−j2βl) + Z0 (1 + e−j2βl)
]
= Z0
ZL + Z0
(1−ej2βz
1+ej2βz
)Z0 + ZL
(1−ej2βz
1+ej2βz
)
= Z0
[ZL − jZ0 tan βzZ0 − jZL tan βz
]
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Voltage Standing Wave Ratio (VSWR)  Definition
VSWR is defined as VmaxVmin
. Since voltage along a TX line is
V (z) = V+e−jβz + V−ejβz,
it can be derived from the Schwartz inequality principle that
V (z) =∣∣∣V+e−jβz + V−ejβz
∣∣∣≤∣∣∣V+e−jβz
∣∣∣+ ∣∣∣V−ejβz∣∣∣
≤∣∣V+
∣∣+ ∣∣V−∣∣ = Vmax.
Similarly, it can also be derived that
V (z) =∣∣∣V+e−jβz + V−ejβz
∣∣∣≥∣∣∣V+e−jβz
∣∣∣− ∣∣∣V−ejβz∣∣∣
≥∣∣V+
∣∣− ∣∣V−∣∣ = Vmin.
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Voltage Standing Wave Ratio (VSWR)  Derivation
So, VSWR is given as
VSWR =VmaxVmin
=V+ + V− V+  − V− 
=1 + V
− V+ 
1− V− V+ 
=1 + Γ01− Γ0
.
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Instantaneous & Time Average Power
Instantaneous power corresponding to the above set of voltage & current is defined as
Pinst (t) = v0i0 cos (ωt + φ1) cos (ωt + φ2)
=v0i0
2[cos (ωt + φ1 + ωt + φ2) + cos (ωt + φ1 −ωt− φ2)]
=v0i0
2[cos (2ωt + φ1 + φ2) + cos (φ1 − φ2)] (17)
Time average power is defined as
Pavg =1
T0
ˆ T0
0Pinst dt =
v0i02
cos (φ1 − φ2) (18)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Time Average Power  Complex Notation
vreal = v0 cos (ωt + φ1) vcomplex = V = v0ej(ωt+φ1)
ireal = i0 cos (ωt + φ2) icomplex = I = i0ej(ωt+φ2)
Pavg =v0 i0
2 cos (φ1 − φ2) Pavg = 12 Re (VI∗) = v0 i0
2 cos (φ1 − φ2)
In the above, does the equation 1
2 Re (VI∗) remind you of some thing ? ... Isn’t it very similar to the
complex Poynting vector 12 Re
(~E× ~H∗
)that you study in EMT course ?!
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Power Transfer along a TX Line  Definitions
+

V (z) = V+e−jβz︸ ︷︷ ︸incident wave
+ V−ejβz︸ ︷︷ ︸reflected wave
I (z) = I+e−jβz︸ ︷︷ ︸incident wave
+ I−ejβz︸ ︷︷ ︸reflected wave
=V+
Z0e−jβz−V−
Z0ejβz
From the above set of equations,
Ptotal =12
Re [V (z)] [I (z)]∗ =12
Re
V+ 2
Z0− V
− 2
Z0−V+V−
Z0e−j2βz +
V+V−
Z0ej2βz︸ ︷︷ ︸
this term is imaginary
Pincident =
12
Re(
V+e−jβz) (
I+e−jβz)∗
=12
Re(V+) (
I+)∗
=12V+ 2
Z0
Preflected = − 12
Re(
V−ejβz) (
I−ejβz)∗
= − 12
Re(V−) (
I−)∗
=12V− 2
Z0. (19)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Return Loss  Definition
+

V (z) = V+e−jβz︸ ︷︷ ︸incident wave
+ V−ejβz︸ ︷︷ ︸reflected wave
I (z) = I+e−jβz︸ ︷︷ ︸incident wave
+ I−ejβz︸ ︷︷ ︸reflected wave
=V+
Z0e−jβz−V−
Z0ejβz
So, from (19)
Ptotal =12
Re [V (z)] [I (z)]∗ =12
Re
[V+ 2
Z0− V
− 2
Z0
]= Pincident − Preflected (20)
From the above equation, return loss is defined as
RL =Pincident
Preflected=V+ 2
V− 2=
1
Γ02=
1
Γ (z)2= −20 log (Γ0) in dB (21)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Important Formulas  Summary
• Γ (z = −l) = Γ0e−j2βl, where Γ0 =ZL−Z0ZL+Z0
• Zin (z = −l) = Z0
(1+Γ0e−j2βl
1−Γ0e−j2βl
)= Z0
[ZL+jZ0 tan βlZ0+jZL tan βl
]• VSWR =
1+Γ01−Γ0
• Ptotal =12 Re [V (z)] [I (z)]∗ = 1
2 Re[V+ 2
Z0− V
− 2Z0
]= Pincident − Preflected
• RL =PincidentPreflected
=V+ 2
V− 2= 1
Γ02= 1Γ(z)2
= −20 log (Γ0) in dB
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Short Circuit & Open Circuit
+

+

Zin (z = −l) = Z0
[ZL + jZ0 tan βlZ0 + jZL tan βl
]From the above formula and L’Hospital’s rule, one gets
ZSCin = jZ0 tan βl and
ZOCin = −jZ0 cot βl. (22)
Also, one can notice from the above equations that
ZSCin × ZOC
in = Z20.
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Quarterwave (λ/4) Transformer
+

Zin (z = −l) = Z0
[ZL + jZ0 tan βlZ0 + jZL tan βl
]From the above formula and L’Hospital’s rule, when l = λ/4,
Zλ/4in =
Z20
ZL. (23)
In other words,
Zλ/4in × ZL = Z2
0.
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
This is how a Smith Chart Looks Like ...
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Would you Like to See a More Complicated SmithChart? ... ,
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Impedance Circles
Zin
Z0= r + jx =
1 + Γ1− Γ
r =1− Γ2
r − Γ2i
(1− Γr)2 + Γ2
i
x =2Γi
(1− Γr)2 + Γ2
i
⇓
[Γr −
r1 + r
]2
+ Γ2i =
[1
1 + r
]2
[Γr − 1]2 +[
Γi −1x
]2
=
[1x
]2
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Impedance Circles
Constant r Circles Constant x Circles
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Admittance Circles
Impedance Circles Plot
Yin
Y0= g + jb =
1− Γ1 + Γ
g =1− Γ2
r − Γ2i
(1 + Γr)2 + Γ2
i
b =−2Γi
(1 + Γr)2 + Γ2
i
⇓
[Γr +
g1 + g
]2
+ Γ2i =
[1
1 + g
]2
[Γr + 1]2 +[
Γi +1b
]2
=
[1b
]2
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Admittance Circles
Constant g Circles Constant b Circles
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Locating a given Load on the Smith Chart
ZL = 100 + j100Ω, Z0 = 50Ω
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Moving towards the Generator using the ConstantVSWR Circle
Γ (z = −l) = Γ0e−j2βl
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Adding a Series Inductor
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Adding a Series Capacitor
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Adding a Parallel Inductor
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Adding a Parallel Capacitor
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
For this section, see the PDF file uploaded separately ...
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Telegraph Equations
From the above figure and Kirchhoff’s circuital laws:
V (z + ∆z)−V (z) = −ZsI (z)∆z
I (z + ∆z)− I (z) = −YpV (z + ∆z)∆z (24)
As ∆z→ 0, the above equations become
lim∆z→0
V (z + ∆z)−V (z)∆z
=∂V (z, t)
∂z= −ZsI (z)
lim∆z→0
I (z + ∆z)− I (z)∆z
=∂I (z, t)
∂z= −YpV (z + ∆z) (25)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Telegraph Equations
If places of Zs and Yp and interchanged along the zdirection as shown in the above figure, thenusing Kirchoff’s circuital laws gives
lim∆z→0
V (z + ∆z)−V (z)∆z
=∂V (z, t)
∂z= −ZsI (z + ∆z)
lim∆z→0
I (z + ∆z)− I (z)∆z
=∂I (z, t)
∂z= −YpV (z) (26)
In either case, ∆z→ 0; so I (z + ∆z)→ I (z) and V (z + ∆z)→ V (z) ... So, the equations (25) and (26)reduce to well known telegraph equations shown below:
∂V (z)∂z
= −ZsI (z) (27)
∂I (z)∂z
= −YpV (z) (28)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Wave Equation
Substituting (27) and (28) into each other gives
d2V (z)dz2 = ZsYpV (z) = γ2V (z)
d2I (z)dz2 = ZsYpI (z) = γ2I (z) (29)
where γ =√
ZsYp. Solving the above second order homogeneous linear differential equation gives
V (z) = V+e−γz + V−e+γz
I (z) = I+e−γz + I−e+γz. (30)
In general, ZS = R + jωL and YP = G + jωC, where R and G indicate conductor and dielectric losses,respectively. So,
γ =√(R + jωL) (G + jωC) (31)
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Characteristic Impedance Z0
Substituting (30) in (28) gives
∂I (z)∂z
=∂ (I+e−γz + I−e+γz)
∂z= −γI+e−γz + γI−e+γz = −YpV (z) = −Yp
(V+e−γz + V−e+γz)
⇓
−γI+ = −YpV+ ⇒ V+
I+= γ
YP=√
ZsYp
=√
R+jωLG+jωC = Z0
γI− = −YpV− ⇒ V−I− = − γ
YP= −
√ZsYp
= −√
R+jωLG+jωC = −Z0
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Special Cases (Lossless & Almost lossless)lossless: For lossless case, R = 0 and G = 0. So, for lossless case,
Z0 =
√LC and
γ = 0 + jω√
LC.
Almostlossless: For almost lossless case, R jωL and G jωC. In such case,
Z0 =
√R + jωLG + jωC
=
√√√√( jωLjωC
) RjωL + 1G
jωC + 1≈√
LC
and
γ =√(R + jωL) (G + jωC) =
√jωL× jωC
(R
jωL+ 1)(
GjωC
+ 1)
= jω√
LC
(√1 +
RjωL
+G
jωC− RG
ω2LC
)≈ jω
√LC
(√1 +
RjωL
+G
jωC
)
≈ jω√
LC[
1 +12
(R
jωL+
GjωC
)]=
12
(RZ0
+ GZ0
)︸ ︷︷ ︸
α
+j ω√
LC︸ ︷︷ ︸β
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Special Cases (Distortionless)Distortionless: For distortionless case, R
L = GC . In such case,
Z0 =
√R + jωLG + jωC
=
√√√√( jωLjωC
) RjωL + 1G
jωC + 1=
√LC
and
γ =√(R + jωL) (G + jωC) =
√jωL× jωC
(R
jωL+ 1)(
GjωC
+ 1)
= jω√
LC
(√1 +
2RjωL− R2
ω2L2
)= jω
√LC×
√(1 +
RjωL
)2
= jω√
LC(
1 +R
jωL
)=
RZ0︸︷︷︸
α
+j ω√
LC︸ ︷︷ ︸β
In the above equation, one can see that β ∝ ω, which is the condition for distortionless
transmission.
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Wave Velocity along a TX Line
For all the cases we considered till now (i.e., lossless, lossy, distortionless, etc.) β = ω√
LC. So,velocity of a wave along the TX line is given as
v =ω
β=
1√LC
Don’t you think this equation is very similar to c = 1√µε ?
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Outline
1 Free Space as a TX Line
2 TX Line Connected to a Load
3 Some Special Cases
4 Smith Chart
5 Impedance Matching
6 Z0&β
7 Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Transmission Lines  Basics
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Smith Chart  Problems
Transmission Line Theory ECE202, School of Electronics Engineering, VIT
Free Space as a TX Line TX Line Connected to a Load Some Special Cases Smith Chart Impedance Matching Z0&β Problems
Transmission Line  Lumped Element Model
Transmission Line Theory ECE202, School of Electronics Engineering, VIT