7. Transmission line analysis
Dr. Rakhesh Singh Kshetrimayum
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
Introduction
High
reactance
effect
Radiation
effect
Size
effect
Transmission line analysis
Telegrapher’s
equations
Lossy line
Smith chart
Ideal
Wave
equation
Terminated
Impedance
Ideal
Line
junction
Terminated
Distributed element concept
2
Lossless line
Admittance
λ/4 transformer
Line
impedance
Fig. 7.1 Transmission line
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
High reactance effect
Consider a 10-V ac source is connected to a 50 Ω load by a
small copper wire of 1 mm radius
Assume that the dc resistance of the wire is R=1m Ω and
inductance of L=0.1 µH
At 10 GHz, inductive reactance is jXL=jωL≈6283 Ω and
hence all the ac signal will die out in the wire itself
The load will not receive any signal
Hence we need special devices which will take these
signals from the source to the load
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
Radiation effect
An accelerating or decelerating charge radiates
electromagnetic energy
Besides the energy radiated from a current carrying
conductor depends on the frequency of current flowing
You might have observed this when you study Herz dipole
(an infinitesimally current carrying element)
2
1  I 0 dl sin θ  β 3
S avg = 
rˆ Watt / m 2

2  4πr  ωε
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Electromagnetic Field Theory by R. S. Kshetrimayum
(
)
5/20/2013
7.1 Introduction
Hence the radiation power loss is directly proportional to the
square of the frequency of the ac current flowing
So there will be high loss of power
We definitely cannot use open wires for transferring energy
or signal
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
7.1.1 Introduction
What is a transmission line?
A structure, which can guide electrical energy from one point to
another
Generally, a transmission line is
a two parallel conductor system
one end of which is connected to a source and
the other end is connected to a load
Examples:
coaxial cables
waveguides
microstrip lines
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
Fig. 7.2 (a) Transmission lines examples (b) General transmission
line structure
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
Two conductor systems could support transverse
electromagnetic (TEM) waves
Both electric and magnetic fields are perpendicular
(transverse) to the direction of the propagation
It is guided wave between these two conductors
Hence the radiation losses are minimized
What is microwave frequency?
300 MHz to 30 GHz (λ=1m to 10 cm)
Nowadays it is meant for frequency up to 300 GHz (λ=1cm)
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
Size effect
Size of commonly used lump elements like capacitor,
inductor and resistor are of the order of cm
Now this size is comparable to the microwave wavelength
Hence the phase {βl=(2πf/c)l} of the electrical signal might
vary along the length of the device
For instance, consider a parallel plate capacitor
We assume capacitor conductor plate is an equipotential
surface
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
But this is not true at microwave frequencies
Besides radiation also increases the problem
So we cannot use such capacitors at high frequencies
So we will see later that a section of a transmission line could
be used as an inductor or resistor or series/shunt RLC
resonators
If we increase the frequency of operation of a circuit,
Usually we require temporal analysis at low frequency
we can’t neglect ‘space’ in the circuit analysis due to size effect
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
7.1.2 Causal effect
What is causal effect?
EM wave requires a finite time to travel along an electrical
circuit
Since no EM wave can travel with infinite velocity
(What is the maximum speed?)
A finite time delay between the 'cause' and the ‘effect’
Also known as the causal effect in physics
When is this effect important?
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
If the time period of the EM wave or signal
(T=1/f) >> the transit time (tr),
we may ignore this effect
1
l
v
T >> tr ⇒ >> ⇒ >> l ⇒ λ >> l
f
v
f
Causal effect becomes important
when the length of the line (l) becomes comparable to the wavelength
(λ)
As the frequency increases,
the wavelength reduces, and
the Causal effect becomes more evident
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
7.1.3 Distributed vs lumped elements
To overcome the effect of transit time or causality or size
effect (more appropriate to use),
one can chop off the transmission line into small sections
such that for each section, this causality effect is minuscule
At high frequencies,
the circuit elements cannot be defined for the whole
transmission line
instead it has to be defined for a unit length of the line
The circuit elements are not located at a point of the line
but are distributed all along the length
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.1 Introduction
Analysis of a transmission lines must be carried out using
the concept of distributed elements not as
lumped elements
as we used to do from our previous circuit analysis at the low
frequencies
But, we can still employ lump element analysis of
transmission lines by
chopping off small sections of the line
so that the Causal effect is negligible in the chopped off sections
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
7.2.1 Lumped element circuit model
Per unit length parameters:
L=Series inductance per unit length
C=Shunt capacitance per unit length
R=Series resistance per unit length
G= Shunt conductance per unit length
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
∆z
Fig. 7.3 (a) Sub-section of
length ∆z of a general
transmission line and its
(b) lumped element
equivalent circuit
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R∆z
L∆z
G∆z
V (z, t ), I( z, t )
Electromagnetic Field Theory by R. S. Kshetrimayum
C∆z V (z + ∆z, t ), I(z + ∆z, t )
5/20/2013
7.2 Telegrapher’s equations
L represents the self-inductance of the two conductors
(magnetic energy storage)
C is due to the close proximity of two conductors (electric
energy storage)
R is due to the finite conductivity of the two conductors
(power loss due to finite conductivity of metallic conductors)
G is due to dielectric loss in the material between the
conductors (power dissipation in lossy dielectric)
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
7.2.2 Telegrapher’s equations
Let the voltage at the input be V and current at the input be I
Due to voltage drop in the series arm,
the output voltage will be different from the input voltage, say
V+∆V
Due to current through the capacitance and the conductance,
the output current will be different from the input current, say
I+∆I
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
Applying Kirchoff’s voltage law (KVL) and Kirchoff’s current
law (KCL)
δi(z, t)
v(z, t) − R∆zi(z, t) − L∆z
− v(z + ∆z, t) = 0
δt
δv(z + ∆z, t)
i(z, t) − G∆zv(z + ∆z, t) − C∆z
− i(z + ∆z, t) = 0
δt
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
Dividing the above two equations by Δz and taking the limit
Δz 0 (What is its implications?)
δv(z, t)
δi(z, t)
= − Ri(z, t) − L
δz
δt
δi(z, t)
δv(z, t)
= −Gv(z, t) − C
δz
δt
Telegrapher’s Equations
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
7.2.3 Wave propagation
For time-harmonic signals, telegrapher’s equation reduces to
dV(z)
= −(R + jωL)I(z)
dz
dI ( z )
= -(G + jωC )V ( z )
dz
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
It is similar to Maxwell’s curl equations, hence, we can get
wave equations
d 2 V(z)
dz 2
− γ 2 V(z) = 0
d 2 I(z)
dz 2
− γ 2 I(z) = 0
γ = α + j β = ( R + jω L)(G + jωC )
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
Transmission line analysis
Traveling wave solutions for the above two equations are
V(z) = V0 + e − γz + V0 − e γz
I ( z ) = I 0+ e − γ z + I 0− eγ z
Point to be noted: current or voltage is wave which is a function of both space
and time unlike the low frequency counter-parts (where is the time dependence?)
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
Physical interpretations
Wave phase has two components:
time phase (ωt) and
space phase (βz)
Since βz is the phase of the wave as function of z,
β represents phase change per unit length of the transmission
line for a traveling wave
phase constant (unit is radians per meter)
Re{V + e − αz e jωt − jβz } = Re{ V + e jφ e − αz e jωt − jβz } = V + e − αz cos(ωt − βz + φ)
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.2 Telegrapher’s equations
For a positive α, the amplitude exponentially decreases as a
function of z
V + e −α z
α represents attenuation of the wave on the transmission line
attenuation constant of the line (unit is Nepers per meter,
1 Neper= 8.68dB)
I(z) =
Z0 =
25
γ
 V0 + e − γz − V0 − e γz 
R + jω L
R + j ωL
R + j ωL
=
γ
G + j ωC
Electromagnetic Field Theory by R. S. Kshetrimayum
V0 +
V0 −
= Z0 = − −
+
I0
I0
5/20/2013
7.2 Telegrapher’s equations
The characteristic impedance Z0 of a transmission line is
defined as the ratio of positively traveling voltage wave to
current wave at any point on the line
Now for a wave the distance over which the phase changes by
2H is called the wavelength 'λ’
phase change per unit length β=2H/λ
2πf ω
=
v p = λf =
β
β
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.1 Ideal lossless line
L∆z
V(z, t ), I(z, t )
C∆z V (z + ∆z, t ), I(z + ∆z, t )
Fig. 7.4 Lumped element equivalent circuit of a sub-section of
length ∆z of a lossless transmission line (R=G=0)
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
γ = α + jβ = jω LC
Z0 =
L
C
+
28
α=0
V(z) = V0 + e − jβz + V0 − e jβz
−
V0 − jβz V0 jβz
I(z) =
e
−
e
Z0
Z0
vp =
β = ω LC
λ=
2π
2π
=
β ω LC
ω
ω
1
=
=
β ω LC
LC
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.2 Terminated lossless lines
Z0 , β
Fig. 7.4 (b) A lossless transmission line terminated with
load impedance ZL
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.3 Reflection coefficient
At the load, z=0,
V (0 ) Vo+ + Vo−
ZL =
=
Z0
+
−
I(0 ) Vo − Vo
(
V (z ) = V0+ e − jβz + Γe jβz
V0−
V0+
)
=Γ=
ZL − Z0
Z L + Z0
V0+ − jβz
I(z ) =
e
− Γe jβz
Z0
(
)
Z L − Z0
V0− e − jβl
− 2 jβ l Γ(0 ) =
z = −l Γ(l ) = + + jβl = Γ(0)e
ZL + Z0
V0 e
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.4 Power flow and return loss
Time average power flow along the line at the point z,
S avg
S avg
31
+ 2
0
1 
1V
*
2

= Re V ( z ) I ( z ) =
Re 1 − Γ*e −2 β z + Γ*e −2 β z − Γ
 2 Z0
2 
+ 2
0
1V
=
2 Z0
{
1− Γ
2
{
}
}
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
When the load is mismatched,
not all of the available power from the generator is delivered
to the load,
this “loss” is called Return loss (RL) and is defined in dB as
7.3.5 Standing wave ratio (SWR)
RL = −20 log Γ
V ( z ) = V0+ 1 + Γe 2 j β z
e 2 jβz = 1 Vmax = V0+ 1 + Γ
+
e 2 jβz = −1 Vmin = V0 1 − Γ
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Electromagnetic Field Theory by R. S. Kshetrimayum
VSWR =
Vmax 1 + Γ
=
Vmin 1 − Γ
5/20/2013
7.3 Lossless line
What is BW?
For acceptable value of VSWR = 2 within the operating
frequency region of a device also known as bandwidth (BW)
Γ=
VSWR − 1 2 − 1 1
1
=
= ; RL = −20 log  = 9.54
VSWR + 1 2 + 1 3
3
Return loss (RL) should be higher than 9.54,
which is approximately 10 dB
RL ≥10 dB has become an acceptable definition for BW
of many devices
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
I max
ISWR =
I min
Vmax I max
2
PSWR =
= PSWR × ISWR = (VSWR )
Vmin I min
2
PL Pi − Pr
4VSWR
2
 VSWR − 1 
=
= 1− (Γ) = 1− 
 =
2
Pi
Pi
 VSWR + 1  (VSWR + 1)
For VSWR=2,
only 89% of the incident power reaches the load
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.3 Lossless line
l = −z
V ( −l ) = V0+ 1 + Γe −2 j β l
 λ
−2 j β  l + 
λ
λ

V  −l −  = V0+ 1 + Γe  2  = V0+ 1 + Γe −2 j β l = V ( −l )
l+
2

2
Point to be noted:
shortest distance between two successive maxima (or
minima) is not λ but λ/2,
it is very important to realize this
since in your experiment on Frequency and Wavelength
measurements,
this is a major mistake most of you make
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
λ
l+
4

λ
−2 j β  l + 
λ

+
V  −l −  = V0 1 + Γe  4  = V0+ 1 − Γe −2 j β l
4

the distance between adjacent maximum and minimum is
λ/4
7.3.6 Transmission line impedance equation
A certain value of load impedance at the end of a particular
transmission line is
transformed into another value of impedance at the input of the
line
impedance transformer
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
Transmission line impedance equation
Z in =
V ( −l )
I ( −l )
= Z0
V0+  e jβ l + Γe − j β l 
V0+ e j β l − Γe− j β l 
 jβ l  Z L − Z 0  − jβ l 
e + 
e 
Z
+
Z
0 
 L

= Z0 
 jβ l  Z L − Z 0  − jβ l 
e − 
e 
Z
+
Z
0 
 L


( Z L + Z 0 ) e jβ l + ( Z L − Z 0 ) e− jβ l 
= Z0
= Z0
− jβ l
jβ l
( Z L + Z 0 ) e − ( Z L − Z 0 ) e 
 Z L ( e jβ l + e− jβ l ) + Z 0 ( e jβ l − e− jβ l )


 Z 0 ( e jβ l + e− jβ l ) + Z L ( e jβ l − e− jβ l )


 Z L ( cos β l ) + jZ 0 ( sin β l ) 
Z + jZ 0 tan( β l)
= Z0
= Z0 L
Z 0 + jZ L tan( β l)
 Z 0 ( cos β l ) + jZ L ( sin β l ) 
37
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
Fig. 7.5 Transmission line impedance
Z in
38
Z0 , β
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.7 Quarter-wave transformer
λ
λ βl = β λ + nβ λ = 2π λ + n 2π λ = π + nπ
l= +n
4
2 λ 4
λ 2 2
4
2
π

Z0 =
⇒ tan ( βl ) = tan  + nπ  = ∞
2

ZL Zin
ZL + jZ0 tan(βl)
jZ0 tan(βl) Z0 2
⇒ Zin = Z0
= Z0
=
Z0 + jZL tan(βl)
jZL tan(βl) ZL
How to do impedance matching for a complex load using quarter-wave
transformer?
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.8 Special cases of lossless terminated lines
Terminated in a short circuit
ZL + jZ0 tan(β l)
sc
Zin = Z0
= jZ0 tan(β l)
Z0 + jZL tan(β l)
Terminated in open circuit
ZL + jZ0 tan(β l)
Z = Z0
= − jZ0 cot(β l)
Z0 + jZL tan(β l)
oc
in
Terminated with matched load
Γ=
40
Z0 − Z0
=0
Z0 + Z0
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
Another important observation is that
if we measure
open and
short circuit input impedances
of a lossless transmission line and
multiply those two values and
take the square root
what we have is the characteristic impedance of the line
(one of the methods for finding the characteristic impedance of
a given line in laboratory)
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
7.3.9 Reflection and transmission at the transmission line
junction
Γ
τ
Fig. 7.7 Junction of two transmission line with different
characteristic impedance
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
For z<0,
characteristic impedance Z0;
z>0,
characteristic impedance Z1 and
the junction of the two transmission lines is at z=0
At the junction,
looking from z<0 towards the right,
it sees an infinite transmission line of characteristic impedance Z1 and
hence it is equivalent to ZL=Z1 for the transmission line z<0
Assuming
ζ is the transmission coefficient and
IL is insertion loss in dB
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Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.3 Lossless line
Γ=
Z1 − Z 0
Z1 + Z 0
(
z < 0 V(z ) = V0+ e − jβz + Γe jβz
)
z>0
V(z) = V0 + τe − jβz
z=0
Z1 − Z 0
2Z 1
τ = 1+ Γ = 1+
=
Z1 + Z 0 Z1 + Z 0
IL = −20 log τ
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
One type of metal loss is I2R loss
In transmission lines,
the resistance of the conductors is never equal to zero
except for superconductors
Whenever current flows through one of these conductors,
some energy is dissipated in the form of heat
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Another type of loss is due to skin effect
Current in the center of the wire becomes smaller and
most of the electron flows on the wire surface
When the frequency applied is in the GHz range,
the electron movement in the center is so small
that the center of the wire could be removed without any
noticeable effect on the current
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Note that the effective cross-sectional area
decreases as the frequency increases
Since resistance is inversely proportional to
the cross-sectional area (R=ρl/A),
the resistance will increase as the frequency is increased
Also, since power loss increases
as resistance increases,
power losses increase with
an increase in frequency
because of the skin effect
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Dielectric losses result from
the heating effect on the dielectric material between the
conductors
Power from the source is
used in heating the dielectric
The heat produced is dissipated into the surrounding medium
When there is no potential difference between two
conductors,
the atoms in the dielectric material between them are normal
and
the orbits of the electrons are circular
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
When there is a potential difference between two
conductors,
the orbits of the electrons change
The excessive negative charge on one conductor
repels electrons on the dielectric toward the positive conductor
and
thus distorts the orbits of the electrons
A change in the path of electrons requires more energy,
introducing a power loss
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Induction losses occur
when the electromagnetic field about a conductor cuts
through any nearby metallic object and
a current is induced in that object
As a result,
power is dissipated in the object and
is lost
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Radiation losses occur
because some magnetic lines of force about a conductor
do not return to the conductor when the cycle alternates
These lines of force are
projected into space as radiation, and
these results in power losses
That is,
power is supplied by the source,
but is not available to the load
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
7.4.1 Ideal lossy line characteristics
γ = α + j β = ( R + jω L)(G + jωC ) = ( jωL)( jωC)  ( R + 1)( G + 1) 
jω L
j ωC


 R
G 
RG 
= jω LC 1 − 
+
j − 2 
 ωL ωC  ω LC 
Low loss case
R << ωL, G << ωC
∴α ≅
52

1  R
G 
RG << ω LC γ ≅ jω LC 1 − 2 j  ω L + ωC  



2

1
C
L  1 R
R
+
G
=
+
GZ



0
2 
L
C  2  Z 0

Electromagnetic Field Theory by R. S. Kshetrimayum
β ≅ ω LC
Z0 =
R + jω L
L
≅
G + jωC
C
5/20/2013
7.4 Lossy lines
7.4.2 Terminated lossy lines
Z0 , γ
(
V (− l ) = V0+ e + γl + Γe −γl
(
V0+ e +γl − Γe −γl
I (− l ) =
Z0
Γ(l ) =
53
V0− e − γl
V0+ e + γl
)
)
Fig. 7.8 (a) A lossy transmission line
terminated with load impedance ZL
= Γ(0 )e − 2 γl = Γ(0)e − 2αl e − 2 jβl = Γe − 2αl e − 2 jβl
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.4 Lossy lines
Zin =
V0+ e γl + Γe −γl 
V(−l)
= Z0 + γl
I(−l)
V0  e − Γe − γl 
 γl  ZL − Z0  − γl 
e + 
e 
 Z L + Z0 


= Z0
 γl  ZL − Z0  −γl 
e − 
e 
 Z L + Z0 


( ZL + Z0 ) e γl + ( ZL − Z0 ) e − γl 
= Z0
= Z0
γl
− γl
( ZL + Z0 ) e − ( ZL − Z0 ) e 
 ZL ( e γl + e − γl ) + Z0 ( e γl − e − γl ) 


 Z0 ( e γl + e −γl ) + ZL ( e γl − e − γl ) 


 ZL ( cosh γl ) + Z0 ( sinh γl ) 
ZL + Z0 tanh γl
= Z0
= Z0
Z0 + ZL tanh γl
 Z0 ( cosh γl ) + ZL ( sinh γl ) 
54
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Pin =
1
Re {V(−l)I∗ (−l)}
2
1  + +γl
= Re V0 e + Γe −γl
2 

(
(
)
{e
2αl
− Γ e − 2αl
(
)]
 V0+ e +γl − Γe −γl

Z0

)



*



2
+
1 V0
2
=
Re eγ l +γ *l − Γ*eγ l −γ *l + Γe −γ l +γ *l − Γ e −γ l −γ *l
2 Z0
{
2
+
V
1 0
2
=
Re e 2α l − Γ*e2 j β l + Γe −2 j β l − Γ e −2α l
2 Z0
{
+ 2
0
1V
2
PL =
1− Γ
2 Z0
(
)
Ploss
}
+ 2
0
1V
= Pin − PL =
2 Z0
}
+
V
1 0
=
2 Z0
[(e
2αl
)
2
2
2
}
− 1 + Γ − e −2αl + 1
What happens to Ploss when α increases?
55
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
7.4.3 Introduction to electromagnetic resonators:
Z in
Z in
λ
λ
4
2
Z0 , γ = α + jβ
ZL = ∞
Z in
Z0 , γ = α + jβ
ZL = 0
Z in
Fig. 7.8 (b) Series RLC resonant circuit (c) Tank or shunt RLC
resonant circuit (d) O.C. terminated transmission line of length λ/4
and (e) S.C. terminated transmission line of length λ/2
56
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Microwave/electromagnetic resonators are used in many
applications:
filters,
oscillators,
frequency meters,
tuned amplifiers, etc.
Its operations are very similar to the
series and
parallel RLC resonant circuits
57
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
We will review the
series and
parallel RLC ciruits and
discuss the implementation of the microwave resonators
using distributive elements such as
microstrip line,
rectangular and
circular waveguides, etc.
Series RLC resonant circuits
Consider the series RLC resonator
The input impedance Zin is given by
58
Electromagnetic Field Theory by R. S. Kshetrimayum
Z in = R + j wL +
1
j wC
5/20/2013
7.4 Lossy lines
The average complex power delivered to the resonator is
The average power dissipated by the resistor is
Ploss =
59
1 2
I R
2
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
The time-averaged energy stored in the inductor is (recall the
energy stored in the inductor)
Wm =
1
LI
4
2
Similarly, the time-averaged energy stored in the capacitor is
We =
60
1
2
C Vc
4
2
2
1C I
1 I
=
=
4 w2C 2
4 w2C
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.4 Lossy lines
The input impedance can then be expressed as follows:
\ Pin = Ploss + 2 j w (W m - W e )
Z in =
Ploss + 2 j w (W m - W e )
Pin
=
2
2
R
R
At resonance,
the average stored magnetic and electric energies are equal,
therefore, we have
Wm = We
61
Electromagnetic Field Theory by R. S. Kshetrimayum
Z in
Ploss
=
= R
1 2
I
2
5/20/2013
7.4 Lossy lines
Hence, the resonance frequency is defined as
The quality factor is defined as the product of
the angular frequency and
the ratio of the
average energy stored to
energy loss per second
62
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Q is a measure of loss of a resonant circuit,
lower loss implies higher Q and
high Q implies narrower bandwidth
As R increases,
power loss increases and
quality factor decreases
Let us see what the approximate Zin near resonance
The input impedance can be rewritten in the following form:
63
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Near by the resonance
The above form is useful for
finding equivalent circuit
near the resonance,
for example,
we can find out
the resistance at resonance and
so as L
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Half power fractional bandwidth
When the real power delivered to the circuit is half that of
the resonance, occurs when
Z in =
65
2R
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.4 Lossy lines
Shunt RLC Resonant Circuits
Now let us turn our attention to the parallel RLC resonator
The input impedance is equal to
66
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
The average complex power delivered to the resonator is
*
1 *
1
V
Pin = V I = V
2
2 Z in *
The average power dissipated by the resistor is
Ploss
67
1V
=
2 R
2
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.4 Lossy lines
The time-averaged energy stored in the inductor is (recall the
energy stored in the inductor)
Wm
1
= L IL
4
2
2
2
1 V
1V
= L 2 2 =
2 wL
2 w 2L
Similarly, the time-averaged energy stored in the capacitor is
1
Wc = C V
4
68
Electromagnetic Field Theory by R. S. Kshetrimayum
2
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7.4 Lossy lines
The input impedance can then be expressed as follows:
Z in
At resonance,
Ploss + 2 j w (W m - W c )
2Pin
=
2 =
1 2
I
I
2
the average stored magnetic and electric energies are equal,
therefore,
we have (same results as in series RLC )
\ Pin = Ploss + 2 j w (W m - W c )
1
w0 =
LC
69
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
The quality factor, however, is different
Contrary to series RLC,
the Q of the parallel RLC increases
as R increases
70
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Similar to series RLC,
we can derive an approximate expression for parallel RLC
near resonance
71
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
As in the series case,
the half-power bandwidth is given by
Z in
72
2
R2
=
2
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.4 Lossy lines
We discuss the use of transmission lines to realize the RLC
resonator
For a resonator, we are interested in Q and
therefore, we need to consider lossy transmission lines
Short-circuited λ/2 line
Note that
tanh(A+B)
=(tanh A + tanh B)/(1+ tanh A tanh B)
73
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
[e jx − e − jx ] / ( 2 j)
tan( x ) =
[e jx + e − jx ] / 2
tanh( x ) =
+
[e x − e − x ] / 2
[e x + e − x ] / 2
Consider the transmission line equation
For a short-circuited line
74
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Our goal here is to
compare the above equation
with input impedance of
Series or
shunt RLC resonant circuit near resonance
so that we can find out the corresponding R, L and C
For a length l=λ/2 of the transmission line,
assuming a TEM line so that
75
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
β = ω µε = ω / v p
l = λ / 2 = πv p / ω o
ωl ω o l ∆ωl
∆ωπ
βl =
+
= π+
=
vp
vp
vp
ωo
For low-loss transmission lines, αl is small, hence
tan βl = tan( π +
76
∆ωπ
∆ωπ
∆ωπ
) = tan(
)≈
ωo
ωo
ωo
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.4 Lossy lines
Note that the loss is usually very small and
therefore, the input impedance can be rewritten as:
77
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
This equation can be compared favorably
with the input impedance of a series RLC resonant circuit
near the resonance
It behaves like a series RLC resonator with
78
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
As α increases,
Q decreases
which is according to our expectation
Open-Circuited λ/4 Line
For a lossy line of length l
with propagation constant γ and
characteristic impedance Z0,
we can find the input impedance for a load of ZL as follows:
79
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
For o.c.,
For
l = λ / 4 = πv p / ( 2ω o )
ω o l ∆ωl π ∆ωπ
ωl
βl =
=
+
= +
2v p 2v p 2v p 2 2ω o
80
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
Knowing that tan d = d when d is small
The input impedance can be written as,
81
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
This equation can be compared favorably
with the input impedance of a series RLC resonant circuit near
the resonance
It behaves like a series RLC resonator with
82
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.4 Lossy lines
As α increases,
Q decreases
which is according to our expectation
We can extend this analysis for a
s.c. λ/4 lines,
o.c. λ/2 lines
and so on
83
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
7.5.1 Impedance Smith chart
Smith chart is
basically a graphical representation of
transmission line impedance transformation formula:
ZL + jZ0 tan(βl)
Zin = Z0
Z0 + jZL tan(βl)
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
If we represent this in x-y coordinates with
x as real part and y as imaginary part of ZL and Zin
then it becomes a semi-infinite plane,
not practical
We know that the
modulus of reflection coefficient (|Γ|) is always less than or
equal to 1
And there is one to one correspondence between Γ and Zin
85
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.5 Smith chart
Γ (l) =
Zin (l) − Z0
Zin (l) + Z0
Zin =
Zin 1 + Γ(l)
=
Z 0 1 − Γ (l )
we will draw
normalized constant resistance and
constant reactance contours
in the reflection coefficient plane which is a circle of Γ ≤ 1
A movement of d distance along the transmission line
is equivalent to e −2 jβd change in the reflection plane
86
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
Distance in movement in terms of wavelength is given in the
circumference of the circle
It could be either
towards load (WTL) or
source (WTG)
At first glance,
Smith chart looks intimidating with so many contours of
constant resistance and
reactance
87
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
Fig. 7.11 Smith
chart
88
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
Smith chart as a polar plot of Γ
(o.c. open circuit and
s.c. short circuit)
It can be simply
interpreted as a polar
plot of
Γ
θ
Γ = Γ e jθ , −1800 ≤ θ ≤ 1800
89
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
The real utility of Smith chart lies
in the fact that we can read the corresponding normalized
impedance value of Γ
from the constant reactance and resistance contours
Zin =
90
Zin
1 + Γ 1 + Γ r + jΓ i
= R in + jX in =
=
Z0
1 − Γ 1 − Γ r − jΓ i
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.5 Smith chart
constant resistance circles
2
2

R in 
1
2


 Γr −
 + ( Γi ) = 

R
+
1
R
+
1
in



in

constant reactance circles

(Γr − 1)2 +  Γi −

91
1
X in
2

 1 

 = 

 X in 
Electromagnetic Field Theory by R. S. Kshetrimayum
2
5/20/2013
7.5 Smith chart
Constant resistance circles
(WTG Wavelength towards generator and
WTL Wavelength towards load)
Rin = 0.5
WTG
-1
Rin = 1
+1
Rin = 2
WTL
92
Electromagnetic Field Theory by R. S. Kshetrimayum
5/20/2013
7.5 Smith chart
Constant reactance circles of an impedance smith chart
X in = 1
X in = 2
X in = 0.5
X in = −0.5
X in = −2
X in = −1
93
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
In many applications,
transmission line and impedances are connected in parallel
(shunt),
then, the admittance analysis is more convenient than the
impedance analysis
1
1
−
ZL − Z0 YL Y0 Y0 − YL 1 − YL
YL − 1
Γ=
=
=
=
=−
1
1
Z L + Z0
Y
+
Y
1 + YL
YL + 1
0
L
+
YL Y0
94
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
Rules for conversion of impedance (say ZL at N) to
admittance (say YL at N )
Γ
95
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
The admittance smith chart is therefore obtained by
rotating the impedance Smith chart by π and
replacing r by g and x by b
Since it is just a matter of rotation,
there is no need to have separate Smith charts for impedance
and admittance
Although r and x can be interchanged with g and b
respectively and
a point (r,x) and (g,b) will have the same spatial location on
the Smith chart for r=g and x=b,
96
Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
But, the physical interpretation corresponding to the two
will not be identical
Upper half of the impedance Smith chart with +jx
represent inductive loads
whereas +jb represents
capacitive load on the admittance Smith chart
Point B on impedance Smith chart
represents s.c.
whereas point B on admittance Smith chart
represent o.c.
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
Interchange on location of o.c./s.c. and
location of VSWR on an impedance Smith chart
Inductive/Capacitive
B
D
C
A
Capacitive/Inductive
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Electromagnetic Field Theory by R. S. Kshetrimayum
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7.5 Smith chart
Point A on impedance Smith chart which represents o.c.
whereas point A on admittance Smith chart which represents
s.c.
Note that the distance between o.c. and s.c. is λ/4
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Electromagnetic Field Theory by R. S. Kshetrimayum
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Transmission line analysis
7.6 Summary
Introduction
Smith chart
Telegrapher’s
equations
Lossy line
Lossless line
Impedance
Transit time
effect
Distributed
element
concept
δv(z, t)
δi(z, t)
= − Ri(z, t) − L
δz
δt
δi(z, t)
δv(z, t)
= −Gv(z, t) − C
δz
δt
Ideal
Z0 =
V(z) = V0 e
d V(z)
dz
2
d 2 I(z)
dz
2
+
I(z) =
+ V0 e
jβ z
− γ V(z) = 0
Line impedance
∴α ≅
R + jω L
L
≅
G + jωC
C

1
C
L  1 R
+G
 R
 =  + GZ 0 
2
L
C  2  Z0

Terminated
V0 − jβz V0 jβz
e
−
e
Z0
Z0
Zin = Z0
Terminated
− γ 2 I(z) = 0
V0 +
V0 −
=
Z
=
−
0
I0 +
I0 −
Z0 =
−
2
γ = α + j β = ( R + jω L)(G + jωC )
100
−
Admittance
β ≅ ω LC
Z0 = ZL Zin
+ − jβ z
2
λ/4
transformer
L
C
γ = α + jβ = jω LC
Wave equation
Ideal
(
)
V+
I(z ) = 0 (e − jβz − Γe jβz )
Z
V(z ) = V0+ e − jβz + Γe jβz
0
1+ Γ
V
VSWR = max =
Vmin 1 − Γ
Electromagnetic Field Theory by R. S. Kshetrimayum
ZL + jZ0 tan(βl)
Z0 + jZL tan(βl)
Line junction
τ = 1+ Γ = 1+
Z1 − Z 0
2Z 1
=
Z1 + Z 0 Z1 + Z 0
(
(e
V (− l ) = V0+ e + γl + Γe −γl
− Γ e − γl
I (− l ) =
Z0
Z + Z tanh (γl )
Z in = Z 0 L 0
Z 0+ Z L tanh (γl )
V0+
+ γl
)
)
Fig. 7.1 Transmission line in a nutshell
5/20/2013