tl theory

1. Fundamental of Transmission Lines
Presenter: Dr. Aijaz Mehdi Zaidi
Assistant Professor
NIT Jalandhar
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2. • Importance of RF Circuit Design
• Why this course
• Lumped no more applicable!
• The solution? -> distributed!!!
Review-Lecture-1

3. • Why distributed?
• current and voltage vary spatially over the component size
• variation in E (or V) and H (or I) fields
Review-Lecture-1

4. Resistors at High Frequency (contd.)
Equivalent Circuit Representation of Thin Film Resistor
L model the leads
Ca models charge separation effects and Cb
models interlead capacitance
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Review-Lecture-1

5. Resistors at High Frequency (contd.)
What is the reason for following behavior of a 2000Ω thin-film resistor?
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Review-Lecture-1

6. Capacitors at High Frequency
Equivalent Circuit Representation of a Capacitor –> for a parallel-plate
A
d
0 r
d
C 
 A
   At high frequency, the dielectric become lossy
ie, there is conduction current through it
Then impedance of capacitor
becomes a parallel combination of C
and conductance Ge
Represents Ge
Accounts for the
losses in the leads
Lead Inductors
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Review-Lecture-1

7. Capacitors at High Frequency (contd.)
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Review-Lecture-1

8. Inductors at High Frequency (contd.)
Equivalent Circuit Representation of an Inductor -> coil type
Composite Effect of all the
distributed Rd
Composite Effect of all
the distributed Cd
Review-Lecture-1

9. Inductors at High Frequency (contd.)
Presence of resonance
Review-Lecture-1

10. Transmission Line
• It is a standard practice to use metallic conductors for transporting
electrical energy from one point of a circuit to another. These conductors
are called interconnects
• Thus cables, wires, conductive tracks on printed circuit boards (PCBs),
sockets, metallic tubes etc are all examples of interconnect
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11. Transmission Line (contd.)
• For short interconnects, the moment the switch is closed, a voltage
will appear across RL as current flows through it. The effect is instantaneous
• Voltage and current are due to electric charge movement along the
interconnect.
• The short interconnect system can be modelled by lumped RLC circuit
Model of
Interconnect
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12. Why Transmission Lines Theory?
• A transmission line is a two-conductor system that is used to
transmit a signal from one point to another point.
• Transmission line theory must be used instead of circuit theory for
any two-conductor system if -> size of the circuit dimensions are
comparable to the wavelength of the traveling wave.
• -> Variations in current and voltage across the circuit dimensions.
• Transmission-line theory is valid at any frequency, and for any type
of waveform (assuming an ideal transmission line).
• Circuit theory does not view two wires as a "transmission line": it
cannot predict effects such as signal propagation, distortion, etc.
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13. Transmission Lines (contd.)
Symbols:
z
Note: We use this schematic to represent a general transmission
line, no matter what the actual shape of the conductors.
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14. Transmission Lines (contd.)
z
Four fundamental parameters characterize any transmission line:
C = capacitance/length [F/m] - Capacitance/m between the two conductors
L = inductance/length [H/m] - Inductance/m due to stored magnetic energy
R = resistance/length [/m] - Resistance/m due to the conductors
G = conductance/length [S/m] - Conductance/m due to the filling material
between the conductors
It is usually assumed that these parameters are constant along the
transmission line (ie, the transmission line is uniform)
Note: most of the times these are “per unit length” parameters.
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15. Transmission Lines (contd.)
• Variations in current and voltage across the circuit dimensions -> KCL
and KVL can’t be directly applied -> This anomaly can be remedied if
the line is subdivided into elements of small (infinitesimal) length over
which the current and voltage do not vary.
z z z z z
Circuit Model:
Rz LZ
Gz Cz
Rz
z
LZ
Gz Cz
Rz LZ
Gz Cz
Rz LZ
Gz Cz
z0
lim  Infinitenumber of infinitesimalsections 15
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16. Transmission Lines (contd.)
Rz Lz
Gz Cz
v(z,t)

v(z  z,t)

i(z,t) i(z  z,t)
Lz
Cz
v(z,t)

v(z  z,t)

i(z  z,t)
i(z,t)
Lossy
Transmission Line
Circuit Model
Lossless Line Circuit
Model i.e, line with
negligible R and G
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17. Transmission Lines (contd.)
Rz Lz
Gz Cz
v(z,t)

v(z  z,t)

i(z,t) i(z  z,t)
Applying KVL:
v(z,t)  v(z  z,t)  Rzi(z,t)  Lz
i(z,t)
t

v(z,t)  v(z  z,t)
 Ri(z,t)  L
i(z,t)
z t

v(z,t)
 Ri(z,t)  L
i(z,t)
z t
For Δz →0
Describes
the voltage
along the
transmission
lines 17
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18. Transmission Lines (contd.)
KCL on this line segment gives:
i(z,t)  i(z  z,t)  Gzv(z  z,t)  Cz
v(z  z,t)
t
Simplification results in:
i(z,t)
 Gv(z,t) C
v(z,t)
z t
For Δz →0
Describes the current along the
transmission lines
These differential equations for current voltages were
derived by Oliver Heavyside. These equations are known
as Telegrapher’s Equations.
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19. Transmission Lines (contd.)
• Finally we can write:
dz
dz
d V(z)
 R  jLI(z)
d I(z)
 G  jCV(z)
These differential
equations can be
solved for the
phasors along the
transmission line
Differentiating with
respect to z gives
2
2
d V(z)
dz2
dz2
I(z)  0
 V(z)  0
d2
I(z)
 2
Here
  (R  jL)(G  jC)
Complex Propagation
Constant
Transmission Line
Wave Equations
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20. Transmission Lines (contd.)
• Complex Propagation Constant:   (R  jL)(G  jC)
    j
Attenuation
Constant
(nepers/m)
Phase
Constant
(radians/m)
• For lossless transmission line (i.e, transmission line where R and G are
negligible) - most common scenario in our transmission line based
circuit design:
  j  j LC
No Attenuation
Phase Constant is
also Propagation
Constant for a
Lossless Line
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21. 2
2
d V(z)
dz2
d2
I(z)
dz2
  V(z)  0
 2
I(z)  0
V(z)  Ae jz
 Bejz
Transmission Lines (contd.)
• For a lossless transmission line the second order differential equation
for phasors are:
   LC
A and B are
complex constants
• Similarly the current phasor for a lossless line can be described:
I(z)  
1 dV(z)
 
1 d

Ae jz
 Bejz


jL dz jL dz
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22. Transmission Lines (contd.)
 Bejz


 I(z) 
 
Ae jz
L
Gives the Definition of
Characteristic Impedance
0
Z 
L

L

L
C
  LC
I(z) 
A
e jz

B
ejz
Z0 Z0
Completely
Dependent on L and C
Characteristic Impedance for a Lossless Line is Real
Opposite Signs in these
Terms Gives a Clue about
Current Flow in Two
Different Directions
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23. 0
Z 
R  jL
G  jC
• Z0 is not an impedance in a conventional circuit sense
• Its definition is based on the incident and reflected voltage and current
waves
• As such, this definition has nothing in common with the total voltage and
current expressions used to define a conventional circuit impedance.
Characteristic Impedance (Z0)
• The characteristic impedance is defined as :
Z0 = (incoming voltage wave) / (incoming current wave)
= (outgoing voltage wave) / (outgoing current wave)
• For a generic transmission line:
The incoming and outgoing voltage and current
waves are position dependent → the ratio of voltage
and current waves are independent of position →
actually is a constant → an important characteristic
of a transmission line → called as Characteristic
Impedance
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24. Line Impedance (Z) – contd.
• Actually, line impedance is the ratio of total complex voltage (incoming
+ outgoing) wave to the total complex current voltage wave
I(z)
Z(z) 
V(z)
 
V (z) V (z)


V 
(z) V 
(z)
 Z 
 0

 Z0
In most of
the cases
• However, the line and characteristic impedance can be equal if
outgoing voltage wave equals ZERO!
• Say, if V-(z) = 0 then:
V 
(z) V 
(z)

 Z0
Z(z) 
V 
(z) V 
(z)
 Z 
 0
 24
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25. Line Impedance (Z) – contd.
Z0 is a transmission line
parameter, depending only
on the transmission line
values R, L, C and G
Whereas, Z(z) depends on the magnitude
and the phase of the two propagating
waves V+(z) and V-(z) → values that
depend not only on the transmission line,
but also on the two things attached to
either end of the transmission line
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26. Terminated Lossless Transmission Line
• High-Frequency electric circuits can be thought as a collection of
finite transmission line sections connected to various discrete active
and passive devices
• The simplest of them is a finite transmission line terminated into a
load impedance
ZL
z = 0
z = -l
Zin 0

Z0
Reflection
Coefficient at Load
Load
Impedance
Characteristic
Impedance
Input
Impedance
to the Line
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27. Terminated Lossless Transmission Line (contd.)
• The load is assumed at z = 0
• The voltage wave couples into the line at z = -l
ZL
z = -l
Zin 0
Z0
V(z)
V(z)  Ae jz Bejz
A.e jz
B.e jz
z = 0
Reflected Wave
Incident Wave
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28. Terminated Lossless Transmission Line (contd.)
Voltage Reflection Coefficient: It is the ratio of reflected to incident
voltage wave.
ZL
z = 0
z = -l
Zin
Z0
0
V(z) V 
e jz
V 
e jz
• It is common to use V+ for constant A and V−
voltage travelling wave. Therefore:
V 
(z)
0 
V 
(z)
At the load
location z =0
V 
e jz
V 
ejz
for constant B in the
V 
(z)
V 
(z)
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29. Terminated Lossless Transmission Line (contd.)
• As a consequence the voltage and current waves can be written as:
• At z = -l , the impedance is Zin whereas the impedance at z = 0 is:
0
L
I(0)
Z(0) 
V(0)
 Z 0
1
0
L 0
 Z
1 0  Z0
  
ZL
Z  Z
More useful representation as it involves
known circuit/system quantities
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30. Terminated Lossless Transmission Line (contd.)
0
L 0
  
ZL  Z0
Z  Z
For Open Circuit
Line: ZL →∞
0 1
That means the wave gets fully
reflected with the same polarity
For Short Circuit
L
Line: Z = 0
0  1
That means the wave gets fully
reflected with inverted amplitude
• When, ZL = Z0 → the transmission line is matched that results in no
reflection → Γ0 = 0 → the incident voltage wave is completely
absorbed by the load → a scenario that says a second transmission
line with the same characteristic impedance but infinite length is
attached at z = 0. 30
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