Download DISP-2003: Introduction to Digital Signal Processing

TELECOMMUNICATIONS
Dr. Hugh Blanton
ENTC 4307/ENTC 5307
Properties of transmission lines
Inductance (L) and Capacitance (C ) per unit length
L
Z
C
•
•
•
•
Characteristic Impedance
Propagation coefficient
Phase Velocity
Effective dielectric constant
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
v p  LC
   LC
 eff
c2

LC
2
RF
• When a capacitor is created
by two parallel conductors,
the dimensions of the
conductors compared to the
actual wavelength (lG)
affects the RF performance
of the component.
• For example:
• If l << lG ,the
capacitance of the
parallel plates may be
treated as a single
(“lumped”) capacitance C
between points x1 and x2.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
3
• If l > 0.05lG, the
capacitance of the
parallel plates must be
treated in “distributed”
form as C1, C2, ... Cn,
including the effects of
the incremental
inductances L1, L2, ...,
Ln-1,etc.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
4
• The physical dimensions of conductors and
components, relative to the effective signal
wavelength, determines the method
required for accurate modeling.
• When conductors are realized on FR-4 type PCboards, the effective wavelength at 1 GHz is
about 10 cm (~ 4”).
• Five percent of that length is 5 mm (~ 200 mils);
therefore if the conductors exceed this length, they
should be analyzed in distributed form at frequencies
above 1 GHz.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
5
• The 5% “border” is just an
approximation, not an absolute rule.
• It is generally used as an upper limit to
which the tangent of an angle changes
in a near linear fashion
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
6
Transmission lines
• At high frequencies
wires become
transmission lines
• Coaxial
• Microstrip
• Coplanar
• Input and Output needs
to be matched.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
7
Types of Transmission Lines
• RF transmission lines
generally consist of two
conductors, one of which
may be a ground plane or
a shield.
• The most commonly used
forms are:
•
•
•
•
twin-leads,
coaxial,
stripline, and
microstrip
Twin leads
Strip line
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
Coaxial
Microstrip line
8
Coaxial
• Coax is the most common form of a
transmission line.
• Note that:
• Stripline is essentially square coax and
• Microstrip is open top coax.
• Most of the magnetic field terminated in the
ground strip.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
9
• Transmission lines may be defined two
ways:
• by physical dimensions (conductor size and
spacing) or
• by electrical parameters (characteristic
impedance and electrical length).
• At higher microwave frequencies single
conductor transmission lines, such as
waveguides may also be used
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
10
• If the dielectric is non-homogeneous, r,
is replaced with the dielectric constant,
eff, which is an average of the dielectric
layers.
lo
l  lG 
 eff
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
11
Characteristic Impedance
• Characteristic
impedance (Zo) of a
uniform, lossy
transmission line is a
complex number;
• it is defined by the ratio
of the series impedance
and shunt admittance of
an incremental line
segment.
ZS
R  j L
Zo 

YP
G  j C
Dr. Blanton - ENTC 4307 -
Z S  R  j L
YP  G  j C
Transmission Lines (cont.)
12
• R and G represent dissipative losses while L
and C are the incremental inductance and
capacitance in the equivalent series and parallel
circuits.
• Characteristic impedance is given by:
Zo 
ZS

YP
R  j L
vT

G  j C
iT
• If the line is lossless (R = 0, G = 0) the impedance
definition is simplified to a real quantity:
Zo 
Dr. Blanton - ENTC 4307 -
L
C
Transmission Lines (cont.)
13
• Characteristic impedance is what the
signal “sees” while traveling through the
transmission line.
• Electrically, it is the ratio of the instantaneous
voltage and current.
• a quantity that is constant throughout a
homogeneous line.
• If high-impedance lines the incremental
inductance is the dominant term of the
impedance expression.
• In low-impedance lines the capacitance term is
relatively large, while inductance is low.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
14
• Thus in high-impedance lines the
incremental inductance is the
dominant term of the impedance
expression.
• In low-impedance lines the capacitance
term is the dominant term in the
impedance expression.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
15
• If the transmission line is lossless (R = 0,
G = 0) and the impedance definition is
simplified to a real quantity:
L
Zo 
C
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
16
Example
• A uniform transmission line has the following incremental lumped
equivalent circuit parameters:
•
•
•
•
R = 0.2W/m
G = 2  10-5 S/m
L = 2.51  10-7 H/m
C = 10  10-12 F/m
R
L
G
C
• Find the characteristic impedance of the line at 1000 Hz and 1
GHz.
• Comment about the nature of the impedance at different
frequencies.
• That is, at 1 KHz, is the circuit a transmission line?
• Is the circuit a transmission line at 1 GHz?
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
17
Electrical Length of Transmission Lines
• The term electrical length refers to the
ratio of the physical length (l) of the
transmission line to the wavelength (lG)
in the applicable dielectric.
360 f GHz  r  cm
E
360   
lG
30cm



 12 f GHz cm  r (degrees)
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
18
Example
• If a 15 cm long coaxial line
is filled with dielectric of r
= 4, what is E at 2 GHz?
360 (2) 4 (15)
E
 720
30
 2 wavelengt hs
lG 
30
f GHz

30

 7.5 cm;
 r 22
15 cm
E
360 
360  720
lG
7.5 cm

• We could also compute the
effective wavelength first
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
19
Physical Forms
• A transmission line may
have various physical
forms:
• The electrical schematic
• The real physical circuit
equivalent in
• coaxial form or
• microstrip form
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
20
Characteristics
• If an ideal transmission
line of characteristic
impedance Zo is terminated
with a complex impedance
ZL, the new input
impedance is
Let Zo = 50 W
⅛l
¼l
Z IN  Z o
Z L  jZo tan 
Z o  jZ L tan 

ZL
ZIN
0
5
5
45
0
j Zo
90
∞
0
90
XL
XC
90
5
500
180
5
5
Impedance Inverter
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
21
Lossy Transmission Lines
• For lossy transmission lines the input
impedance is a more complicated
function:
Z IN
Z L  jZo tanh g 
 Zo
Z o  jZ L tanh g 
• where
• g is the propagation constant (a  j
• l is the physical length of the transmission line.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
22
• If ZL = Zo, the input impedance of the
transmission line is always equal to ZL
and is not a function of the line length.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
23
• In a uniform transmission line the current
flow is determined by the ratio of the
instantaneous voltage and characteristic
impedance.
• Load current depends on the voltage and load
impedance.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
24

Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
VR Z L  Z o

VF Z L  Z o
25
• When a uniform transmission line is
terminated with a load impedance other
than its characteristic impedance,
reflected waves are created.
• The ratio of the reflected and forward
voltages is called the reflection coefficient
and is denoted by .
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
26
RF Parameters
• As frequencies reach 100 MHz, the
voltages and currents are difficult to
measure.
• A more practical set of parameters can be
defined in terms of traveling waves.
• Four such parameters are:
•
•
•
•
Reflection Coefficient
Return Loss
Voltage Standing Wave Ratio
Mismatch Loss
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
27
Reflection Coefficient
• The Reflection Coefficient  shows
what fraction of an applied signal is
reflected when a Zo source drives a load
of ZL.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
28
Return Loss
• The Return Loss RL shows the level of
reflected wave referenced to the incident
wave, expressed in dB.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
29
Reflection Coefficient ()
• If the load impedance
differs from the
characteristic impedance
of the line then part of
the wave is reflected.
• The ratio of the incident
voltage to the reflected
voltage is 
Z
Z ZZ0
L
   Z  Z
Z L  Z0
L
o
L
o
Sometimes specified by the return loss = 20 log ()
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
30
VSWR
• The Voltage Standing Wave Ratio
VSWR compares the maximum and
minimum values of a “standing wave”
pattern, caused by wave reflection.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
31
Mismatch Loss
• The Mismatch Loss ML is the power
lost between two interconnected ports,
due to mismatch.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
32
• The four circuit parameters (, RL,
VSWR, and ML) are interrelated.
• Knowing one, the magnitudes of the others
can be computed.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
33
• When EM waves propagate in two
directions inside a transmission line, a
“standing wave” pattern is formed.
Dr. Blanton - ENTC 4307 -
Transmission Lines (cont.)
34
• Voltage Standing Wave Ratio (VSWR) is
by definition the ratio of maximum
(Vmax) and minimum (Vmin) voltages
of the standing wave function.
Vmax VF  VR 1  
VSWR 


Vmin VF  VR 1  
Dr. Blanton - ENTC 4307 -

Transmission Lines (cont.)
VSWR  1

VSWR  1
35