Part 1: Transmission Lines Basics

Chapter 2 Part 1: Transmission Lines Basics (Note: bring a thumb drive to the lab next Tuesday)  Definition: A transmission line is a mechanical structure that guides EM energy in a desired direction.  Basic properties of transmission lines: . TEM mode transmission lines: E and H are in the transverse plane, a plane that is perpendicular to the direction of the EM wave propagation. Commonly used lines consisting a pair of parallel conductors usually belong to this class. . Higher-order transmission lines: One of the field components is in the direction of the EM wave propagation. Waveguide types consisting a hollow conductor usually belong to this class (next semester). . Dispersion in most commonly used transmission lines such as coaxial lines is negligible. However, the line loss is usually high especially at microwave frequencies. . On the other hand, waveguides suffer much lower losses at microwave frequencies but dispersion can be a problem.  The most popular TEM lines are (Fig. 2-4, P. 38): . Coaxial cable: Used mainly in the high-frequency regime (MHz and up) at low- to medium -power levels. . Parallel-wire: Used mainly for lower frequency transmission (AC power line) at high-power levels. . Microstrip: Used mainly for microwave IC and PC board fabrications.  The most popular higher order transmission lines are: . Waveguide: Hollow conductors used at high frequencies and high-power. . Optical fiber: Concentric dielectric layers used at optical frequencies.  Differences and similarities between a transmission line and a waveguide: . They both are mechanical structure used to carry EM energy. . Transmission line has two parallel conductors; waveguide has only one, usually hollow, conductor. . Waves inside transmission line are TEM; waveguide waves are TE or TM. . Transmission line has higher loss, especially at high frequencies. . Waveguide waves may be distorted due to dispersion. . Transmission line has no low-frequency limit; waveguide acts like a high- pass filter.  In general, the impact on the signal integrity is much severe from the effect of distortion than from the effect of losses.  Beside guiding EM energy, short sections of transmission line can sometimes be used as high-frequency discrete circuit components such as Z transformer, filter, …  Simplifications: . Source: A voltage source and a series resistor are used to represent the generator (Thevenin’s theorem) . Load: A resistance (impedance) is used to represent the load. . Transmission line: A two-port network is used to describe a general transmission line.  The effect of short  (transmission line effect): . Guideline: l > 0.01  (1% of the wavelength) . Phase shift: The voltage and current along a transmission line generally vary as a function of position. Short 

Long 

. Reflection: If the line is not terminated properly (with a match load) portion of the energy will be reflected back to the generator. This is usually bad because: 1) Reflected energy is wasted 2) Generator may be damaged if the reflection is significant. . Power loss: Energy is dissipated due to conduction loss and dielectric loss. . Distortion: A signal containing more than one frequency component may be distorted due to dispersion—different frequency components traveling at different speeds in a waveguide.  Lumped-element model . We will first consider the general properties of a TEM mode line. . Similar to your calculus classes, we are going to cut a thin slice (differential section) of the line and examine its electrical properties.  Transmission line parameters: . Series L’ and R’ per unit length . Parallel C’ and G’ per unit length

 Transmission line equations (Telegrapher’s equations): We can derive a set of circuit equations for a differential slice of the above model and from them a set differential equations can be developed to describe the voltage and current in a transmission line.

~ ~ ~ ~ ~ dV ~ V  (V  dV )  I (R' j L')dz    (R' j L')I dz ~ ~ ~ ~ ~ ~ dI ~ I  (I  dI )  (V  dV )(G' jC')dz    (G' jC')V dz  Wave equations: Combining the two 1st order/2-unknown equations we have a pair of 2nd order/1-unknown equations. ~ ~ d 2V ~ d 2 I ~   2V  0   2 I  0     j  (R' j L')(G' jC') dz 2 dz 2 ~   z   z ~   z   z V  V0 e  V0 e I  I0 e  I0 e   V0 V0 R' j L'      Z0  u p  f  I0 I0 G' jC'   – propagation constant  – attenuation constant = Re{},  – phase constant = Im{}, up – phase velocity, Z0 – characteristic impedance = v / i for a single wave  The time domain solution to the voltage wave equation due to sinusoidal excitation is: v(z,t)  V  e z cos(t   z   )  V  e z cos(t   z   )  0           0          Travels in the z direction Travels in the z direction  Electrical length: . The electrical length of a line is the ratio of the physical length to the wavelength of the line signal. As a result, a physically long line may have a short electrical length if a low-frequency signal (long ) is being transmitted and vice versa. . Electrically short lines: For example, AC power lines less than 50 miles long are usually considered electrically short. For such cables we can consider them as discrete passive components.

R + jX

. Medium-length lines: In general, power lines between 50 miles and 150 miles long are considered to be medium-length lines. We can also consider them as discrete passive components with a more complex structure.

Z/2 Z/2

. Short and medium lines are important in low-frequency applications such as power transmission. They are handled with lumped-circuit theory. . Electrically long lines: A line is considered to be electrically long if its physical length is comparable to the wavelength. For example, a 15 cm coaxial cable would be considered electrically long at 3 GHz ( = c/f = 3E8 / 3E9 = 0.1 m = 10 cm). . Because at high frequencies the physical length of an electrically long line is often very short, the series resistance becomes negligible. . Since low-loss materials are usually required to fabricate high-frequency cables, the shunt conductance can also be neglected.  Lossless (ideal) line (R’ = G’ = 0   = 0):

~ ~ d 2V ~ d 2 I ~   2V  0   2 I  0    L'C' dz 2 dz 2 ~   j z   j z ~   j z   j z V  V0 e  V0 e I  I0 e  I0 e   V0 V0 L'      Z0  u p  f  I0 I0 C'  . For all TEM lines: G'  L'C'  and  C'  1 c      and u p     r

0 c    where 0  (free space wavelength)  r f . Dispersion refers to the spreading of a signal due to the differences in speed between different frequency components. It is often detrimental to digital communication since digital pulses contain multiple frequency components. . Since the phase velocity is independent of frequency, ideal TEM lines are nondispersive.  Terminating an ideal line . The characteristic impedance Z0 of a transmission line is not a real resistance even though it has a unit of ohm. Z0 of a line merely represents the ratio of the voltage and current of one traveling wave along the line. If more than one wave (for example, incident and reflected waves) are present in the line, the ratio of the total voltage and total current would not equal to Z0. I-,V- I+,V+ Z0 ZL

    VL  V0 V0 I L  I0  I0 V V  V  V  V  V  V   L  0  0  0 0  0 0 Z L Z0 Z0 Z L Z0 Z L 1 V  I  Z  Z Z     e j r  0   0  L 0  0 V  I  Z  Z Z 0 0 L 0 L 1 Z0

. If a transmission line is terminated with a load resistance ZL equal to its characteristic impedance Z0 there will be no energy reflected from the load. In other words, a matched load acts like an infinitely long line with the same Z0. . For a mismatch load, part of the energy will be reflected back to the generator. The amplitude of the reflected wave is according to the above formula. . The portion of power reflected back is 2. .  = 1 for o/c and  = -1 for s/c . || = 1 for purely reactive loads. . Examples: Z0 = 50 

1. ZL = 55   = (ZL – Z0) / (ZL + Z0) = (55 – 50) / (55 + 50) = 5/105 = 0.048 2 = 0.0482 =0.0023 = 0.23%

2. ZL = 45   = (ZL – Z0) / (ZL + Z0) = (45 – 50) / (45 + 50) = -5/95 = -0.053 2 = 0.0532 =0.0028 = 0.28%

3. ZL = 5   = (ZL – Z0) / (ZL + Z0) = (5 – 50) / (5 + 50) = -45/55 = -0.82 2 = 0.822 =0.67 = 67%

4. ZL = 5 K  = (ZL – Z0) / (ZL + Z0) = (5000 – 50) / (5000 + 50) = 4950/5050 = 0.98 2 = 0.982 =0.96 = 96% 5. Z = 75 + j25  Z0L  50 ZL  75  25j ZL  Z0 Z  Z  25  25i   L 0 Z  Z L 0 ZL  Z0  125  25i

ZL  Z0  35.355 

argZL  Z0  45 deg

ZL  Z0  127.475 

argZL  Z0  11.31 deg 35.355  0.277 45  11.31  33.69 127.475   0.231  0.154i arg  33.69deg 2   0.277     7.692%

(Excel programs)  Impedance transformation: . The input impedance is the ratio of the total line voltage to the total line current. This ratio in general varies along the line and thus the input impedance is a function of the distance between the measurement point and the load. . For an ideal line (lossless) terminated with a match load the input impedance Zin measured at any distance always equals to Z0.

. On the other hand, if ZL  Z0 then Zin will vary as a function of the distance away from the load.

Zin Z0 ZL

. The input impedance Zin: ~   j z   j z  j z  j z V (z) V0 e  V0 e e  e  Zin (z)  ~    j z   j z  Z0   j z  j z  I (z) V0 e  V0 e e  e 

Z0 Let z  l e j l  e j l  1 e2 j l  Zin (l)  Z0  j l  j l   Z0  2 j l  e  e  1 e 

. The input impedance Zin measured at any distance l from the load is:

Z L cos l  jZ0 sin l Z L  jZ0 tan l Zin (l)  Z0  Z0 Z0 cos l  jZ L sin l Z0  jZ L tan l . As a result, the input impedance of an unmatched line can vary drastically as a function of l.  l l 0 0 90 /4 180 /2 270 3/4 360 

. Examples: Z0 = 50 , f = 3 GHz

1) ZL = 0  and  = 3E8 / 3E9 = 10 cm Zin = jZ0 (sin  l / cos  l)

Zin = 0 for  l = 0, 180, 360, … Zin =  for  l = 90, 270, … A shorted line 2.5 cm long sometimes looks like an open.

2) ZL =   and  = 3E8 / 3E9 = 10 cm Zin = jZ0 (cos  l / sin  l)

Zin =  for  l = 0, 180, 360, … Zin = 0 for  l = 90, 270, … An open line 2.5 cm long sometimes looks like a short. 3) For ZL = 30  and v = 0.8 c, find Zin for a 5.5m cm line 0.83108 Z  50 Z  30 f  3109Hz s 0 L l  5.510 2m     0.08m 360deg f l  l l  247.5 deg l   0.687 360deg ZLcosl  jZ0sinl Zin  Z0 Z0cosl  jZLsinl

ZLcosl  11.481 jZ0sinl  46.194i Z0cosl  19.134 jZLsinl  27.716i

ZLcosl  jZ0sinl  11.481  46.194i Z0cosl  jZLsinl  19.134  27.716i

11.481  46.194i  47.599 19.134  27.716i  33.679 arg(11.481  46.194i)  103.957 deg arg(19.134  27.716i)  124.62 deg

ZLcosl  jZ0sinl  ZLcosl  jZ0sinl   1.413arg   20.663 deg Z cosl  jZ sinl Z cosl  jZ sinl 0 L  0 L  Zin  66.119  24.935i 24.935  L   9 2f L  1.323  10 H Zin  70.665 argZin  20.663 deg

4) For ZL = 30  and v = 0.8 c, find Zin for a 2.2 cm line 8  2 0.8310 Z0  50 ZL  30 l  2.210     0.08 3109 l l  360deg l  99 deg 

ZLcosl  jZ0sinl Zin  Z0 Z0cosl  jZLsinl

ZLcosl  4.693 jZ0sinl  49.384i Z0cosl  7.822 jZLsinl  29.631i

ZLcosl  jZ0sinl  4.693  49.384i Z0cosl  jZLsinl  7.822  29.631i 11.481  46.194i  47.599 19.134  27.716i  33.679 arg(11.481  46.194i)  103.957 deg arg(19.134  27.716i)  124.62 deg

ZLcosl  jZ0sinl  ZLcosl  jZ0sinl   1.619 arg   9.359 deg Z0cosl  jZLsinl  Z0cosl  jZLsinl 

Zin  79.859  13.161i Zin  80.936 argZin  9.359 deg  1 C  2310913.161 C  4.031  10 12

. From the examples we conclude that a short section of transmission line can transform a short to an open and vice versa. It can also transform a capacitance to an inductance and vice versa. . Properly terminating the output of microwave amplifiers and oscillators is important. . Tune circuits at microwave frequencies usually require L and C with extremely small values. Using small segments of transmission line is one way to accomplish this requirement.  Complete solution of the wave equations: ~  Vg Zin   1  V   g     0  Z  Z  jl  jl  g in   e  e   Standing waves (ppt): . Transmission line waves (incident and reflected) are traveling waves in opposite directions. It is usually difficult to observe them separately. . The combined wave displays an alternating pattern (repeats at every ½ ) of constructive and destructive interference between the incident and reflective waves. . The envelop of this pattern is called the standing wave. . Match load: Since no reflection exists in the line, no standing wave pattern is observed (0 V swing between maximum and minimum). + - . Open-circuit (o/c) load: Since V and V are the same at ZL, we expect the

load voltage to be doubled at ZL and complete cancellation at ¼  away (2 V swing). This pattern repeats every ½ . + - . Short-circuited (s/c) load: Since V and V are opposite at ZL, we expect

complete cancellation at ZL and the voltage to be doubled at ¼  away (2 V swing). This pattern repeats every ½ . . In summary, voltage maxima are located at n  ½  (1/2 , 1 , 1 1/2 ,

…) for resistive loads larger than Z0 and at ¼  + n  ½  (1/4 , 3/4 , 1

1/4 , …) for resistive loads smaller than Z0. . Minima are located at ¼  away from maxima. . The locations on the line corresponding to voltage maxima also correspond to current minima, and vice versa. . In general, voltage maxima occur at locations where the incident and reflective waves are in phase (constructive interference). . In general, voltage minima occur at locations where the incident and reflective waves are out of phase (destructive interference). . VSWR (Voltage Standing Wave Ratio): A quantity that can be measured easily. The reflective coefficient and input impedance can be computed from the VSWR and the locations of the first minimum or first maximum. ~ Vmax 1  S   ~ 1 Vmin   S 1   S 1 1  Z L  Z0 1    Special cases: . Short-circuited line: A short-circuited line behaves like a discrete reactive component with: SC Zin  jZ 0 tan l For length between 0 and ¼  it is inductive with equivalent L: Z tan l 1 L  0 1 eq  Leq   l  tan      Z0  For length between ¼  and ½  it is capacitive with equivalent C: 1 1   1  C    l    tan 1  eq Z  tan l  C Z  0   eq 0  The reactive type repeats beyond l > ½  following the same pattern. . Open-circuited line: An open-circuited line behaves like a discrete reactive component with:

OC  jZ 0 Zin   jZ 0 cot l  tan  l For length between 0 and ¼  it is capacitive with equivalent C:

tan l 1 1 Ceq   l  tan Ceq Z0  Z0  For length between ¼  and ½  it is with inductive equivalent L: Z 1   Z  L   0  l    tan 1 0  eq  tan  l  2  L    eq  The reactive type repeats beyond l > ½  following the same pattern. . In microwave and high-speed circuits small C and L are often needed. It is much easier to fabricate them using transmission lines segments.

. Transmission parameters (Z0 and ) can be determined from the results of OC and SC measurements. 1   Z SC  Z  Z SC  Z OC   tan 1 in  0 in in l  Z OC   in  . 1/2 wave window: A 1/2  section of transmission line, regardless of characteristic impedance, can be inserted into an existing transmission line system without disturbing the electrical properties of the system.

Z L cos l  jZ1 sin l Z L (1)  jZ1 (0) Zin  Z1  Z1  Z L Z1 cos l  jZ L sin l Z1 (1)  jZ L (0) Zin = ZL

Z0 Z1 ZL

l /2

. 1/4 wave transformer: A 1/4  section of transmission line (Z02) can be inserted into used to match any Z01 to ZL.

Z L cos l  jZ02 sin l Z L (0)  jZ 02 (1) jZ02 (1) Zin  Z02  Z02  Z02  Z01 Z02 cos l  jZ L sin l Z02 (0)  jZ L (1) jZ L (1)

 Z02  Z01  Z L Zin = Z01

Z01 Z02 ZL

l /4

Example: match a 50- line to a 200- load at 100 MHz (v = c/8). /4 = (c/8) / f / 4 = 0.094 m 1/2 Z1 = (50  200) = 100 