Transmission Line Theory

Transmission Line Theory
ECE Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE Notes 1 Transmission Line Theory

Overview In this set of notes we first overview different types of waveguiding systems, including transmission lines. We then develop transmission line theory in detail. We discuss several types of practical transmission lines. We discuss baluns.

Waveguiding Structures
A waveguiding structure is one that carries a signal (or power) from one point to another. There are three common types: Transmission lines Fiber-optic guides Waveguides Note: An alternative to using waveguiding structures is wireless transmission using antennas. (Antennas are discussed in ECE 5318.)

(shown connected to a 4:1 impedance-transforming balun)
Transmission Line Properties Has two conductors running parallel (one can be inside the other) Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss (but there always is) May or may not be immune to interference (shielding property) Does not have Ez or Hz components of the fields (TEMz) Twin lead (shown connected to a 4:1 impedance-transforming balun) Coaxial cable (coax)

Transmission Line (cont.)
CAT 5 cable (twisted pair) The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities.

Transmission lines commonly met on printed-circuit boards Microstrip h w er er w Stripline h Coplanar strips h er w Coplanar waveguide (CPW) h er w

Transmission lines are commonly met on printed-circuit boards. Microstrip line A microwave integrated circuit.

Waveguides Properties
Consists of a single hollow metal pipe Can propagate a signal only at high frequency: f > fc The width must be at least one-half of a wavelength Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz)

Fiber-Optic Guide Properties Uses a dielectric rod
Can propagate a signal at any frequency (in theory) Can be made to have very low loss Has minimal signal distortion Has a very high bandwidth Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields (“hybrid mode”)

Fiber-Optic Guide (cont.)
Two types of fiber-optic guides: 1) Single-mode fiber Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength. 2) Multi-mode fiber Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect). Note: The carrier is at optical frequency, but the modulating (baseband) signal is often at microwave frequencies (e.g., TV, internet signal).

Fiber-Optic Guide (cont.)
Multi-mode Fiber Higher index core region

Transmission-Line Theory
We need transmission-line theory whenever the length of a line is significant compared to a wavelength.

Transmission Line C = capacitance/length [F/m]
2 conductors 4 per-unit-length parameters: C = capacitance/length [F/m] L = inductance/length [H/m] R = resistance/length [/m] G = conductance/length [ /m or S/m] Dz 

x B + - Note: There are equal and opposite currents on the two conductors. (We only need to work with the current on the top conductor, since we have chosen to put all of the series elements there.)

+ -

TEM Transmission Line (cont.)
Hence Now let Dz  0: “Telegrapher’sEquations”

To combine these, take the derivative of the first one with respect to z: Switch the order of the derivatives.

Hence, we have: The same differential equation also holds for i.

Time-Harmonic Waves:

Note that = series impedance/length = parallel admittance/length Then we can write:

Define Then Solution:  is called the “propagation constant”. We have: Question: Which sign of the square root is correct?

We choose the principal square root. Principal square root: (Note the square-root (“radical”) symbol here.) Hence

Denote:

Re Im Re Im There are two possible locations for  : it must be in the first quadrant ( > 0). Re Im Hence:

Forward travelling wave (a wave traveling in the positive z direction): The wave “repeats” when: Hence:

Phase Velocity Let’s track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest of the wave. (phase velocity)

Phase Velocity (cont.) Set In expanded form: Hence

Characteristic Impedance Z0
Assumption: A wave is traveling in the positive z direction. so (Note: Z0 is a number, not a function of z.)

Characteristic Impedance Z0 (cont.)
Use first Telegrapher’s Equation: so Recall: Hence

From this we have: Use: Both are in the first quadrant (principle square root)

Hence, we have

General Case (Waves in Both Directions)
Wave in +z direction Wave in -z direction In the time domain:

Backward-Traveling Wave
z A wave is traveling in the negative z direction. so Note: The reference directions for voltage and current are chosen the same as for the forward wave.

General Case Most general case:
z Most general case: A general superposition of forward and backward traveling waves: Note: The reference directions for voltage and current are the same for forward and backward waves.

Summary of Basic TL formulas
Guided wavelength Phase velocity

Lossless Case so (real and independent of freq.)

Lossless Case (cont.) If the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that (proof omitted): (proof given later) The speed of light in a dielectric medium is Hence, we have that The phase velocity does not depend on the frequency, and it is always the speed of light (in the material).

Terminated Transmission Line
Terminating impedance (load) Amplitude of voltage wave propagating in positive z direction at z = 0. Amplitude of voltage wave propagating in negative z direction at z = 0. Where do we assign z = 0 ? The usual choice is at the load.

Terminated Transmission Line (cont.)
Terminating impedance (load) Can we use z = z0 as a reference plane? Hence

Terminating impedance (load) Compare: Note: This is simply a change of reference plane, from z = 0 to z = z0.

Terminating impedance (load) What is V(-d) ? Propagating forwards Propagating backwards The current at z = -d is d  distance away from load (This does not necessarily have to be the length of the entire line.)

or L  Load reflection coefficient Similarly,

Z(-d) = impedance seen “looking” towards load at z = -d. Note: If we are at the beginning of the line, we will call this “input impedance”.

At the load (d = 0): Recall Thus,

Simplifying, we have: Hence, we have

Terminated Lossless Transmission Line
Impedance is periodic with period g/2: The tan function repeats when Note:

Summary for Lossless Transmission Line
For the remainder of our transmission line discussion we will assume that the transmission line is lossless.

Matched Load (ZL=Z0) No reflection from the load

Short-Circuit Load (ZL=0)
Always imaginary!

Short-Circuit Load (ZL=0)
Inductive Capacitive Note: S.C. can become an O.C. with a g/4 transmission line.

Open-Circuit Load (ZL=)
or Always imaginary!

Open-Circuit Load (ZL=)
Note: Inductive Capacitive O.C. can become a S.C. with a g/4 transmission line.

Using Transmission Lines to Synthesize Loads
We can obtain any reactance that we want from a short or open transmission line. This is very useful in microwave engineering. A microwave filter constructed from microstrip line.

Voltage on a Transmission Line
Find the voltage at any point on the line. Note: For a lossy line, replace: At the input:

Voltage on a Transmission Line (cont.)
Incident wave (not the same as the initial wave!) At z = -d : Hence

Let’s derive an alternative form of the previous result. Start with:

Hence, we have where (source reflection coefficient) Substitute Recall: Therefore, we have the following alternative form for the result:

This term accounts for the multiple (infinite) bounces. The “initial” voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load).

Wave-bounce method (illustrated for z = -d ): We add up all of the bouncing waves.

Group together alternating terms: Geometric series:

Hence or This agrees with the previous result (setting z = -d ). Note: The wave-bounce method is a very tedious method – not recommended.

Time-Average Power Flow
P(z) = power flowing in + z direction At a distance d from the load: Note: If Z0  real (low-loss transmission line)

Time-Average Power Flow (cont.)
Low-loss line Lossless line ( = 0) Note: In the general lossy case (very lossy), we cannot say that the total power is the difference of the powers in the two waves.

Quarter-Wave Transformer
Lossless line Matching condition (This requires ZL to be real.) so Hence

Match a 100  load to a 50  transmission Line at a given frequency.
Example Match a 100  load to a 50  transmission Line at a given frequency.

Voltage Standing Wave

Voltage Standing Wave Ratio

Coaxial Cable Find C, L, G, R
Here we present a “case study” of one particular transmission line, the coaxial cable. Find C, L, G, R For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics. We will assume no variation in the z direction, and take a length of one meter in the z direction in order to calculate the per-unit-length parameters.

Find C (capacitance / length)
Coaxial Cable (cont.) Coaxial cable h = 1 [m] Find C (capacitance / length) From Gauss’s law: -l0 l0 a b

Coaxial Cable (cont.) Coaxial cable Hence We then have:

Find L (inductance / length)
Coaxial Cable (cont.) Find L (inductance / length) From Ampere’s law: Coaxial cable Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency. S Center conductor Magnetic flux:

Coaxial Cable (cont.) Hence

This result actually holds for any transmission line (proof omitted).
Coaxial Cable (cont.) Observations: (independent of frequency) (independent of frequency) This result actually holds for any transmission line (proof omitted).

Coaxial Cable (cont.) For a lossless cable:

Find G (conductance / length)
Coaxial Cable (cont.) Find G (conductance / length) Coaxial cable From Gauss’s law:

Coaxial Cable (cont.) We then have or

This result actually holds for any transmission line (proof omitted).
Coaxial Cable (cont.) Observation: This result actually holds for any transmission line (proof omitted).

This is the loss tangent that would arise from conductivity.
Coaxial Cable (cont.) As just derived, Hence: This is the loss tangent that would arise from conductivity.

(This is discussed later.)
Coaxial Cable (cont.) Find R (resistance / length) Coaxial cable Rs = surface resistance of metal (This is discussed later.)

General Transmission Line Formulas
These formulas hold for any shape of line (not restricted to coax). (1) (2) (4) (3) Equations (1) and (2) can be used to find L and C if we know the material properties and the characteristic impedance of the lossless line. Equation (3) can then be used to find G if we know the material loss tangent. Equation (4) can be used to find R (discussed later).

General Transmission Line Formulas
All four per-unit-length parameters (R, L, G, C) can be found from Note:

Accounting for Dielectric Loss
Complex Permittivity The permittivity becomes complex when there is polarization (molecular friction) loss. Loss term due to polarization (molecular friction) Example: Distilled water heats up in a microwave oven, even though there is essentially no conductivity!

Accounting for Dielectric Loss (cont.)
Effective Complex Permittivity The effective permittivity accounts for conductive loss. Effective permittivity that accounts for conductivity Hence We then have

Most general expression for loss tangent: Loss due to conductivity Loss due to molecular friction The loss tangent accounts for both molecular friction and conductivity.

For most practical insulators (e.g., Teflon), we have Note: The loss tangent is usually (approximately) constant for practical insulating materials, over a wide range of frequencies. Typical microwave insulating material (e.g., Teflon): tan =

Here we allow for dielectric polarization loss.
General Transmission Line Formulas Here we allow for dielectric polarization loss. These formulas hold for any shape of line (not restricted to coax). (1) (2) (4) (3) Note: Note: tan accounts for both polarization loss and conductivity.

General Transmission Line Formulas (cont.)
Here we allow for dielectric polarization loss. All four per-unit-length parameters (R, L, G, C) can be found from Note:

Common Transmission Lines
Coax Twin-lead

Common Transmission Lines (cont.)
Microstrip Approximate CAD formula er

Limitations of Transmission-Line Theory
At high frequency, discontinuity effects can become important. Transmitted Incident Bend Reflected The simple TL model does not account for the bend.

Limitations of Transmission-Line Theory (cont.)
At high frequency, radiation effects can also become important. We want energy to travel from the generator to the load, without radiating. When will radiation occur? This is explored next.

Coaxial Cable The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, as long as the metal thickness is large compared with a skin depth. The fields are confined to the region between the two conductors.

The twin lead is an open type of transmission line – the fields extend out to infinity. The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair”.) Having fields that extend to infinity is not the same thing as having radiation, however!

The infinite twin lead will not radiate by itself, regardless of how far apart the lines are (this is true for any transmission line). Incident Reflected No attenuation on an infinite lossless line S The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.

A discontinuity on the twin lead will cause radiation to occur. Incident wave Pipe Obstacle Reflected wave Bend Incident wave Reflected wave Note: Radiation effects usually increase as the frequency increases.

To reduce radiation effects of the twin lead at discontinuities: Reduce the separation distance h (keep h << ). Twist the lines (twisted pair). CAT 5 cable (twisted pair)

Baluns Baluns are used to connect coaxial cables to twin leads.
They suppress the common mode currents on the transmission lines. Balun: “Balanced to “unbalanced” Coaxial cable: an “unbalanced” transmission line Twin lead: a “balanced” transmission line 4:1 impedance-transforming baluns, connecting 75  TV coax to 300  TV twin lead

Baluns (cont.) Baluns are also used to connect coax (unbalanced line) to dipole antennas (balanced). “Baluns are present in radars, transmitters, satellites, in every telephone network, and probably in most wireless network modem/routers used in homes.”

Baluns (cont.) What does “balanced” and “unbalanced” mean? + - +
Ground (zero volts) + Ground (zero volts) + - Twin Lead Coax The voltages are balanced with respect to ground. The voltages are unbalanced with respect to ground. Note: Even with coax, one can assume balanced voltages with respect to the ground plane if one wants to, so the distinction is somewhat vague.

Baluns (cont.) Baluns are necessary because, in practice, the two transmission lines are always both running over a ground plane. If there were no ground plane, and you only had the two lines connected to each other, then a balun would not be necessary. (But you would still want to have a matching network between the two lines if they have different characteristic impedances.) Coax Twin Lead No ground plane

Baluns (cont.) When a ground plane is present, we really have three conductors, forming a multiconductor transmission line, and this system supports two different modes. + - + Differential mode Common mode (desired) (undesired) The differential and common modes on a twin lead over ground In the common mode, both conductors of the transmission line act as one net conductor, while the ground plane acts as the other conductor (return path for the current).

Baluns (cont.) + - + - Differential mode Common mode (desired)
(undesired) The differential and common modes on a coax over ground

Baluns (cont.) Because of the asymmetry, a common mode current will get excited at the junction between the two lines, and propagate away from the junction. Coax Twin Lead Ground Common mode No Balun Differential mode Note: If the height above the ground plane goes to infinity, the characteristic impedance of the common mode goes to infinity, and there is no common mode current – we would not need a balun.

Baluns (cont.) A balun prevents common modes from being excited at the junction between a coax and a twin lead. With Balun Coax Twin Lead Ground Differential mode Balun

Baluns (cont.) From another point of view, a balun prevents currents from flowing on the outside of the coax. Coax Twin Lead Ground No Balun I1 I2 I2- I1 (current on outside of the coax)

Baluns (cont.) A simple model for the mode excitation at the junction
The coax is replaced by a solid tube with a voltage source at the end. Solid tube Twin Lead Ground + -

Next, we use superposition.
Baluns (cont.) Next, we use superposition. Solid tube Twin Lead Ground + - Differential mode + Solid tube Twin Lead Ground + - Common mode

One type of balun uses an isolation transformer.
Baluns (cont.) One type of balun uses an isolation transformer. The input and output are isolated from each other, so the input can be grounded if desired (e.g., when using a coax feed for the primary) while the output is not (e.g., when using a twin lead feed for the secondary). The common mode must be zero at the transformer, since a net current cannot flow off of the windings into the transformer core. This balun uses a 4:1 autotransformer, having three taps on a single winding, on a ferrite rod. The center tap on the input can be grounded if desired, while the balanced output is not. The common mode is choked off due to the high impedance of the coil.

Here is a more exotic type of 4:1 impedance transforming balun.
Baluns (cont.) Here is a more exotic type of 4:1 impedance transforming balun. “Guanella balun” This balun uses two 1:1 transformers to achieve symmetric output voltages.

Inside of 4:1 impedance transforming VHF/UHF balun for TV
Baluns (cont.) Inside of 4:1 impedance transforming VHF/UHF balun for TV Ferrite core

Another type of balun uses a choke to “choke off” the common mode.
Baluns (cont.) Another type of balun uses a choke to “choke off” the common mode. A coax is wound around a ferrite core. This creates a large inductance for the common mode, while it does not affect the differential mode (whose fields are confined inside the coax).

Baluns (cont.) Coax can be used to feed a microstrip line on a PCB without needing a balun. Both the microstrip line (with ground plane) and the coax are two-conductor transmission line systems, so there is no common mode.

Baluns (cont.) Baluns are very useful for feeding differential circuits with coax on PCBs. CPS line No balun Without a balun, there would be a common mode current on the coplanar strips (CPS) line.

= + Baluns (cont.) Superposition allows us to see what the problem is.
- + - + Excites undesired common mode Excites desired differential mode + -

Baluns can be implemented in microstrip form.
Baluns (cont.) Baluns can be implemented in microstrip form. Coax feed port Paired-strip output Tapered balun Balun feeding a microstrip yagi antenna

Baluns can be implemented in microstrip form.
Baluns (cont.) Baluns can be implemented in microstrip form. Marchand balun Null output Input #2 (balanced) Input #1 (balanced) Output (unbalanced) 180o hybrid rat-race coupler used as a balun

Baluns (cont.) If you try to feed a dipole antenna directly with coax, there will be a common mode current on the coax. No balun

Baluns are commonly used to feed dipole antennas from coax.
Baluns (cont.) Baluns are commonly used to feed dipole antennas from coax. A balun first converts the coax to a twin lead, and then the twin lead feeds the dipole.

Baluns (cont.) Baluns are commonly used to feed dipole antennas from coax. A “sleeve balun” directly connects a coax to a dipole. Coaxial “sleeve” region Coaxial “sleeve balun” The current on the outside of the sleeve is forced to be zero at the top of the sleeve. This makes it small everywhere on the outside.

Baluns are commonly used to feed dipole antennas from coax.
Baluns (cont.) Baluns are commonly used to feed dipole antennas from coax. This acts like voltage source for the dipole, which forces the differential mode. The dipole is loaded by a short-circuited section of twin lead – an open circuit because of the /4 height.

A balun is not always necessary.
Baluns (cont.) A balun is not always necessary. Dipole antenna Twin lead The system stays balanced. Monopole antenna Coax Infinite ground plane Current cannot flow on the outside of the coax.

Baluns (cont.) Baluns can be important when connecting to
unbalanced devices like oscilloscopes. OSC Coax input Measuring probe Coax leads No Balun There may be a strong common mode current on the coax leads. The coax leads will act as an antenna! You may not really be measuring the field from the probe.

Baluns (cont.) Baluns can be important when connecting to
unbalanced devices like oscilloscopes. OSC Coax input Measuring probe Coax leads Balun With Balun Now there is only a differential mode on the coax leads, and the coax leads act as a transmission line. We measure what is coming from the probe.

Transmission Line Theory

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