Transmission Lines No. 1Seattle Pacific University

Transmission Lines

Kevin BoldingElectrical Engineering

Seattle Pacific University

Transmission Lines No. 2Seattle Pacific University

Transmission Lines

• Long electrical wires are known as transmission lines

• In short lines, it is safe to assume that wires have the same voltage and current at all points

• In transmission lines, the voltage and current may vary along the length of the line

How long is “long”?Wires longer than1/10 of the wavelength of the signal applied. (Assume travelling at the speed of light, 3.0x108 m/s)At 1MHz, this is 30m.At 3GHz, this is 1cm.

Transmission Lines No. 3Seattle Pacific University

Transmission Line Thought Experiments3,000,000,000 m

+-

When the switch is closed, what current flows?

5V5

i

What if the switch is closed for 1 second, then opened?

Transmission Lines No. 4Seattle Pacific University

Transmission Line Model

• Any wire has inductance• Inductors oppose changes in current

+-5V

i

Infinitely Long

• Any wire has resistance… but let’s pretend it doesn’t• Any pair of wires has some capacitance between them

• Charging the capacitors will take current.

• A balance will be reached, with some resultant current, i• This implies an resistance, R = V/i

Transmission Lines No. 5Seattle Pacific University

Characteristic Impedance• A transmission line appears to have a resistance that is related to the distributed capacitance and

inductance

Characteristic Impedance – Z0

Apparent resistance seen at the input of a transmission line.Determined primarily by the capacitance and inductance of the line, especially for high frequency signals.Applies to infinitely long lines, but also to lines with AC inputs if the line is “long” compared with the AC wavelength.

• Z0 cannot be measured in traditional ways• If we put an Ohmmeter on an open transmission line, it will measure “open” or high resistance because we’re making a DC measurement

• An Ohmmeter would work if we had an infinite line…

• AC instruments (scopes) can be used

Transmission Lines No. 6Seattle Pacific University

Factors affecting Characteristic Impedance

• Z0 depends on C, L, and R per length• Higher C Lower Z0 (More ability to “soak up” charge)

• C ↑ Z0 ↓

• Higher L Higher Z0 (More opposition to current)

• L ↑ Z0 ↑

• Higher R Higher Z0 (Duh!)

• R ↑ Z0 ↑

+-5V

Transmission Lines No. 7Seattle Pacific University

Factors affecting Characteristic Impedance

• Capacitance per length depends on:• Conductor spacing – d – Larger spacing means less capacitance

• d ↑ C ↓, but C ↑ Z0 ↓, thus d ↑ Z0 ↑

• Relative permittivity of the dielectric separating the conductors

• Inductance per length depends on:• Conductor radius – r – Larger radius means less inductance

• r ↑ L ↓, but L↑ Z0 ↑, thus r ↑ Z0 ↓

• Conductor spacing – d – When conductors are close together, the magnetic fields cancel out• d ↑ L ↑, and L↑ Z0 ↑, thus d ↑ Z0 ↑

Conductors Dielectric Insulator d

r

Transmission Lines No. 8Seattle Pacific University

Determining Characteristic Impedance

• For two parallel wires:

ConductorsDielectric Insulator

with relative permittivity kd

r

r

d

kZ 100 log

276

Looking for k? Try “relative static permittivity” on wikipedia.

Dielectric Insulatorwith relative permittivity k

d1 (inside diameter of outer conductor)d2

• For coaxial cable:

2

1100 log

138

d

d

kZ

Signal Propagation speed (c = speed oflight in vacuum): k

cv1

Transmission Lines No. 9Seattle Pacific University

Z0 and Resistance

ConductorsDielectric Insulator

with relative permittivity kd

r

• Characteristic Impedance looks like a resistance, but it isn’t!• Composed of C’s and L’s: No resistance (ideally)

• Z0 doesn’t convert electrical energy into heat like a resistance

• Z0 does not contribute to attenuation

• Z0 does not vary with the length of the line

• Real transmission lines have real resistance• Loss due to metal resistance - proportional to square root of frequency; goes up with line length

• Loss due to dielectric resistance – proportional to frequency; goes up with line length

• It is these two resistances that are the primary causes of attenuation

Transmission Lines No. 10Seattle Pacific University

The end of the line

+-5V

i

NOT Infinitely Long

• After switch is closed, a wave travels down the t-line

• What happens when it hits the open end?

open

• What if the end is shorted?

• Open or shorted t-lines will cause (unwanted) reflections.

shorted

Transmission Lines No. 11Seattle Pacific University

Termination

+-5V

i

NOT Infinitely Long

• Place a resistor of value Z0 at the end of the line

• What happens when the wave hits the terminated end?

R = Z0

• A properly terminated line will appear to be an infinite line and have no reflections

• Any impedance change will cause reflections

Transmission Lines No. 12Seattle Pacific University

Why we dislike Reflections

• Reflections can mess up values on the line in locations between the source and end

• If a periodic signal is applied to an un-terminated transmission line, the reflections will set up a standing wave pattern• The amplitude of the standing wave may be very large and will

radiate power away

• This is great if we’re building an antenna, but we’re not!

• Un-terminated lines are a major source of Electromagnetic Interference (EMI)

Transmission Lines No. 13Seattle Pacific University

Creating Reflections – Things to Avoid

• Obviously, un-terminated t-lines• Remember, a line should be treated as a t-line if it is longer than 1/10 of

the wavelength of the applied signal

• Any change in impedance of the line will create reflections• Change of cable type

• Change of the separation between conductors

• Change in the width of conductors

• Change in the dielectric between conductors

• Use of standard cable with standard connectors avoids problems• Match the impedance of the cable to the equipment being used

Transmission Lines No. 14Seattle Pacific University

Creating Reflections – Things to Avoid

• Printed Circuit Boards (PCBs) present a challenge

If the spacing between conductors changes, the impedance changes!

If the width of the conductors changes, the impedance changes!

Square corners not only change the spacing between conductors, but the width of the conductors too!

Ground planes help a lot because the signal’s return path will follow directly underneath the signal. However, the impedance of the line will depend on the thickness and type of material in the PCB.

signal

ground

signal

ground

signal

ground