Transmission Line “Definition”General transmission line: a closed system in which power is transmitted from a source to a

destination

Our class: only TEM mode transmission linesA two conductor wire system with the wires in close proximity, providing relative impedance,

velocity and closed current return path to the source.Characteristic impedance is the ratio of the voltage and current waves at any one position on the

transmission line

Propagation velocity is the speed with which signals are transmitted through the transmission line in its surrounding medium.

I

VZ 0

r

cv

Presence of Electric and Magnetic Fields

Both Electric and Magnetic fields are present in the transmission lines

These fields are perpendicular to each other and to the direction of wave propagation for TEM mode waves, which is the simplest mode, and assumed for most simulators(except for microstrip lines which assume “quasi-TEM”, which is an approximated equivalent for transient response calculations).

Electric field is established by a potential difference between two conductors.

Implies equivalent circuit model must contain capacitor. Magnetic field induced by current flowing on the line

Implies equivalent circuit model must contain inductor.

V

I

I

E

+

-

+

-

+

-

+

-

V + DV

I + DI

I + DI

V

IH

IH

V + DV

I + DI

I + DI

General Characteristics of Transmission Line Propagation delay per unit length (T0) { time/distance}

[ps/in] Or Velocity (v0) {distance/ time} [in/ps]

Characteristic Impedance (Z0) Per-unit-length Capacitance (C0) [pf/in] Per-unit-length Inductance (L0) [nf/in] Per-unit-length (Series) Resistance (R0) [/in] Per-unit-length (Parallel) Conductance (G0) [S/in]

T-Line Equivalent Circuit

lL0lR0

lC0lG0lL0lR0

lC0lG0

Ideal T Line Ideal (lossless) Characteristics of Transmission Line

Ideal TL assumes: Uniform line Perfect (lossless) conductor (R00) Perfect (lossless) dielectric (G00)

We only consider T0, Z0 , C0, and L0.

A transmission line can be represented by a cascaded network (subsections) of these equivalent models. The smaller the subsection the more accurate the model

The delay for each subsection should be no larger than 1/10th the signal rise time.

lL0

lC0

lL0

lC0

Signal Frequency and Edge Rate vs.

Lumped or T line Models

In theory, all circuits that deliver transient power from one point to another are transmission lines, but if the signal frequency(s) is low compared to the size of the circuit (small), a reasonable approximation can be used to simplify the circuit for calculation of the circuit transient (time vs. voltage or time vs. current) response.

T Line Rules of Thumb

Td < .1 Tx

Td < .4 Tx

May treat as lumped Capacitance Use this 10:1 ratio for accurate modeling of transmission lines

May treat as RC on-chip, and treat as LC for PC board interconnect

So, what are the rules of thumb to use?

Other “Rules of Thumb”

Frequency knee (Fknee) = 0.35/Tr (so if Tr is 1nS, Fknee is 350MHz)

This is the frequency at which most energy is below

Tr is the 10-90% edge rate of the signal Assignment: At what frequency can your

thumb be used to determine which elements are lumped?Assume 150 ps/in

When does a T-line become a T-Line?

Whether it is a bump or a mountain depends on the ratio of its size (tline) to the size of the vehicle (signal wavelength)

When do we need When do we need to use to use

transmission line transmission line analysis analysis

techniques vs. techniques vs. lumped circuit lumped circuit

analysis? analysis?

TlineWavelength/edge rate

Similarly, whether or not a line is to be considered as a transmission line depends on the ratio of length of the line (delay) to the wavelength of the applied frequency or the rise/fall edge of the signal

Equations & Formulas

How to model & explain transmission line behavior

Relevant Transmission Line Equations

Propagation equation

jCjGLjR ))((

)(

)(0

CjG

LjRZ

Characteristic Impedance equation

In class problem: Derive the high frequency, lossless approximation for Z0

is the attenuation (loss) factor

is the phase (velocity) factor

Ideal Transmission Line Parameters

Knowing any two out of Z0, Td, C0, and L0, the other two can be calculated.

C0 and L0 are reciprocal functions of the line cross-sectional dimensions and are related by constants: is electric permittivity

0= 8.85 X 10-12 F/m (free space)

ri s relative dielectric constant

is magnetic permeability 0= 4p X 10-7 H/m (free space)

r is relative permeability.;

;;1

;;

;;

00

000

000

0

0

0

00d

0

0

0

rr

LCv

TZLZ

TC

CLTC

LZ

Don’t forget these relationships and what they mean!Don’t forget these relationships and what they mean!

Parallel Plate Approximation

Assumptions TEM conditions Uniform dielectric ( )

between conductors TC<< TD; WC>> TD

T-line characteristics are function of: Material electric and

magnetic properties Dielectric Thickness (TD) Width of conductor (WC)

Trade-off TD ; C0 , L0 , Z0 WC ; C0 , L0 , Z0

TD

TC

WC

To a first order, t-line capacitance and inductance can be approximated using the parallel plate approximation.

d

PlateAreaC

* Base

equation

C0 WC

TD

F

m

8.85 rWC

TD

pF

m

L0 TD

WC

F

m

0.4 rTD

WC

H

m

Z0 377TD

WC

r

r

Improved Microstrip Formula

Parallel Plate Assumptions + Large ground plane with

zero thickness To accurately predict microstrip

impedance, you must calculate the effectiveeffective dielectric constant.

TD

TC

WC

From Hall, Hall & McCall:

CC

D

r TW

TZ

8.0

98.5ln

41.1

870

DC

Cr

C

D

rre

TW

TF

W

T1217.0

1212

1

2

1

2

1102.0

D

Cr

T

WF

1D

C

T

Wfor

0 1D

C

T

Wfor

Valid when: 0.1 < WC/TD < 2.0 and 1 < r < 15

Improved Stripline Formulas

Same assumptions as used for microstrip apply here

TD2

TC

WCTD1

From Hall, Hall & McCall:

)8.0(67.0

)(4ln

60 110

CC

DD

r

symTW

TTZ

Symmetric (balanced) Stripline Case TD1 = TD2

),,,2(),,,2(

),,,2(),,,2(2

00

000

rCCsymrCCsym

rCCsymrCCsymoffset

TWBZTWAZ

TWBZTWAZZ

Offset (unbalanced) Stripline Case TD1 > TD2

Valid when WC/(TD1+TD2) < 0.35 and TC/(TD1+TD2) < 0.25

Refection coefficient

Signal on a transmission line can be analyzed by keeping track of and adding reflections and transmissions from the “bumps” (discontinuities)

Refection coefficient Amount of signal reflected from the “bump” Frequency domain =sign(S11)*|S11| If at load or source the reflection may be called gamma (L or

s) Time domain is only defined a location

The “bump” Time domain analysis is causal. Frequency domain is for all time. We use similar terms – be careful

Reflection diagrams – more later

Reflection and Transmission

Incident

Reflected

Transmitted

Special Cases to Remember

1

Zo

Zo

0

ZoZo

ZoZo

1

0

0

Zo

Zo

Vs

ZsZo Zo

A: Terminated in Zo

Vs

ZsZo

B: Short Circuit

Vs

ZsZo

C: Open Circuit

Transmission Lines, Reflections, and Termination

Short Circuited Line

Transmission Lines, Reflections, and Termination

Open Circuited Line

Transmission Lines, Reflections, and Termination

If the source impedance (Rsrc) does not equal Z0, then reflections occur at the near end of the line as well as at the far end. Each end of the line has its own value of ρ. The principle of superposition applies, so that voltage at any point on the line and instant in time is the sum of that point’s initial condition and all waves that have passed it so far.

The end of a transmission line is said to bematched if it is terminated with its

characteristic impedance.

Transmission Lines, Reflections, and Termination

Mismatched Line

Transmission Lines, Reflections, and Termination

Matched Line

Model

http://www.fourier-series.com/rf-concepts/flash_programs/Reflection/index.html