Prof. D. R. Wilton

Note 2 Note 2 Transmission LinesTransmission Lines

(Time Domain)(Time Domain)

ECE 3317

Disclaimer:

Transmission lines is the subject of Chapter 6 in the book. However, the subject of wave propagation on transmission line in the time domain is not treated very thoroughly there, appearing only in the latter half of section 6.5.

Therefore, the material of this Note is roughly independent of the book.

Note about Notes 2

Approach:

Transmission line theory can be developed starting from either circuit theory or from Maxwell’s equations directly. We’ll use the former approach because it is simpler, though it doesn’t reveal the approximations or limitations of the approach.

A transmission line is a two-conductor system that is used to transmit a signal from one point to another point.

Transmission Lines

Two common examples:

coaxial cable twin line

r a

bz

A transmission line is normally used in the balanced mode, meaning equal and opposite currents (and charges) on the two conductors.

Transmission Lines (cont.)

coaxial cable

Here’s what they look like in real-life.

twin line

Another common example (for printed circuit boards):

microstrip line

w

hr

Transmission Lines (cont.)

microstrip line

Transmission Lines (cont.)

Transmission Lines (cont.)

Some practical notes:

Coaxial cable is a perfectly shielded system (no interference).

Twin line is not a shielded system – more susceptible to noise and interference.

Twin line may be improved by using a form known as “twisted pair.”

l

-l

+

++

+

++

-

---

- -

E

coax twin line

E

Transmission Lines (cont.)

Transmission line theory must be used instead of circuit theory for any two-conductor system if the travel time across the line, TL, (at the speed of light in the medium) is a significant fraction of a signal’s period T or rise time for periodic or pulse signals, respectively.

LoadLV

LI

GV

GI

LT

L T

speed of light

3[m] 30[cm] 300[ m] 3[ m]

10[nS]* 100[pS]* 1[pS]* 10[f S]*

60 [Hz] 16.7[mS]

1 [kHz] 1[mS]

1 [MHz] 1[ S]

1 [GHz] 1[nS]

1 [THz] 1[pS]

L L L LT T T T

L L L L

Frequency Period

X X

X X X

Transmission line theory

Circuit theory

X * 1.0r r assumed

4 parameters

symbol:

z

Note: We use this schematic to represent a general transmission line, no matter what the actual shape of the conductors.

Transmission Lines (cont.)

4 parameters

C = capacitance/length [F/m]

L = inductance/length [H/m]

R = resistance/length [/m]

G = conductance/length [S/m]

Four fundamental parameters that characterize any transmission line:

z These are “per unit length” parameters.

capacitance between the two wires

inductance due to stored magnetic energy

resistance due to the conductors

conductance due to the filling material between the wires

Transmission Lines (cont.)

Circuit Model

Circuit Model:

Rz Lz

Gz Cz

z

z z

z

Coaxial Cable

Example (coaxial cable)

r a

bz

0

0

2F/m

ln

ln H/m2

rCba

bL

a

2S/m

ln

1 1 1/m

2 2

d

m

Gba

Ra b

2

m

(skin depth of metal)

d = conductivity of dielectric [S/m].

m = conductivity of metal [S/m].

Coaxial Cable (cont.)

Overview of derivation: capacitance per unit length

l

-l

+

++

+

++

-

---

- -

E

1QC

V z V

0 0

ln2 2

b b

r ra a

bV E d d

a

02

ln

rCba

Coaxial Cable (cont.)

Overview of derivation: inductance per unit length

y

E

x

Js

1L

I z

0 0

0

2

ln2

b b

a a

Iz H d z d

I bz

a

0 ln2

bL

a

Coaxial Cable (cont.)

Overview of derivation: conductance per unit length

2

ln

dGba

d

C G

d

RC Analogy:

2

lnC

ba

Coaxial Cable (cont.)

Relation between L and C:

0 02F/m ln H/m

2ln

r bC L

b aa

0 0 rLC

Speed of light in dielectric medium: 1

dc

2

1

d

LCc

Hence: This is true for ALL two-conductortransmission lines.

Telegrapher’s Equations

Apply KVL and KCL laws to a small slice of line:

+ V (z,t)-

Rz Lz

Gz Cz

I (z,t)

+ V (z+z,t)-

I (z+z,t)

zz z+z

( , )KVL : ( , ) ( , ) ( , )

( , )KCL : ( , ) ( , ) ( , )

I z tV z t V z z t I z t R z L z

tV z z t

I z t I z z t V z z t G z C zt

Hence

( , ) ( , ) ( , )( , )

( , ) ( , ) ( , )( , )

V z z t V z t I z tRI z t L

z tI z z t I z t V z z t

GV z z t Cz t

Now let z 0:

V IRI L

z tI V

GV Cz t

“Telegrapher’s Equations (TE)”

Telegrapher’s Equations (cont.)

To combine these, take the derivative of the first one

with respect to z:

2

2

2

2

V I IR L

z z z t

I IR L

z t z

V V VR GV C L G C

t t t

Take the derivative of the first TE with respect to z.

Substitute in from the second TE.

Telegrapher’s Equations (cont.)

2 2

2 2( ) 0

V V VRG V RC LG LC

z t t

The same equation also holds for i.

Hence, we have:

2 2

2 2

V V V VR GV C L G C

z t t t

There is no exact solution to this differential equation, except for the lossless case.

Telegrapher’s Equations (cont.)

2 2

2 2( ) 0

V V VRG V RC LG LC

z t t

The same equation also holds for i.

Lossless case:

2 2

2 20

V VLC

z t

0R G

Note: The current satisfies the same differential equation:

Telegrapher’s Equations (cont.)

2 2

2 20

I ILC

z t

The same equation also holds for i.

2 2

2 2 2

10,

d

V V

z c t

Hence we have

Solution:

,

,.

d d

d

V z t V z c t V z c t

V V

z c t I z t

where and are

func

tions of (Similarly for !)

arbitrary

Solution to Telegrapher's Equations

This is called the D’Alembert solution to the Telegrapher's Equations (the solution is in the form of traveling waves).

2 2

2 2 2

10

d

I I

z c t

Traveling Waves

2 2

2 2 2

1

d

V V

z c t

,d d

V z t V z c t V z c t

Proof of solution:

2

2

22 2

2

,

,

d d

d d d d

V z tV z c t V z c t

zV z t

c V z c t c V z c tt

It is seen that the differential equation is satisfied by the general solution.

General solution:

Example:

z

z0

z

t = 0 t = t1 > 0 t = t2 > t1

z0 + cd t1z0 z0 + cd t2

V(z,t)

Traveling Waves

,d

V z t V z c t

( )V z

Example:

zz0

z

t = 0t = t1 > 0t = t2 > t1

z0 - cd t1 z0z0 - cd t2

V(z,t)

Traveling Waves

,d

V z t V z c t

( )V z

Loss causes an attenuation in the signal level, and it also causes distortion (the pulse changes shape and usually becomes broader).

z

t = 0 t = t1 > 0t = t2 > t1

z0 + cd t1z0 z0 + cd t2

V(z,t)

Traveling Waves (cont.)

(These effects can be studied numerically.)

CurrentV I

RI Lz t

V IL

z t

lossless

,d d

V z t V z c t V z c t

,d d

V z tV z c t V z c t

z

,d d

I z t I z c t I z c t

,d d d d

I z tc I z c t c I z c t

t

(first TE)

Our goal is to now solve for the current on the line.

Assume the following forms:

The derivatives are:

Current (cont.)

d d d d d dV z c t V z c t L c I z c t c I z c t

V IL

z t

d d dV z c t L c I z c t

d d dV z c t L c I z c t

1, , Constant

d

I z t V z tLc

1, , Constant

d

I z t V z tLc

This becomes

Equating terms with the same space and time variation, we have

Hence we have

Constants represent independent DC voltages or currents on the line. Assuming no initial line voltage or Current we conclude Constants =0

1 1d

LLc L L

CLC

Then

0

0

1, ,

1, ,

I z t V z tZ

I z t V z tZ

Define characteristic impedance Z0:

0

LZ

C The units of Z0 are Ohms.

Current (cont.)

Observation about term:

or

0

0

, ,

, ,

V z t Z I z t

V z t Z I z t

General solution:

0

,

1,

d d

d d

V z t V z c t V z c t

I z t V z c t V z c tZ

For a forward wave, the current waveform is the same as the voltage, but reduced in amplitude by a factor of Z0.

For a backward traveling wave, there is a minus sign as well.

Note that without this minus sign, the ratio of voltage to current would be constant rather than varying from point-to-point and over time along the line as is generally the case!

Current (cont.)

Current (cont.)

z

Picture for a forward-traveling wave:

0

,

1,

d

d

V z t V z c t

I z t V z c tZ

+

- ,V z t

,I z t

forward-traveling wave

z

Physical interpretation of minus sign for the backward-traveling wave:

0

,

1,

d

d

V z t V z c t

I z t V z c tZ

+

- ,V z t

,I z t

backward-traveling wave

,I z t

The minus sign arises from the reference direction for the current.

Current (cont.)

Coaxial Cable

Example: Find the characteristic impedance of a coax.

0

0

2F/m

ln

ln H/m2

rCba

bL

a

0

00

ln2

2

ln

r

bL a

ZC

ba

00

0

1 1ln

2r

bZ

a

r a

bz

Coaxial Cable (cont.)

r a

bz

0 0

1 1ln

2r

bZ

a

00

0

0 376.7303

-120

-70

8.8541878 10 [F/m]

= 4 10 [H/m] ( )µ

exact

(intrinsic impedance of free space)

Twin Line

10 0

1 1cosh

2r

dZ

a

d

a = radius of wires

r

10 0

1

F/m cosh H/m2

cosh2

r dC L

d aa

0 0

1 1ln

r

dZ

a

a d

1 2cosh ln 1 ln 2xx x x x

Twin Line (cont.)

coaxial cable

0 300Z

twin line

0 75Z

These are the common values used for TV.

75-300 [] transformer

300 [] twin line 75 [] coax

w

hr

Microstrip Line

0

0

,

,

r

wC w h

hh

L w hw

0 0

1,

r

hZ

w

w h

parallel-plate formulas:

Microstrip Line (cont.)

More accurate CAD formulas:

w

hr

w

hr

0

120

/ 1.393 0.667 ln / 1.444effr

Zw h w h

1 1 11 /

2 2 4.6 /1 12 /

eff r r rr

t h

w hh w

( / 1)w h

( / 1)w h

Note: the effective relative permittivity accounts for the fact that some of the field exists outside the substrate, in the air region. The effective width w' accounts for the strip thickness.

21 ln

t hw w

t

t = strip thickness

Some Comments

Transmission-line theory is valid at any frequency, and for any type of waveform (assuming an ideal transmission line).

Transmission-line theory is perfectly consistent with Maxwell's equations (although we work with voltage and current, rather than electric and magnetic fields).

Circuit theory does not view two wires as a "transmission line": it cannot predict effects such as signal propagation, distortion, etc.