ECE 3317
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ECE 3317. Prof. D. R. Wilton. Note 2 Transmission Lines (Time Domain). Note about Notes 2. Disclaimer:

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Prof. D. R. Wilton

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Prof d r wilton

ECE 3317

Prof. D. R. Wilton

Note 2 Transmission Lines(Time Domain)


Prof d r wilton

Note about Notes 2

Disclaimer:

Transmission lines is the subject of Chapter 6 in the book. However, the subject of wave propagation on transmission lines 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.

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.


Prof d r wilton

Transmission Lines

a

z

b

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

Two common examples:

twin line

coaxial cable

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


Prof d r wilton

Transmission Lines (cont.)

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

coax to twin line matching section

coaxial cable

twin line


Prof d r wilton

Transmission Lines (cont.)

Another common example (for printed circuit boards):

w

r

h

microstrip line


Prof d r wilton

Transmission Lines (cont.)

microstrip line


Prof d r wilton

Transmission Lines (cont.)

E

-

l

-

-

+

+

+

+

+

-

+

-

-

-l

Some practical notes:

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

  • Two-wire (twin) lines do not form a shielded system – more susceptible to noise and interference.

  • The coupling between two-wire lines may be reduced by using a form known as a “twisted pair.”

E

twin line

coax


Prof d r wilton

Transmission Lines (cont.)

Load

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


Prof d r wilton

Transmission Lines (cont.)

symbols:

z

4 parameters

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


Prof d r wilton

Transmission Lines (cont.)

z

Capacitance/m between the two conductors

Inductance/m due to stored magnetic energy

Resistance/m due to the conductors

Conductance/m due to the filling material between the conductors

Four fundamental parameters characterize any transmission line:

These are “per unit length” parameters.

4 parameters

C= capacitance/length [F/m]

L= inductance/length [H/m]

R= resistance/length [/m]

G= conductance/length [S/m]


Prof d r wilton

Circuit Model

Dz

z

RDz

LDz

CDz

GDz

Circuit Model:

Dz

z


Prof d r wilton

Circuit Model (cont.)

z

CDz

CDz

CDz

GDz

GDz

GDz

Dz

Dz

Dz

Dz

Circuit Model:

Dz

RDz

RDz

LDZ

RDz

LDZ

RDz

LDZ

LDZ

CDz

GDz

z


Prof d r wilton

Coaxial Cable

a

z

b

Example: coaxial cable

d = conductivity of dielectric [S/m].

m = conductivity of metal [S/m].

(skin depth of metal)


Prof d r wilton

Coaxial Cable (cont.)

E

-

l

-

-

+

+

+

+

+

-

+

-

-

-l

Overview of derivation: capacitance per unit length


Prof d r wilton

Coaxial Cable (cont.)

y

x

E

Js

Overview of derivation: inductance per unit length


Prof d r wilton

Coaxial Cable (cont.)

Overview of derivation: conductance per unit length

RC Analogy:


Prof d r wilton

Coaxial Cable (cont.)

Relation between L and C:

Speed of light in dielectric medium:

This is true for ALL two-conductor

transmission lines.

Hence:


Prof d r wilton

Telegrapher’s Equations

RDz

LDz

I(z+Dz,t)

I(z,t)

+

V(z+Dz,t)

-

+

V(z,t)

-

CDz

GDz

z

z+Dz

z

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


Prof d r wilton

Telegrapher’s Equations (cont.)

Hence

Now let Dz 0:

“Telegrapher’s Equations (TE)”


Prof d r wilton

Telegrapher’s Equations (cont.)

To combine these, take the derivative of the first one with respect to z:

To obtain an equation in V alone, eliminate I between eqs.:

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

  • Substitute in from the second TE.


Prof d r wilton

Telegrapher’s Equations (cont.)

Hence, we have:

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

The same equation also holds for i.


Prof d r wilton

Telegrapher’s Equations (cont.)

Lossless case:

Note: The current satisfies the same differential equation:

The same equation also holds for i.


Prof d r wilton

Solution to Telegrapher's Equations

Hence we have

Solution:

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

The same equation also holds for i.


Prof d r wilton

Traveling Waves

Proof of solution:

General solution:

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


Prof d r wilton

Traveling Waves

Example:

z

z0

t = t2 > t1

t = 0

t = t1 > 0

V(z,t)

z

z0

z0 + cdt1

z0 + cdt2


Prof d r wilton

Traveling Waves

Example:

z

z0

t = 0

t = t2 > t1

t = t1 > 0

V(z,t)

z

z0 - cdt1

z0 - cdt2

z0


Prof d r wilton

Traveling Waves (cont.)

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

t = 0

V(z,t)

t = t1 > 0

t = t2 > t1

z

z0

z0 + cdt1

z0 + cdt2

(These effects can be studied numerically.)


Prof d r wilton

Current

(first TE)

lossless

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

Assume the following forms:

The derivatives are:


Prof d r wilton

Current (cont.)

This becomes

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

Hence we have

Constants C1, C2 represent time and

space-independent DC voltages or

currents on the line. Assuming no

initial line voltage or current we

conclude C1, C2=0


Prof d r wilton

Current (cont.)

Observation about term:

Define (real) characteristic impedance Z0:

The units of Z0are Ohms.

Then

or


Prof d r wilton

Current (cont.)

General solution:

OR

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!


Prof d r wilton

Current (cont.)

Picture for a forward-traveling wave:

forward-traveling wave

z

+

-


Prof d r wilton

Current (cont.)

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

backward-traveling wave

z

+

-

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


Prof d r wilton

Coaxial Cable

a

z

b

Example: Find the characteristic impedance of a coax.


Prof d r wilton

Coaxial Cable (cont.)

a

z

b

(intrinsic impedance of free space)


Prof d r wilton

Twin Line

d

a = radius of wires


Prof d r wilton

Twin Line (cont.)

These are the common values used for TV.

75-300 [] transformer

75 [] coax

300 [] twin line

twin line

coaxial cable


Prof d r wilton

Microstrip Line

w

r

h

parallel-plate formulas:


Prof d r wilton

Microstrip Line (cont.)

t = strip thickness

More accurate CAD formulas:

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


Prof d r wilton

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.


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