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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

Note 2 Transmission Lines(Time Domain)

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.

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.

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

coax to twin line matching section

coaxial cable

twin line

Another common example (for printed circuit boards):

w

r

h

microstrip line

microstrip line

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

• 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.

symbols:

z

4 parameters

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

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]

Dz

z

RDz

LDz

CDz

GDz

Circuit Model:

Dz

z

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

a

z

b

Example: coaxial cable

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

m = conductivity of metal [S/m].

(skin depth of metal)

E

-

l

-

-

+

+

+

+

+

-

+

-

-

-l

Overview of derivation: capacitance per unit length

y

x

E

Js

Overview of derivation: inductance per unit length

Overview of derivation: conductance per unit length

RC Analogy:

Relation between L and C:

Speed of light in dielectric medium:

This is true for ALL two-conductor

transmission lines.

Hence:

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:

Hence

Now let Dz 0:

“Telegrapher’s Equations (TE)”

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.

Hence, we have:

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

The same equation also holds for i.

Lossless case:

Note: The current satisfies the same differential equation:

The same equation also holds for i.

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.

Proof of solution:

General solution:

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

Example:

z

z0

t = t2 > t1

t = 0

t = t1 > 0

V(z,t)

z

z0

z0 + cdt1

z0 + cdt2

Example:

z

z0

t = 0

t = t2 > t1

t = t1 > 0

V(z,t)

z

z0 - cdt1

z0 - cdt2

z0

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.)

(first TE)

lossless

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

Assume the following forms:

The derivatives are:

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

Define (real) characteristic impedance Z0:

The units of Z0are Ohms.

Then

or

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!

Picture for a forward-traveling wave:

forward-traveling wave

z

+

-

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.

a

z

b

Example: Find the characteristic impedance of a coax.

a

z

b

(intrinsic impedance of free space)

d

These are the common values used for TV.

75-300 [] transformer

75 [] coax

300 [] twin line

twin line

coaxial cable

w

r

h

parallel-plate formulas:

t = strip thickness