Co-Axial Cable Analysis

Construction Details

Question 1

What is the fundamental equation relating the magnetic field surrounding a conductor and the current in the conductor?

i H dS

Ampere’s Law:

Internal Magnetic Field

and

Question 2

By what method is the total flux passing through a given area computed?

A

B dA

Total Flux in Short Section

Inductance per Unit Length

Incremental Inductance:

Definition of inductance:

Ratio of total flux to total linked current causing the flux.

Question 3

What is the fundamental equation relating the electric field surrounding a charged body and the charge on that body?

A

q D dA

Gauss’s Law:

Internal Electric Field

Question 4

What is the fundamental equation relating the electric field in the region about two bodies and the potential difference (voltage) between those two bodies?

a

ba

b

v E dS

Total Voltage

E Field Relation to Voltage

We showed previously that:

..and since...

..therefore...

Incremental Capacitance

Capacitance per Unit Length

Definition of capacitance:

Ratio of total charge to the voltage resulting from the charge.

Question 5

What is the fundamental relationship between the magnitudes of Electric and Magnetic fields when Energy is propagating through a medium?

HE where

is the intrinsic impedance of the dielectric material

Ohm’s Law

Induced Co-ax VoltageWe previously determined that the magnetic field

strength associated with a current in the co-ax is given by:

r

irH

2)( , thus ( )

2

iE r

r

..and the voltage between inner and outer conductor will be:

Characteristic ImpedanceWe see now that the ratio of voltage to current associated

with energy propagating in a coaxial cable is:

.. but, from our previous discussion of inductance and capacitance per unit length,

20

2

0

1

0

1

0

1

0

0 ln2

1

ln

2

ln2

Zr

r

r

r

r

r

C

L

00

0

LvZ

i C

Recap

0 1 10

0 0 0

1 1ln ln

2 2

L r rvZ

i C r r

1

0

2

2ln

iH r

r

i vE r H r

r rr

r

Question 6

What is the fundamental equation relating the Power density flowing through a region and the fields in that region?

HEP

Power TransferThe Poynting Vector is used to represent the

power transferred by electromagnetic fields:

HEP

If the fields are perpendicular, as they are in this case, then

2

2

r

irHrErP in watts per square meter

Question 7

How do we compute the total power flowing through a surface if we know the power density at all points on that surface?

dArPPr

r

T 1

0

Power Transfer (cont)

We’ll integrate using a ring of

thickness dr ...

ivZiP

r

ri

r

driurdr

r

iuP

T

r

r

r

r

T

02

0

122

2

2

ln2

1

22

1

2

1

0

1

0

To find the total power transfer (watts) we must integrate P(r) over the entire cross section of the dielectric, between r0 and r1. . .

Power Flow Through Dielectric

2

0

1

0

1

1

ln2 2

ln

)()()(r

rr

vi

r

i

rr

r

vrHrErP

1

0

1

0

21

ln2

)(2

0

1

r

r

r

r

T rdrr

r

r

vidArPP

vir

dr

r

r

vir

r

1

0

0

1ln

Traveling WavesIf one applies Kirchhoff’s Laws to a differential length of transmission line having Inductance and Capacitance per unit length of L0 and C0 respectively, and excited by a source with radian frequency , solution of the resulting differential equations yields a solution for the voltage function of the form:

, j t xiV t x V e

Vi represents a complex amplitude.

The + preceding the t term indicates that solutions will exist in complex conjugates to yield a real valued time function. As per our long standing convention, we will only explicitly carry the + term through our derivations.

The + preceding the x term indicates solutions exist representing waves traveling in the positive and negative directions. Let’s see how this works.

Traveling Waves (cont)

Consider the solution having the phase term (t-x). This represents the instantaneous phase of the voltage function.

Now consider The waveform peaks, where the instantaneous phase equals 2N (or any point of constant phase). If we solve for x, we get

2 2N

N tx N t

Two important observations can be made.

1.The distance between adjacent peaks (wavelength) is

2.The position of the peaks is increasing at a velocity

2 pv

f

0 0

1 1pv

L C