ELEC351 Lecture Notes Set 1There is a tutorial on Monday September 10! You will do the first “workshop problem” in the tutorial. Bring a calculator.

The course web site is:www.ece.concordia.ca/~trueman/web_page_351.htm

• Course outline• Lecture notes• Software: BOUNCE, TRLINE, WAVES

PART 1Waves on Transmission Lines

in the Time Domain

• Time delay• Wave equation and travelling waves• Solving transmission line circuits.

4

Coaxial CableUlaby page 53, 212, 267

Coaxial Cable Transmission Line

Source: Wikipediahttps://en.wikipedia.org/wiki/Coaxial_cable

RG-59A:Outer plastic sheathB:Woven copper shieldC:Inner dielectric insulatorD:Copper core

Transmission Line Properties

• 𝑐𝑐 = capacitance per unit length, F/m• ℓ = inductance per unit length, H/m• 𝑟𝑟 = series resistance per unit length, ohm/m• 𝑔𝑔 = shunt conductance per unit length,

Siemens /m

7

Coaxial Cable from ELEC251

•You should know how to derive these formulas from ELEC251.•If you don’t remember this, you should review it.

Ulaby page 212

Ulaby page 267

Coaxial Cable - LossesShunt Conductance • The dielectric filling a coaxial cable has

conductivity 𝜎𝜎 S/m.• If the center conductor has charge

density 𝜌𝜌ℓ Coul/meter, then the electric field in the dielectric is 𝐸𝐸𝑟𝑟 = 𝜌𝜌ℓ

2𝜋𝜋𝜋𝜋𝑟𝑟V/m

• The voltage between the outer and inner conductor is

𝑉𝑉 = −�𝑏𝑏

𝑎𝑎 𝜌𝜌ℓ2𝜋𝜋𝜋𝜋𝑟𝑟

𝑑𝑑𝑟𝑟 =𝜌𝜌ℓ2𝜋𝜋𝜋𝜋

ln𝑏𝑏𝑎𝑎

Shunt Conductance, continued• Since 𝐽𝐽 = 𝜎𝜎𝐸𝐸, the current density in the

dielectric is

𝐽𝐽𝑟𝑟 = 𝜎𝜎𝐸𝐸𝑟𝑟 =𝜎𝜎𝜌𝜌ℓ

2𝜋𝜋𝜋𝜋𝑟𝑟• The current flowing through the surface of a

cylinder of length 1 meter and radius 𝑟𝑟 is

𝐼𝐼 = 1 meter 𝑥𝑥 �0

2𝜋𝜋𝐽𝐽𝑟𝑟𝑟𝑟𝑑𝑑𝜙𝜙

= �0

2𝜋𝜋 𝜎𝜎𝜌𝜌ℓ2𝜋𝜋𝜋𝜋𝑟𝑟

𝑟𝑟𝑑𝑑𝜙𝜙 = 2𝜋𝜋𝜎𝜎𝜌𝜌ℓ2𝜋𝜋𝜋𝜋

=𝜎𝜎𝜌𝜌ℓ𝜋𝜋

• The resistance of a 1 m length of cable is

𝑅𝑅 =𝑉𝑉𝐼𝐼

=𝜌𝜌ℓ2𝜋𝜋𝜋𝜋 ln 𝑏𝑏𝑎𝑎𝜎𝜎𝜌𝜌ℓ𝜋𝜋

=ln 𝑏𝑏𝑎𝑎2𝜋𝜋𝜎𝜎

• Hence the conductance of a 1 m length of cable is

𝑔𝑔 = 1𝑅𝑅

= 2𝜋𝜋𝜎𝜎

ln𝑏𝑏𝑎𝑎

Series Resistance of Coaxial CableThe Surface Resistance • If a conductor has conductivity 𝜎𝜎𝑐𝑐 and permeability 𝜇𝜇𝑐𝑐 then the ratio of the electric

field to the magnetic field at the surface of the conductor is defined as the “surface resistance”

𝑅𝑅𝑠𝑠 = 𝐸𝐸𝐻𝐻

• It May Be Shown (Ulaby page 341) that 𝑅𝑅𝑠𝑠 = 𝜋𝜋𝜋𝜋𝜇𝜇𝑐𝑐/𝜎𝜎𝑐𝑐

• We will talk about this later in the course. • If we know the magnetic field tangent to the surface

𝐻𝐻, then we can find the electric field as 𝐸𝐸 = 𝑅𝑅𝑠𝑠𝐻𝐻

• If the center conductor carries current 𝐼𝐼 flowing out of the page, then the magnetic field in the coaxial cable is

𝐻𝐻 =𝐼𝐼2𝜋𝜋𝑟𝑟

• What is the series resistance of the outer conductor of a coaxial cable?

Series Resistance of Coaxial Cable, continued• At the surface of the outer conductor 𝑟𝑟 = 𝑏𝑏 the

magnetic field is

𝐻𝐻 =𝐼𝐼

2𝜋𝜋𝑏𝑏• The electric field is

𝐸𝐸 = 𝑅𝑅𝑠𝑠𝐻𝐻 = 𝑅𝑅𝑠𝑠𝐼𝐼

2𝜋𝜋𝑏𝑏• The field is oriented axially along the cable.• The voltage across a 1 meter length of cable is

𝑉𝑉 = electric field 𝑥𝑥 distance

= 𝑅𝑅𝑠𝑠𝐼𝐼

2𝜋𝜋𝑏𝑏𝑥𝑥1 meter = 𝑅𝑅𝑠𝑠

𝐼𝐼2𝜋𝜋𝑏𝑏

• The resistance of the 1 meter length of outer conductor is

𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑟𝑟 =𝑉𝑉𝐼𝐼

=𝑅𝑅𝑠𝑠

𝐼𝐼2𝜋𝜋𝑏𝑏𝐼𝐼

=𝑅𝑅𝑠𝑠2𝜋𝜋𝑏𝑏

• Similarly the resistance of the inner conductor is

𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑜𝑜𝑟𝑟 =𝑅𝑅𝑠𝑠2𝜋𝜋𝑎𝑎

Series Resistance of Coaxial Cable, continued• The resistance of the outer

conductor is

𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑟𝑟 =𝑅𝑅𝑠𝑠2𝜋𝜋𝑏𝑏

• The resistance of the inner conductor is

𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑜𝑜𝑟𝑟 =𝑅𝑅𝑠𝑠2𝜋𝜋𝑎𝑎

• Since the current flows from the generator to the load on the outer conductor and back from the load to the generator on the inner conductor, the conductors are in series and the net resistance for a 1 meter length of cable is𝑟𝑟 = 𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑟𝑟 + 𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑜𝑜𝑟𝑟

=𝑅𝑅𝑠𝑠2𝜋𝜋𝑏𝑏

+𝑅𝑅𝑠𝑠2𝜋𝜋𝑎𝑎

=𝑅𝑅𝑠𝑠2𝜋𝜋

1𝑎𝑎

+1𝑏𝑏 𝑟𝑟 =

𝑅𝑅𝑠𝑠2𝜋𝜋

1𝑎𝑎

+1𝑏𝑏

13

Coaxial CableWe will show that:• The speed of travel on a transmission line is u = 1

ℓ𝑐𝑐m/s

• The “characteristic impedance” of a transmission line is

𝑍𝑍𝑜𝑜 = ℓ𝑐𝑐

ohms

Coaxial Filters: Low Pass Filter

http://www.atlantarf.com/CLPfilter.php

𝑗𝑗𝜔𝜔𝐿𝐿

1𝑗𝑗𝜔𝜔𝜔𝜔

matched

matched

15

Two-conductor transmission lines are also called TEM lines.

“TEM” stands for “transverse electric magnetic”. Both the electric field and the magnetic field vectors lie in the transverse plane, with no axial component.

Hollow tubes such as rectangular waveguide do not support a TEM mode!

Ulaby Fig. 2-4

Transmission LinesReading Assignment:Ulaby Chapter 1 Waves and PhasorsUlaby Chapter 2 Transmission Lines

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Ulaby Table 2-1

Notation:

𝑅𝑅𝑅 = 𝑟𝑟

𝐿𝐿′ = ℓ

𝐺𝐺′ = 𝑔𝑔

𝜔𝜔′ = 𝑐𝑐

Twin Lead Transmission Line

Source: Wikipediahttps://en.wikipedia.org/wiki/Twin-lead

300-ohm twin lead

Balance to unbalance transformer for connecting 300-ohm twin lead to 75 ohm coaxial cable.

Stripline and MicrostripInan, Inan and Said Figure 2.1

Stripline Microstrip

Microstrip Transmission Line

Cross-section of microstripgeometry. Conductor (A) is separated from ground plane (D) by dielectric substrate (C). Upper dielectric (B) is typically air.[1]

[1] Source: Wikipediahttps://en.wikipedia.org/wiki/Microstrip

Microstrip 90° mitred bend. The percentage mitre is 100x/d [1]

Microstrip

Design Formulas for Microstrip

Pozar Fig. 3.25

Characteristic impedance:

Speed of travel:

Effective permittivity:

25

Hairpin Filter in Microstrip

Stripline Transmission Line

[1] Source: Wikipediahttps://en.wikipedia.org/wiki/Stripline

Cross-section diagram of striplinegeometry. Central conductor (A) is sandwiched between ground planes (B and D). Structure is supported by dielectric (C). [1]

40

Transmission Line Circuit

• Account for the capacitance-per-unit-length 𝑐𝑐 and the inductance-per-unit-length ℓ

• Account for the speed of travel• Hence account for the time delay• Find the voltage at the input and the voltage at the

load.

cu

1=

uLT =

Ulaby Fig. 2-5

How do we:

Typical Response of a Transmission Line Circuit

Parameters:

step function from 0 to 10 volts starting at t=0

internal resistance of the generator, use 1 ohm

length of the interconnection path, say 2 cm

“characteristic resistance” of the interconnection path (?????????)Use 50 ohms.

speed of travel of a voltage on the interconnection path (???????) Use 20 cm/ns

input resistance of the load, which is ideally “matched” (𝑅𝑅𝐿𝐿 = 𝑅𝑅𝑐𝑐)but for this example we will use a high-resistance load, 𝑅𝑅𝐿𝐿 = 1000 ohms.

=sV

=sR

=L

== cRZ0

=u

=LR

𝐿𝐿

𝑉𝑉𝐿𝐿𝑅𝑅𝐿𝐿𝑅𝑅𝑠𝑠

𝑉𝑉𝑠𝑠 𝑉𝑉0Transmission Line

𝑅𝑅𝑐𝑐, 𝑢𝑢

Step Response

The slide shows one frame from an animation of the response of the circuit, obtained from the BOUNCE program.

Questions:

• What is the physics of this behavior?• How does it arise from circuit analysis?• How do you solve circuits such as this one?

Time domain?Frequency domain?

• What determines the speed of travel?• What is ‘characteristic resistance’?• How do you design the circuit to avoid this

behavior?