lesson 2 transmission lines fundamentals - …sdyang/courses/em/lesson02_slides.pdflesson 2...

Lesson 2Transmission Lines Fundamentals

楊尚達 Shang-Da YangInstitute of Photonics TechnologiesDepartment of Electrical EngineeringNational Tsing Hua University, Taiwan

Sec. 2-1Introduction

1. Why to discuss TX lines (distributed circuits)?

2. Criteria of using distributed circuits3. Background knowledge

Why to discuss transmission lines-1

Current electronic technology is based on lumped circuit theory (simple, powerful):

1. Lumped elements (R, C, L, dependent sources) are connected in series or shunt

2. Conducting wires play no role


In fact, elements and wires provide a framework over which electric charges can move and set up the (total) EM vector fields. The behavior of circuit is thus determined.


Distributed circuit (transmission lines) theory is somewhere in between:

1. Can describe some wave properties (wavelength, phase velocity, reflection, …), which are crucial in power transmission, high-speed IC, ….

2. Only deal with scalar quantities (v, i), require no complicated vector analysis

Criteria of using distributed circuits

sigd tt << sigd tt ≈

Finite speed of propagation ncv =

sigt sigt

Example 2-1: Power lines

( )tVtV AA ⋅⋅=′ 602cos)( 0 π

[ ])(602cos)( 0 dBB ttVtV −⋅⋅=′ π vltd =

Lumped circuit is inadequate when(rule of thumb), ⇒

Ttd 01.0>

sec 601

=T

km 50>l

Example 2-2: Interconnection of ICs

Lumped circuit is inadequate when(rule of thumb). Fast CMOS can have !

rise time

ps 1655.2 ≈< dr tt

1-cm silica interconnection

ps 67≈= vltd

ps 100≈rt

Logic 1

Logic 0

Background knowledge-1

Models of linear circuit elements:

iRv ⋅= vdtdCi ⋅= i

dtdLv ⋅=


Kirchhoff’s laws :

0=∑k

ki

0=∑k

kv


Phasors of sinusoidal functions:

( ) { }tjeVtVtv ωφω ⋅=+= Recos)( 0

( )φφφ sincos00 jVeVV j +==

{ }tjtj eVjedtdVtv

dtd ωω ω ⋅=

⎭⎬⎫

⎩⎨⎧ ⋅= ReRe)(

nn

n

jdtd )( ω→ Derivatives → Algebraic multiplication

Sec. 2-2Equivalent Circuit and Equations of Transmission Lines

1. Geometry of typical TX lines2. Equivalent circuit3. TX line equations4. Solutions

Geometry of typical transmission lines

Two long conductors, separated by some insulating material

across, along the conductors ),( 0 tzv ),( 0 tzi±

Equivalent circuit-1

Since the voltage, current can vary with z, ⇒ use of distributed circuit model, i.e. a TX line consists of infinitely many coupled lines of infinitesimal length Δz


Currents on a short line set up magnetic field (Ampere’s law ), ⇒ magnetic fluxTime-varying current (flux), ⇒ voltage changes along the line (Faraday’s law) to counter the change of current (Lenz’s law), ⇒ series inductor

idtdLv ⋅=Δ


The upper and lower conductors of the adjacent short lines are connected respectively, ⇒ shunt capacitor

vdtdCi ⋅=Δ


Imperfect conducting materials, ⇒ voltage drop along the conducting line, ⇒ series resistorImperfect insulating materials, ⇒ leakage current between conductors, ⇒ shunt conductor


Complete model: {R, L, G, C} as {resistance, inductance, conductance, capacitance} per unit length


By the equivalent circuit, the behavior of vector EM fields can be described by scalar voltage and current.Values of R, L, G, C depend on the geometry and materials of the transmission line.

Transmission line equations-1

Assume R=0, G=0: by KVL

),()(),(),( tzit

zLtzvtzzv∂∂

Δ−=Δ+

( ) )()( tidtdzLtv LL ⋅Δ=


),()(),(),( tzit

zLtzvtzzv∂∂

Δ−=Δ+

,0Let →Δz

),(),( tzit

Ltzvz ∂

∂−=

∂∂ (2.1)

),()(),(),( tzit

zLtzvtzzv∂∂

Δ−=−Δ+⇒

),()(),(),( tzit

Lz

tzvtzzv∂∂

−=Δ

−Δ+⇒

⇒

1st-order PDE with 2 unknown functions v and i


By KCL:( ) )()( tv

dtdzCti CC ⋅Δ=

),()(),(),( tzzvt

zCtzzitzi Δ+∂∂

Δ+Δ+=


,0Let →Δz

),()(),(),( tzzvt

zCtzzitzi Δ+∂∂

Δ+Δ+=

),(),( tzvt

Ctziz ∂

∂−=

∂∂ (2.2)

),()(),(),( tzzvt

zCtzzitzi Δ+∂∂

Δ=Δ+−⇒

),()(),(),( tzzvt

Cz

tzzitziΔ+

∂∂

=Δ

Δ+−⇒

⇒

1st-order PDE with 2 unknown functions v and i

Taking for both sides of eq. (2.2),


Taking for both sides of eq. (2.1),z∂∂

⇒∂∂

−=∂∂ ),,(),( tzi

tLtzv

z),(),(

2

2

2

tzitz

Ltzvz ∂∂

∂−=

∂∂

⇒∂∂

−=∂∂ ),,(),( tzv

tCtzi

z

t∂∂

),(),( 2

22

tzvt

Ctzitz ∂

∂−=

∂∂∂


Combine the two equations:

),(),( 2

2

2

2

tzvt

LCtzvz ∂

∂=

∂∂

),(),( 2

2

2

2

tzit

LCtziz ∂

∂=

∂∂

),(),( 2

2

2

2

tzvt

LCtzvz ∂

∂=

∂∂

⇒

2nd-order PDE with 1 unknown function v

(2.3)

(2.4)

Solutions to the transmission line equations-1

),(),( 2

2

2

2

tzvt

LCtzvz ∂

∂=

∂∂Compare

),(),( 2

22

2

2

txux

ctxut ∂

∂=

∂∂with

(TX line eq.)

(1-D wave eq.)

LCvp

1≡),( tzv ~ a wave propagating with velocity


D’Alembart’s solution to wave equation:

any function of variable )(⋅fpvzt ±=τ

( )pvztf − ~ distortion-free wave traveling in the+z direction with velocity pv


( )pvztf + ~ distortion-free wave traveling in the-z direction with velocity pv

( ) ( )pp vztfvztftzv ++−= −+),(

General solution to the voltage (superposition):

Can be totally different functionsdetermined by BCs and ICs.Their superposition may have “distortion”.

(2.6)

Comments

Phase velocity only depends on

the insulating materials,LC

vp1

=

though L, C also depend on the geometry of the lines


( ) ( )pp vztfvztftzv ++−= −+),(Substitute voltage solution

into the 1st-order PDE: ),(),( tzit

Ltzvz ∂

∂−=

∂∂

( ) ( )tiL

ddf

vddf

vzv

pp ∂∂

−=+−=∂∂

⇒−+

ττ

ττ 11

( ) ( ) td

dfd

dfLv

tzv

Ltzi

p

∂⎥⎦

⎤⎢⎣

⎡−=∂

∂∂

−=⇒ ∫∫−+

ττ

ττ11),(

⇒=∂∂ ,1

tτ

Q( ) ( ) τ

ττ

ττ d

ddf

ddf

LCtzi ∫ ⎥

⎦

⎤⎢⎣

⎡−=

−+

),(


( ) ( )[ ] ),(),(1),(0

tzitzivztfvztfZ

tzi pp−+−+ +=+−−=⇒

CLZ =0

( ) ( ) ),(),(),( tzvtzvvztfvztftzv pp−+−+ +=++−=

Characteristic impedance(not the resistance of the conductor or insulator)

( )( )

( )( )

( )( )tzi

tzvtzitzv

tzitzvZ

,,

,,

,,

0 ≠−== −

−

+

+

Physical meaning

Example 2-3:

Infinitely long line, no reflected wave

TX line ~ a load of resistance CLZ =0

( )tzv ,−

00

0)0( VRZ

ZVtvs

ss +=== +

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