Lesson 2 Transmission Lines Fundamentals - …sdyang/Courses/EM/Lesson02_Slides.pdfLesson 2...
Transcript of 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
Why to discuss transmission lines-2
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.
Why to discuss transmission lines-3
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 ⋅=
Background knowledge-2
Kirchhoff’s laws :
0=∑k
ki
0=∑k
kv
Background knowledge-3
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
Equivalent circuit-2
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
Equivalent circuit-3
The upper and lower conductors of the adjacent short lines are connected respectively, ⇒ shunt capacitor
vdtdCi ⋅=Δ
Equivalent circuit-4
Imperfect conducting materials, ⇒ voltage drop along the conducting line, ⇒ series resistorImperfect insulating materials, ⇒ leakage current between conductors, ⇒ shunt conductor
Equivalent circuit-5
Complete model: {R, L, G, C} as {resistance, inductance, conductance, capacitance} per unit length
Equivalent circuit-6
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=
Transmission line equations-2
),()(),(),( 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
Transmission line equations-3
By KCL:( ) )()( tv
dtdzCti CC ⋅Δ=
),()(),(),( tzzvt
zCtzzitzi Δ+∂∂
Δ+Δ+=
Transmission line equations-4
,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),
Transmission line equations-5
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 ∂
∂−=
∂∂∂
Transmission line equations-6
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
Solutions to the transmission line equations-2
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
Solutions to the transmission line equations-3
( )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
Solutions to the transmission line equations-4
( ) ( )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 ∫ ⎥
⎦
⎤⎢⎣
⎡−=
−+
),(
Solutions to the transmission line equations-5
( ) ( )[ ] ),(),(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 +=== +