Distributed constants

• Lumped constants are inadequate models of extended circuit elements at high frequency.

• Examples are telephone lines, guitar strings, and organ pipes.

• We shall develop the model for an electrical transmission line.

• Also a model for sound traveling in a pipe, and show how they are equivalent.

Ideal transmission line model

A voltage v is applied between two wires comprising a transmission line. A current i+ enters one wire. An equal current i- returns from the other. This voltage and current generate electric and magnetic fields around the wires, shown respectively as dashed and dot-dashed lines in a cross sectional view on the right. In turn, these fields manifest themselves as a series inductance L and a parallel capacitance C per length x of the transmission line, depicted by the circuit component overlay on the left.

i+

i-

+v-

xL

C

L

C

L

C

An infinitesimal length of line

),/1/(1 21 dCjZdLjZ

As dx0, the second term goes to zero, while the constant first term is simply L/C, the inductance and capacitance per unit length of the line. Thus, which is called the characteristic impedance of the transmission line.

dLZjdC

dLZ

dCj

dLjdLZjZ 0

200

20 0

Consider a short segment dx of an infinite line. Its measurable input impedance Z1 is identical to Z2, that of the next segment. Thus we write, then let Z2 = Z1 = Z0 and simplify:

Z2dC

dLZ1

,/0 CLZ

Transmission lines with loss

• Wires (except superconductors) have some resistance R per unit length.

• Likewise, most insulators have some conductance G per unit length.

G C

LR

CjG

LjRZ

0

Characteristic impedanceInfinitesimal line model

Signal propagation on a line (1)

)( )2(

)( )1(

2

1

CjGvdx

di

LjRidx

dv

G C

LR

dv/dx = v2 - v1

i1

di/dx = i1 - i2v1

i2

v2

Take the first derivative of (1),

.))((

, )(

2

2

2

2

vCjGLjRdx

vd

dx

diLjR

dx

vd

then substitute (2) to get,

.)( Similarly, .)( are, solutions The

. so ,))(( Let 22

2

xB

xA

xB

xA eIeIxieVeVxv

vdx

vdCjGLjRn constantpropagatio

Signal propagation on a line (2)IA and IB are related to VA and VB as follows. From eq(1),

).(1

)()(

))((

)()(

so ,)(

0

xB

xA

xB

xA

xB

xA

xB

xA

eVeVZ

eVeVLjR

CjGLjR

eVeVLjR

i

LjRieVeVdx

dv

The transmission line equations for sinusoidal signals are,

),(),( ,)(),(0

xB

xA

tjx

Bx

Atj eVeV

Z

etxieVeVetxv

where the explicit time dependence (usually ignored) is .tje

Traveling waves

The propagation constant = + i where is the attenuation constant and is the phase constant, provides a complete description of a wave on a transmission line. If VB = 0, we have a pure (forward) traveling wave. The amplitude of such a wave is plottedhere over a 1 [m] length of transmission line for = 2 [nepers/m] and = 16[radians/m]. The neper is a dimensionless natural logarithmic unit of measure. Thus, specifies the exponential decay rate of a wave, while specifies its spatial angular frequency.

Boundary conditionsRs

Vs Zl

Vo Vl

x=0 x=l

Transmission line, Zo,

VA is the amplitude of the forward traveling wave so if Rs = Z0, we can write . At the load, the voltage and current are related by the load impedance,

2/sA VV

. )(

)(

0

020 ZZ

ZZeVV

eVeV

eVeVZ

I

VZ

l

llABx

Bx

A

xB

xA

l

ll

Notice that the reverse traveling wave vanishes iff that is if the transmission line and the load impedances match.

,0ZZ l

Standing waves.1 clearly, ; The

0

0

vl

lv ZZ

ZZoefficientflection cvoltage re

If Zl = Z0 at the load, i.e., v = 0, we have seen that no signal energy is reflected. Conversely, if Zl = (open circuit) or 0 (short circuit), v = ±1, respectively; in either case, all the energy is reflected, resulting in a pure standing wave, which over time appears not to move, rather just to oscillate in place. For intermediate values of v, the voltage standing wave ratio VSWR = (1 + |v|)/(1 - |v|) is the ratio between the max and min of the voltage envelope. A plot for v = 0.5 is shown.Figure 2-3 from Matick, Transmission Lines for Digital and Communication Networks, Mcgraw Hill, 1969.

The wave equation

form. thisof is equation lineion transmissThe

function. spatial just the is where,

or , ,dependence timeout the

Factoring . form theof are Solutions

. dimension, onein or ,

22

2

)(22

2

)(22

2

2)(

)()()(

2

22

2

2

2

222

vdx

vd

ekdx

d

ess

k

dx

de

eee

ts

k

xts

k

kxxx

x

xstxst

stkxstkx

Acoustic wave in a pipe (1)

. pressure catmospherisay value,intial its

from by dropsy accordingl pressure the

and ,by increasescylinder in theair

of volumeThe . distance athrough

move area of tubeain piston aLet

ap

dp

dV

Vd

A

. can write, weTherefore constant). is (

pressure total theof portion acoustic therepresents and

, variation thesound,For . Thus,

air. theofstiffness"" theis The

V

dVKpp

p

pdpV

dVKdp

dV

dpVKusbulk modul

a

a

dV

V

A

d

Acoustic wave in a pipe (2)

dx dxx

x

dx

1

A

. so

,But .or

,1 volume

a thereforeand ,1length a hascolumn

displaced theThus .by endother theand

, distance aby column theof end one displacesit ,air ofcolumn short a through passesit As

. area of tubeadown traveling waveaConsider

xKp

V

dVKp

xV

dVx

AdxdVV

xdx

dxx

dxA

Acoustic wave in a pipe (3)

.)( forces these

of sum The air.adjacent by thecolumn theof endeach on exerted is

of forceA air. ofdensity theis where, mass a has alsoair ofcolumn The

21 x

pAdxppAf

Apf

Vm

ii

. gives ngsubstituti and respect to

with thisatingDifferenti equation. waveldimensiona one

theis which ,get we, ngSubstituti

. e therefor, Now

2

2

2

2

2

2

2

2

2

2

2

2

2

2

x

pK

t

p

K

p

xx

x

K

txKp

tx

p

tAdx

tVmaf

dx

Am

f1 f2

Acoustic plane wave propagation

• Same as wave in a pipe:• c2 = K/; where c is the speed of sound.

– c = 332 [m/s] at 0 [degC] at 50% RH.– dc/dT = 0.551 [m/s/degC].

• Solutions are of the form– where s = j for temporal sine waves.– Factoring out time, just as for transmission lines.

,kxstkxst BeAep

,kxkx BeAep

.2

22

2

2

x

pc

t

p

Reflection of sound

• Wave impedance of a medium

• Reflection of a wave normal to an interface between media is,

which is the same as v for transmission lines

.

pZ i

,12

12

ZZ

ZZ

A

B

Homework problem

10 [m]

100 [Ohms]

1 [V] 318 [pF]Vin

10 [MHz]

A 10 [m] length of transmission line with 100 [pF/m] capacitance and 250 [nH/m] inductance in series with a 1 [/m] wire resistance is driven by a 10 [MHz] sine wave from a 1 [V] 100 [Ohm] Thevenin source, and loaded by a 318 [pF] capacitor. Calculate the complex input voltage Vin of the transmission line.

Extra credit: plot the magnitude and phase of the voltage as a function of position on the transmission line.