ECE 451 Advanced Microwave Measurements

Transmission Lines

Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois [email protected]

ECE 451 –Jose Schutt‐Aine 1 Maxwell’s Equations

B E  Faraday’s Law of Induction t

D HJ   Ampère’s Law t

D   Gauss’ Law for electric field

B 0 Gauss’ Law for magnetic field

Constitutive Relations

BH  D   E

ECE 451 –Jose Schutt‐Aine 2 Why Transmission Line?



 z

Wavelength : 

propagation velocity  = frequency

ECE 451 –Jose Schutt‐Aine 3 Why Transmission Line?

In Free Space

At 10 KHz : = 30 km

At 10 GHz : = 3 cm

Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength.

ECE 451 –Jose Schutt‐Aine 4 Justification for Transmission Line

Let d be the largest dimension of a circuit

circuit

z



If d << , a lumped model for the circuit can be used

ECE 451 –Jose Schutt‐Aine 5 Justification for Transmission Line

circuit

z



If d ≈ , or d >  then use transmission line model

ECE 451 –Jose Schutt‐Aine 6 Telegraphers’ Equations

L I

+

V C -

z L: Inductance per unit length.

C: Capacitance per unit length. VI V L Assume jLI zt time-harmonic z dependence IV I C jt jCV zt VI,~ e z

ECE 451 –Jose Schutt‐Aine 7 TL Solutions

L I

+

V C -

z

2 2V VV   I  2 LCV  jLjLjCV 2 zz  z  z z

2 2 I II   V  2CLI  jCjLjCI 2 zz  z  z z

ECE 451 –Jose Schutt‐Aine 8 TL Solutions (Frequency Domain)

z

Zo  forward wave

backward wave

Forward Wave Backward  Wave  LC jz jz Vz() Ve Ve

L VVjz jz Z  Iz() e e o ZZ C oo   Forward Wave Backward Wave

ECE 451 –Jose Schutt‐Aine 9 TL Solutions 1 Propagation constant  LC Propagation velocity v  LC v L Wavelength   Characteristic impedance Z  f o C

Forward Wave  Backward Wave 

Vzt(,) V cos t z V cos t  z

VV Izt(,) cos t z cos t z ZZ oo Forward Wave Backward Wave

ECE 451 –Jose Schutt‐Aine 10 Reflection Coefficient

At z=0, we have V(0)=ZRI(0) But from the TL equations:

VVV(0)  ZR VV   VV VV I(0)  Zo ZooZ

Which gives VV  R 

ZR  Zo where R is the load reflection coefficient ZR  Zo

ECE 451 –Jose Schutt‐Aine 11 Reflection Coefficient

-If ZR = Zo, R=0, no reflection, the line is matched

-If ZR = 0, short circuit at the load, R=-1

-If ZR inf, open circuit at the load, R=+1

V and I can be written in terms of R

jz jz jz 2 jz Vz() V  eR e Vz() Ve  1 R e  V Ve jz  jz jz   2 jz  Iz() eR e Iz() 1 R e  Zo Zo

ECE 451 –Jose Schutt‐Aine 12 Generalized Impedance

Vz() eejz jz Zz() Z R o  jz jz Iz()  eR e 

Impedance  ZRo jZtan  l  Zl Zo   transformation Z  jZtan  l  oR  equation

ECE 451 –Jose Schutt‐Aine 13 Generalized Impedance

- Short circuit ZR=0, line appears inductive for 0 < l < /2

Z()ljZl o tan

- Open circuit ZR  inf, line appears capacitive for 0 < l < /2 Z Zl() o j tan l -If l = /4, the line is a quarter-wave transformer Z 2 Zl()o ZR

ECE 451 –Jose Schutt‐Aine 14 Generalized Reflection Coefficient

Backward traveling wave at z Vz() ()z b Forward traveling wave atz Vzf ( ) 

 jz Ve V 22jz jz ()zee jz  R Ve V

Reflection coefficient ()le2 jl transformation equation R

1() z Z()zZ o Zz() Zo ()z 1() z Z()zZ o

ECE 451 –Jose Schutt‐Aine 15 Voltage Standing Wave Ratio (VSWR)

jz2 jz We follow the magnitude of the Vz() Ve  1 R e voltage along the TL

jz 22 jz  jz Vz() Ve 1RR e V 1 e Maximum and minimum magnitudes given by

Vmax  1R 

Vmin  1R 

ECE 451 –Jose Schutt‐Aine 16 Voltage Standing Wave Ratio (VSWR)

Define Voltage Standing Wave Ratio as:

V 1  VSWR max R Vmin 1 R

It is a measure of the interaction between forward and backward waves

ECE 451 –Jose Schutt‐Aine 17 VSWR – Arbitrary Load

Shows variation of amplitude along line

ECE 451 –Jose Schutt‐Aine 18 VSWR –For Short Circuit Load

Voltage minimum is reached at load

ECE 451 –Jose Schutt‐Aine 19 VSWR –For Open Circuit Load

Voltage maximum is reached at load

ECE 451 –Jose Schutt‐Aine 20 VSWR –For Open Matched Load

No variation in amplitude along line

ECE 451 –Jose Schutt‐Aine 21 Application: Slotted‐Line Measurement

‐ Measure VSWR = Vmax/Vmin ‐ Measure location of first minimum

ECE 451 –Jose Schutt‐Aine 22 Application: Slotted‐Line Measurement

At minimum, (z ) pure real R

2 jd min Therefore, ()demin R R

2 jd min So, RR e

VSWR 1 VSWR 1 2 jd min Since R then R e VSWR 1 VSWR 1

1 R and ZZRo  1 R

ECE 451 –Jose Schutt‐Aine 23 Summary of TL Equations Voltage Current V jz2 jz Iz() ejz 1 e2 jz Vz() Ve  1 R e  R  Zo

 ZRo jZtan  l  Impedance Transformation  Zl Zo    ZoR jZtan  l 

2 jl Reflection Coefficient Transformation  ()leR

1()  z Reflection Coefficient –to Impedance  Zz() Z o 1()  z

Z()zZ Impedance to Reflection Coefficient  ()z o Z()zZ o

ECE 451 –Jose Schutt‐Aine 24 Example 1

 Ig

Zg =(10+j10)  + Z in Zo = 50  ZR=(30+j40)  -  Vg =100  V

d=l d=0 Consider the transmission line system shown in the figure above.

(a) Find the input impedance Zin.

ZRo Z (30 j 40) 50 R  0.5  90  ZZR o (30 j 40) 50

4  j  .0.725 2 jl (dl )R e  0.5  90 e  0.5  72

ECE 451 –Jose Schutt‐Aine 25 Example 1 –Cont’

1()l  10.572    ZZ50  in o   1()l  10.572   

Zjin 64.361 51.738 39.86 50.54 

(b) Find the current drawn from the generator

Vg 100 0 I g  1.552 39.11 A ZZgin(10 j 10)(39.86  j 50.54)

I g 1.207jA 0.981

ECE 451 –Jose Schutt‐Aine 26 Example 1 –Cont’

c) Find the time‐average power delivered to the load 1 Vl( ) ZI 64.361  51.73 1.5562 39.11 100.159 12.62 in g 2 11 PVlI Real ( )*  Real 100.159  12.62  1.5562  39.11  22g

1 PVlIWReal ( )*  48.26 2 g

ECE 451 –Jose Schutt‐Aine 27 Example 1 –Cont’

R=50  + Zo = 50  L=1 H - V g C=100 pF

d=l d=0

In the system shown above, a line of characteristic impedance 50  is terminated by a series R, L, C circuit having the values R = 50 , L = 1 H, and C = 100 pF.

(d) Find the source frequency fo for which there are no standing waves on the line.

ECE 451 –Jose Schutt‐Aine 28 Example 1 –Cont’

No standing wave on the line if ZR = R = Zo which occurs at the resonant frequency of the RLC circuit. Thus, 

11108 fo  Hz15.92 MHz 2 LC 21010 610 2

ECE 451 –Jose Schutt‐Aine 29