sign in sign up
<<
Home , Port (circuit theory), Stripline, Unbalanced line

1. INTRODUCTION TO DISTRIBUTED CIRCUIT DESIGN . circuit elements and analysis • Frequency bands [COLLIN 1.1] • RF circuit analysis [COLLIN 1.3] . Transmission lines • types • Propagation equations [COLLIN 3.1] [POZAR 3.1] forward and reverse o lossless lines propagating waves o lossy lines attenuation o low-loss lines • Reflection coefficient [POZAR 3.3] impedance of transmission lines • Power and losses: return loss [POZAR 3.3] • standing wave ratio [POZAR 3.3] • Impedance [POZAR 3.3] • Generator mismatch [POZAR 3.6] • Smith Chart [COLLIN 5.1] • [COLLIN 5.2-5.5] o lumped elements o Single- matching o Double-stub matching o Quarter-wave impedance

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 1 EETAC-UPC 1. INTRODUCTION TO DISTRIBUTED CIRCUIT DESIGN . Transmission line design • Balanced and unbalanced lines • Homogeneous and non-homogeneous lines • Coupled lines [POZAR 8.6] • Line design . Application notes • Coaxial cables • Connectors

[COLLIN] R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York) [POZAR] D.M. Pozar, Microwave Engineering, Addison-Wesley Publishing Company, 2nd Edition, 1993 (Reading, Massachusetts)

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 2 EETAC-UPC GLOSSARY •  : attenuation constant [m-1] •  : phase constant [rad·m-1]

• Cd : distributed per unit length [F/m] -12 • 0 : electric permittivity of vacuum [8.85·10 F/m] • f0 : frequency [Hz] •  : propagation constant [m-1] • G : distributed conductance per unit length [S/m] • i(z,t) : current in time domain [V] + • I0 : current amplitude of progressive wave at z=0 [A] • l : transmission line length [m]

• Ld : distributed per unit length [H/m] •  : wavelength [m] -7 • 0 : magnetic permeability of vacuum [4·10 H/m] •  : angular frequency [rad/s] • R : distributed resistance per unit length [/m] • T : period [s]

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 3 EETAC-UPC GLOSSARY

• RS : surface resistivity [/square] • S : skin depth [m] •  : conductivity [Sm]

• d : dielectric conductivity [Sm] • tan  : loss tangent [adim] • RL : return loss [dB] •  : (voltage) reflection coefficient [adim]

• G : (voltage) generator reflection coefficient [adim] • IN : (voltage) reflection coefficient at input port [adim] • L : (voltage) load reflection coefficient [adim] • v(z,t) : voltage in time domain [V]

• VG : voltage at generator [V] + • V0 : voltage amplitude of progressive wave at z=0 [V] • vp : phase velocity [m/s] • VSWR : Voltage Standing Wave Ratio [adim]

• ZG : generator impedance [] • ZIN : impedance at the input port of the transmission line [] • ZL : load impedance [] • Z0 : transmission line characteristic impedance [] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 4 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS Frequency bands

• International classification of the frequency bands.

• Radar classification of frequency bands. The old one (WW II) is still widely used.

[Tables taken from: R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York)]

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 5 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis • At frequencies where  is several orders of magnitude larger than the greatest dimension of the circuit or system: . To transmit, receive, and/or process data the basic building blocks are capacitors, inductors, resistors, and . . Loop currents and node are enough to analyse the circuits. . To analyse the circuits no propagation effects have to be considered: the delay in the propagation of signals at different points in the circuit is negligible compared with the period of the applied signal. . Lumped circuit models are valid.

• At microwave frequencies  is compared with the circuit dimensions and: . Propagation effects can not be ignored: there is a delay in the propagation of signals among different points in a circuit. . There are distributed and in the circuit. . There is an increase in the impedance of terminals and connectors. . Unshielded circuits with dimensions compared with  become effective radiators. . Distributed circuit models are used.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 6 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis

• Wave propagation along a line considering the propagation delays: L: line length t = 0 Source voltage:

v: propagation speed vG t A cos0t propagation delay v (t) G Voltage:   z  vz,t  A cos0 t     v  z = 0 z • Some remarks: . Wave propagation along a line considering the propagation delays. . Each point in the line has a different voltage/current at the same time t. . Periodicity in time or period T. 2 2 v T     . Spatial periodicity or wavelength . 0 0 v f0 . Dimensions use to be defined with respect to .

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 7 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis

Example: Propagation at Low Frequencies. Consider a circuit having a transmission line length of 0.003 (3 lines, 0.001-length each) fed with a sinusoidal wave of 1 GHz.

V1 V2 V3 V4 VtSine R TRANSIENT SRC1 R TLIN TLIN TLIN R2 Vdc=0 V R1 TL1 TL2 TL3 R=50 Ohm Tran Amplitude=1 V R=50 Ohm Z=50.0 Ohm Z=50.0 Ohm Z=50.0 Ohm Tran1 Freq=1 GHz E=0.36 E=0.36 E=0.36 StopTime=10.0 nsec Delay=0 nsec F=1 GHz F=1 GHz F=1 GHz MaxTimeStep=1.0 psec

600 600 600 600

400 400 400 400

200 200 200 200

0 0 0 0 V4, mV V1, mV -200 V2, mV -200 V3, mV -200 -200

-400 -400 -400 -400

-600 -600 -600 -600 01024 6 8 0102468 0102468 0102468 time, nsec time, nsec time, nsec time, nsec

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 8 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis

Example: Propagation at High Frequencies. Consider a circuit having a transmission line length of 15 (3 lines, 5-length each) fed with a sinusoidal wave of 1 GHz.

V1 V2 V3 V4 VtSine R TRANSIENT SRC1 R TLIN TLIN TLIN R2 Vdc=0 V R1 TL1 TL2 TL3 R=50 Ohm Tran Amplitude=1 V R=50 Ohm Z=50.0 Ohm Z=50.0 Ohm Z=50.0 Ohm Tran1 Freq=1 GHz E=1800 E=1800 E=1800 StopTime=10.0 nsec Delay=0 nsec F=1 GHz F=1 GHz F=1 GHz MaxTimeStep=1.0 psec

600 600 600 600

400 400 400 400

200 200 200 200

0 0 0 0 V4, mV V1, mV V1, -200 mV V2, -200 mV V3, -200 -200

-400 -400 -400 -400

-600 -600 -600 -600 01024 6 8 01024 6 8 01024 6 8 01024 6 8 time, nsec time, nsec time, nsec time, nsec

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 9 EETAC-UPC TRANSMISSION LINES Transmission line types • Transmission lines are physical devices whose purpose is to guide electromagnetic waves (carry RF power) from one place to another. • They are capable of guiding TEM waves (TEM waves can only exist in structures containing two or more separated conductors). • Two-wire transmission lines are inefficient for transfering electromagnetic energy at high frequencies due to the lack of confinement in all directions. • Coaxials are more efficient than two-wire lines in those cases. two-wire ribbon line (twin lead)

shielded pair air coaxial line

flexible coaxial line

[Images from: http://www.techlearner.com/Apps/TransandGuides.pdf] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 10 EETAC-UPC TRANSMISSION LINES Transmission line types • There are lots of planar structures used as transmission lines. Metallic parts are supported by dielectrics (fiberglass, ceramics, foams,...).

coplanar transmission line

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 11 EETAC-UPC TRANSMISSION LINES Transmission line types

• are the most efficient. They are fabricated with just one conductor. Waveguides do not support TEM waves. • Two-wire lines are less bulky and less expensive than waveguides.

waveguides

[Image from: http://www.techlearner.com/Apps/TransandGuides.pdf]

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 12 EETAC-UPC TRANSMISSION LINES Propagation equations

• Field modes: electromagnetic fields configurations supported by a structure. • A coaxial transmission line supports a TEM mode (electric field orientation, magnetic field orientation, and energy propagation direction for a triad). z z  H + iz,t L vz,t b  E -

L a abstract model (ideal transmission line)

electromagnetic field distribution   physical structure vz,t  Ez,t dl (TEM mode)     i z,t   H z,t dl

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 13 EETAC-UPC TRANSMISSION LINES Propagation equations: lossless lines • The knowledge of voltage and current waves propagating along the transmission line allows the use of a distributed circuit model to analyze its performance. • The model represents an infinitesimally short segment of the transmission line • This model is convenient to explore properties of lines without knowing the fields in detail. However, the structures should be analyzed in detail if accurate performances have to be known. iz,t iz  z,t iz,t iz  z,t + + + L + vz,t vz  z,t vz,t vz  z,t C - - - - L  L z z d or dz in case that z  0 C  Cd z

• Ld and Cd are the distributed inductance [H/m] and capacitance [F/m] associated to the coaxial structure and materials. No losses are assumed in this example (meaning that there is no distributed resistance). Radiofrequency Engineering C. Collado, J.M. González-Arbesú 14 EETAC-UPC TRANSMISSION LINES Propagation equations: lossless lines • Applying Kirchhoff’s voltage and current laws: iz,t iz  z,t i z,t  vz,t  Ld z  vz  z,t + + t L  vz,t vz  z,t  v z  z,t C i z,t  C z  iz  z,t - - d t L  Ld z

C  Cd z • Dividing by z and taking • Considering sinusoidal steady-state condition the limit z 0: (cosine based phasors) (TRANSIENTS NOT vz,t iz,t CONSIDERED): V  L  j L I z d t z d iz,t vz,t I  C   j C V z d t z d Radiofrequency Engineering C. Collado, J.M. González-Arbesú 15 EETAC-UPC TRANSMISSION LINES Propagation equations: lossless lines  2V  - 2 V • The wave equations 2  z being: can be solved   Ld Cd  2 I simultaneously:  - 2 I z 2 phase constant     V0  V0   j z   j z being: I0  I0   V z V0 e  V0 e Z Z • Solutions are: 0 0    j z  jz I z  I0 e  I0 e Ld and: Z0  Cd characteristic impedance

• Meaning that there are waves propagating in V  V  opposite directions along the transmission line (positive and negative waves).

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 16 EETAC-UPC   TRANSMISSION LINES Propagation equations: lossless lines  • Wave propagation solutions in time domain are:      vz,t V0 cos t  z  argV0  V0 cos t  z  argV0 

     iz,t  I0 cost  z  argI0  I0 cost  z  argI0

the wave propagation on the line lengths l are given in terms means a delay (in fact a phase delay) of  or in degrees (l)

• Wavelength and phase velocity on the line are: vp does not change with 2  1 frequency: NO DISTORTION (each frequency component of   vp    f   L C a signal travels at the same vp d d along the line) • Do not forget… that the electrical model parameters depend on line geometry.  E distributed L electrical  physical model d model structure  parameters H parameters Cd Z0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 17 EETAC-UPC Take your time…

Characteristic impedance. Given the solutions of the voltage and current wave equations derived for a two-wire lossless transmission line differential equations:

 V    j Ld I   j z   j z z V z V0 e  V0 e I    j z  jz I z  I0 e  I0 e   j CdV z  being:   Ld Cd

Find the characteristic impedance of the line and the relation between the voltage and current waves.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 18 EETAC-UPC Take your time…

Solution: Characteristic impedance.  Sustituting the solution for the current into the differential equation for the voltage:   V   j Ld I z     j z  j z   j z  jz  j V0 e  j V0 e   jLd I0 e  I0 e  

  j z   j z V z  V0 e  V0 e    j z  jz I z  I0 e  I0 e     V0 V0 V0 I0      Ld Ld Z0 C d characteristic  impedance  V0 I0   Z0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 19 EETAC-UPC Take your time…

Lossy two-wire transmission line. In previous slides we presented the model of a two- wire lossless transmission line. Suggest a model for a lossy two-wire transmission line (e.g. lossy ).

metallic shield and core (lossy) dielectric insulator (lossy)

Hints: 1 Losses in metals are characterized by their electrical conductivity  (units: -1m=Sm) or R  by their surface resistivity R (units: /square).R represents the mean power S  S S S absorbed by a unit area (1 m2) and can be used on planar surfaces or surfaces with  curvature radius smaller than the skin depth  . Currents on metals flow mainly in their 1 S  outer “skin” an a level called skin depth. S f

Losses in dielectrics are characterized by their loss tangent tan . It represents the  '' tangent of the angle in the complex plane between the electric field resistive losses and tan  d its lossless reactive component.  '

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 20 EETAC-UPC Take your time…

Solution. Lossy two-wire transmission line.

depends on 

and/or RS depends on

tan S R Ld

G Cd

z

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 21 EETAC-UPC Take your time…

Telegrapher’s equations. Derive the propagation equations for a lossy two-wire transmission line having the lumped-circuit model of the figure. The resulting set of differential equations are called the telegrapher’s equations and are due to Oliver

Heaviside in 1880. ]

R Ld

G Cd

z [Image from: [Image http://es.wikipedia.org/wiki/Oliver_Heaviside Ld distributed inductance per unit length [H/m] Cd distributed capacitance per unit length [F/m] R distibuted resistance per unit length, for both conductors [/m] G distibuted conductance per unit length, for both conductors [S/m]

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 22 EETAC-UPC Take your time…

Solution: Telegrapher’s equations. iz,t iz  z,t • According to Kirchhoff’s laws: R Ld iz,t vz,t  L z  Rz iz,t  v z  z,t  + + d t vz  z,t vz,t    v z  z,t i z,t  Gz v z  z,t  C z  iz  z,t G Cd d t - -

z • Dividing by z and taking the limit z 0: vz,t iz,t  R iz,t  L z d t iz,t v z,t  G vz,t  C  z d t Telegrapher’s equations

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 23 EETAC-UPC TRANSMISSION LINES Propagation equations: lossy lines • From the lumped-circuit model of a lossy two-wire transmission line we get the telegrapher’s equations. iz,t iz  z,t R L d vz,t i z,t  R iz,t  L  + + d vz  z,t z t vz,t iz,t v z,t  G Cd  G vz,t  C - - z d t

z

V  • Considering sinusoidal steady-state  R  j Ld I condition (cosine based phasors): z I  G  j C V z d

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 24 EETAC-UPC   TRANSMISSION LINES Propagation equations: lossy lines • Wave equations are now:  2  V 2  V being:   j  R  j Ld  G  jCd  z 2 phase constant 2 complex attenuation  I 2   I  propagation z 2 constant  vp depends with frequency: DISTORTION  (each frequency component of a signal • The solutions are: travels at different v along the line)  p   z  z V z  V0 e V0 e   R  j Ld being: Z0   1    z  z   z  z G  jCd I z  I0 e  I0 e  V0 e V0 e Z0 • And in time domain:

  z   z   vz,t V0 e cos t  z  argV0  V0 e cos t  z  argV0 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 25 EETAC-UPC TRANSMISSION LINES Propagation equations: lossy lines

• Lumped-circuit parameters required to model some common lines as a function of

their dimensions, surface resistivity (RS), and materials filling the space between the conductors (permitivitty  = ’-j’’ and permeability  = 0r). • Here follows a table showing some classical transmission line:

parameters for other transmission lines are found in the literature or computed by using specific software

[Table from: [POZAR]] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 26 EETAC-UPC     TRANSMISSION LINES Propagation equations: low-loss lines   • In practical lines losses are small. The equations for attenuation and propagation factor can be simplified. . The complex propagation factor can be re-arranged:   R G  RG   j  R  j L G  j C  j L C 1 j     d d d d  L C   2 L C    d d  d d

. For a low-loss line we can assume: R    L d and G  Cd  R G   j  R G        j Ld Cd 1 j    j Ld Cd 1     Ld Cd   2  Ld Cd  R  j L L Z  d  d 0 same values than lossless lines G  jCd Cd 1  C L   L . Finally:    R d  G d    L C Z  d   d d 0 2  Ld Cd  Cd DISTORTIONLESS Radiofrequency Engineering C. Collado, J.M. González-Arbesú 27 EETAC-UPC TRANSMISSION LINES Reflection coefficient • Wave reflection on a transmission line can be illustrated by considering a lossless

transmission line loaded with an arbitrary impedance ZL. Z0 is the characteristic impedance of the transmission line. ZG=Z0 Z0

   jz   jz V  V0 e V  V0 e VG ZL

z=-l z=0 • A voltage reflection coefficient can be defined for any point in the V  z z   line as the amplitude of the reflected voltage wave normalized to V  z  the amplitude of the incident voltage wave.

• Because at the load (z=0) the impedance of the line is ZL: V 0 V  V   Z  Z Z   0 0 Z  0  L 0 L    0 L I 0 V0 V0 Z L  Z0 voltage reflection coefficient at load Radiofrequency Engineering C. Collado, J.M. González-Arbesú 28 EETAC-UPC TRANSMISSION LINES Reflection coefficient • One has to be careful with the coordinate axis chosen to define the reflection coeficient along the line. ZG=Z0 Z0 . by definition:     jz   jz V z  V  V0 e V  V0 e z  V ZL V  z  G

z=-l z=0 z=0 z=l    V0 Z L  Z0  V0 j2l Z L  Z0 . coefficient at load: L  0    L  l   e  V0 Z L  Z0 V0 Z L  Z0

   V  j2l  V . coefficient at input port:   l  0 e  0  0 IN V  IN V  0 0 . in both cases:  j2l IN  Le   . however: j2z  j2 lz  z Le z Le

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 29 EETAC-UPC    TRANSMISSION LINES Reflection coefficient  • Because at the input port of the line (z=-l) the impedance is:   V   l V e j l   l   0  0 e2 j l   e2 jl IN   j l L V  l V0 e

 2 jl IN  L e

voltage reflection coefficient at input port  2 jz • At any point in the line: z L e • Reflection coefficient is a complex number. • For a passive load the magnitude of the reflection coefficient is always lower than 1. 0   1

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 30 EETAC-UPC TRANSMISSION LINES Reflection coefficient

Example: Reflection coefficient of a loaded lossy transmission line. Taking into consideration the equations of the voltage and current waves flowing in a lossy  transmission line find the equation for the input impedance on the line.  V   l V e l    l   0   e2 le2 jl IN   l L V  l V0 e

The reflection coefficient is attenuated when the line increases its length.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 31 EETAC-UPC Take your time…

Standard loads. Which is the reflection coefficient corresponding to, respectively, an

open circuit, short circuit, and reference impedance (Z0)?

L L L Z L  Z L 0 Z L Z0

Line transition. Is the voltage reflection coefficient the same at both sides of the transitions between the transmission lines?

' Z0 Z0

1 2

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 32 EETAC-UPC Take your time…

Mismatched lossless transmission line. By appliying boundary conditions to the ports

of the lossless transmission line of the figure, find the voltage reflection coefficients IN + and L, and the magnitude of the progressive wave V0 .

Z V  V e jz V  V e jz VG G  0  0 ZL Z0

z=0 z=l

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 33 EETAC-UPC Take your time…

Solution. Mismatched lossless transmission line.

   jz   jz ZG V  V0 e V  V0 e ZL • and at the input port: V Z0 G . voltage and current:   V 0V 0 VG  I 0 ZG      z=0 z=l V 0 V 0    I0  • at the output port of the line: Z0 Z0  . defining IN… . voltage:   Z V  0 V  l V  l  V e j l V e j l  I 0 Z  L V e j l V e jl     e j2l 0 0 L 0 0 IN  L Z0 V 0 . defining the load reflection coefficient  … . and G… L ZG  Z0   G  V l V 2 jl    0 e ZG  Z0 L V  l V  0 . …we get: . … we get: Z L  Z0  Z0 1 L  V0  VG  Z L  Z0 Z0  ZG 1 IN G

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 34 EETAC-UPC TRANSMISSION LINES Power and losses: return loss

2 2 • The time-average power flow V  V  1 * 0 0   along the line at point z is: Pav  ReV z I z    P  P 2 2Z0 2Z0

 2 V0 2 Pav  1 z 2Z0

 2 V0 • Incident power to the line: Pinc  2Z0  2 V0 2 • Reflected power at the load: Pref  L 2Z0 • To avoid the existence of reflected waves (=0) on a loaded transmission line

the load impedance ZL should be equal to the characteristic impedance Z0. Such a load is said to be a matched load.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 35 EETAC-UPC TRANSMISSION LINES Power and losses: return loss

• Some considerations derive from the previous equations: power on the load   Z L  Z0 P  0 PL  P   Z L  0, P  P PL  0 • The power carried by the positive flowing wave can be greater that the average power flowing on the line

• When the load is mismatched not all the available power is the delivered to the load. This loss is called return loss (RL):

RL  20log L dB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 36 EETAC-UPC TRANSMISSION LINES Power and losses: return loss

• What happens in case of a lossy transmission line? For a forward propagating wave in a transmission line with length l:  V z  V e z 1 attenuation 0 P l  ReV  I *  P 0 e2l  1  z 2 I z  V0 e incoming power Z0 at input port  Wave attenuation (L) between planes separated l:  P 0  . in decibels [dB]: L dB 10log   l 20log e  8.686 l     P l  1  P 0 . in Nepers [Neper]: L Nepers  ln   l   2  P l 

1 Neper = 8.686 dB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 37 EETAC-UPC Take your time…

Power losses: dB and Nepers. A RG59/U coaxial cable has an attenuation of 39.3 dB/100m (this means 39.3 dB each 100 m of cable) at 1 GHz. Find the attenuation in Nepers when having 200 m of cable, and the value of the attenuation coefficient  (in Np/m).

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 38 EETAC-UPC Take your time…

Solution. Power losses: dB and Nepers.

Attenuation in dB for 200 m of cable: dB LdB  0.393 200 m  78.6 dB    m  Attenuation in Nepers for 200 m of cable: 78.6 dB LNp   9.05 Np  dB  8.686    Np Attenuation coefficient: 9.05 Np  Np     0.045 200 m  m 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 39 EETAC-UPC Take your time…

Feeding and antenna. A 50- antenna is fed with a signal of 100 W by means of a 75  cable. Find the power finally arriving to the antenna. a) 4 W b) 20 W c) 80 W d) 96 W

Phasors. Find the phasor of the current flowing along the open transmission line of the figure. ZG

Z0  VG

z=0 z=l    jl  V0 a) Iz 2 jV0 e sin z  l b) Iz  2 cosz Z0 V   V   c) Iz  2 j 0 e jl sinz  l d) Iz  2 0 cosz  l Z0 Z0 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 40 EETAC-UPC TRANSMISSION LINES Voltage standing wave ratio • The voltage on each point of a line depends on the V z  V  1  e j2z load attached at its end: 0 L . When the transmission line is matched, the magnitude of the  V z  V0 voltage on the line is constant:

 . When the transmission line is not matched the overlap of an Vmax  V0 1 L  incoming and a reflected wave leads to a standing wave whose magnitude oscillates with the position on the line:  Vmin  V0 1 L   • A measure of the line mismatch is the Vmax 1 L Voltage Standing Wave Ratio (VSWR): VSWR   Vmin 1 L • SWR is a real positive number: 1 VSWR  

matched load L  0 VSWR 1

total reflection L 1 VSWR  

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 41 EETAC-UPC TRANSMISSION LINES Voltage standing wave ratio

Example: Voltage on a transmission line. Plot the normalized voltage and current on a transmission line 2 in length as a function of position on the line (z). Consider the j90 j180 j270 following loads: L=0,0.2,0.4,0.6,0.8,1 and L=0.8, 0.8 e , 0.8 e , 0.8 e .

V z decreasing   1  e j2z L  L V0 0 90

 180 270

Iz  1  e j 2z  L decreasing L I0 180 270

0 90 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 42 EETAC-UPC  TRANSMISSION LINES Impedance   • The impedance in every position (z) of the line is:   V z V  e j z V  e j z 1 e j2 z Zz   Z 0 0  Z L  0   j z  j z 0 j2z I z V0 e V0 e 1 L e

• The input impedance of a line loaded with ZL is: V  l Z cosl jZ sin l Z  jZ tanl Z Z L 0 Z L 0 IN   0   0  I  l Z0 cos l  jZ Lsin l Z0  jZ L tan l

Z0 Z0

 Z IN Zz L ZL Z IN Zz

z=-lz z=0 z=-lz z=0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 43 EETAC-UPC TRANSMISSION LINES Impedance

• The case of highly reflective loads:

short-circuited stub: Z IN  jZ0tanl inductive impedance

Z open-circuited stub: Z  0 capacitive impedance IN jtanl • Quarter-wave long transmission lines perform as quarter-wave (impedance) or impedance inverters, being the input impedance inversely proportional to the load impedance. 2   Z0 When: l   n n  0,1,2,... Z IN  4 2 Z L • Otherwise, any line whose length is any multiple of /2 does not transform the load impedance, regardless of the characteristic impedance.  When: l  n n 1,2,3,... Z  Z 2 IN L

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 44 EETAC-UPC TRANSMISSION LINES Impedance

Example: Emulating lumped elements with electrically-short transmission lines. Suggest what lumped elements can be simulated by means of short transmission lines when these line end, respectively, with an open circuit and with a short circuit.

short-circuited short-stub: l  

Z l Z jZ l j L being: 0 IN  0   L  Z IN Z Z IN v 0 p L

open-circuited short-stub: l  

Z 1 l Z  0  being:C  IN Z IN Z IN j l jC v Z Z0 p 0 C

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 45 EETAC-UPC TRANSMISSION LINES Impedance

Example: Input impedance of a loaded lossy transmission line. Taking into consideration the equations of the voltage and current waves flowing in a lossy transmission line, find the equation for the input impedance on the line.  Z 0    z  z V z V0 e V0 e  Z Zz  L Z IN L    z  z 1   z  z I z  I0 e  I0 e  V0 e V0 e Z0 at z=-l z=-lz z=0 that is, also  Z IN  Z0 tanhl  Z Z coshl Z sinhl Z  Z L 0  Z  IN 0 Z cosh l  Z sinh l 1 L  0 L  Z  tanh    Z0 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 46 EETAC-UPC TRANSMISSION LINES Generator mismatch

• The voltage on the input port of the line can be calculated from the source voltage and depends on the load impedance. ZG ZIN Z0

VG ZL G IN L

z=0 z=l

 Z 1 Z  Z V  V 0 G 0 0 G  being: G  Z0  ZG 1 IN G ZG  Z0

Z IN    at z=0: V 0  VG  V0 V0  V0 1 IN Z IN  ZG V 0 1  Z Z IN and being: IN   0 I 0 1 IN

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 47 EETAC-UPC TRANSMISSION LINES Generator mismatch

• Consequently: V when: Z  Z V   G G 0 0 2

 Z0 Z IN  Z0 V0  VG Z0  ZG the power delivered from the source to the line (and if it has 2 1 * 1 VG no losses, delivered to the load): Pav  ReV 0 I 0  ReZ IN 2 2 Z IN  ZG 2 being: Z  R  jX V R IN IN IN P  G IN and: Z  R  jX av 2 2 G G G 2 RIN  RG  X IN  X G

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 48 EETAC-UPC Take your time…

Power delivered to the load. Consider a lossless transmission line connected to a fixed

source impedance ZG=RG+jXG. Find the power delivered to the load (or the power delivered to the transmission line) in the following two cases: when the load is matched

to the line (ZL=Z0); and when the generator is matched to the loaded line (ZG=ZIN).

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 49 EETAC-UPC Take your time…

Solution: Power delivered to the load.

• Load matched to the line (ZL=Z0).

2 V Z P  G 0 av 2 2 2 Z0  RG  X G

• Generator matched to the loaded line (ZG=ZIN). 2 VG RG Pav  2 2 8 RG  X G

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 50 EETAC-UPC Take your time…

Impedance for maximum power transfer or available power. Assuming that the

generator series impedance is fixed, find the input impedance ZIN to achieve the maximum power transfer to the load (lossless transmission line). In that case, find the power delivered to the load.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 51 EETAC-UPC Take your time…

Solution: Impedance for maximum power transfer. • Maximizing the power transfer… P  av  0  2 2 2 RIN  R  R  X  X   0 RIN  RG   G IN G IN   Z  Z * P R X  X  0  X  X IN G av  0 IN G IN  IN G   X IN  … we get that the input impedance of the line should be the complex conjugate of the source impedance. This condition is called conjugate matching.

• In this case, the power transferred to the load is (lossless transmission line): 2 VG Pav  8RG

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 52 EETAC-UPC TRANSMISSION LINES Smith Chart

•A graphical tool very helpful when dealing with impedance transformation and matching network design is the Smith Chart (invented by Phillip H. Smith in 1939 while working for Bell Laboratories). . Because working with (almost) infinite values for resistances and reactances is usual in , a graphical plot of these impedances is not practical in a rectangular coordinate system. However, operating with the reflection coefficient associated to a given normalized passive impedance (impedance

Z with possitive resistance normalized to the characteristic impedance of the line Z0) leads to a graphical representation of the loads inside a circle of unity radius.

• The transformation between impedances and reflection coefficients leads to a Z-chart: z 1 1  Z   z  being: z  z 1 1  Z0

• The transformation between admitances and reflection coefficients leads to a Y-chart. 1 y 1  Y   y  being: y   YZ0 1 y 1  Y0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 53 EETAC-UPC TRANSMISSION LINES Smith Chart • In the Z-(Smith) Chart, lines of constant reactance map into circumferences.

Im

impedances z 1 corresponding to   active loads x z 1 d e e x  0 Z-Chart d r c Re z  r  jx c fghi b 1  b x  0 a z  1  a fghi

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 54 EETAC-UPC TRANSMISSION LINES Smith Chart • In the Y-(Smith) Chart, lines of constant susceptance map into circumferences

Im

1 y b   1 y f a b g e g h Y-Chart d i c b  0 Re c 1  y  g  jb b y  d b  0 a 1  e

fghi

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 55 EETAC-UPC TRANSMISSION LINES Smith Chart • Z-Smith chart: main loads and their correspondence with the reflection coefficient.

[Images from: http://upload.wikimedia.org/wikipedia/commons/d/df/Smith_chart_explanation.svg] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 56 EETAC-UPC TRANSMISSION LINES Smith Chart • Z-Smith chart. shift towards shift towards GENERATOR LOAD

  j2l IN  Le phase of reflection coefficient in degrees (periodicity 180)  angle transmission line length in wavelengths (periodicity /2)

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 57 EETAC-UPC TRANSMISSION LINES Smith Chart

Example: Impedance on Smith Chart. Plot 63.5º

the load ZL=50+j50  on the Smith chart (reference impedance 50 ). Indicate the value of the reflection coefficient, the return loss, and the VSWR.

Z  50  j50  z 1 j Z  50  0  Graphically from the Smith Chart:  0.46 angle()  63.5º VSWR  2.6 VSWR  8.5 dB RL  7 dB Numerically from equations: z 1    0.45 e j63.4º z 1 1 VSWR   2.62  8.36 dB 1  RL  20log  6.99 dB Radiofrequency Engineering C. Collado, J.M. González-Arbesú 58 EETAC-UPC TRANSMISSION LINES Smith Chart

Example: Loaded coaxial. A 2 cm-length coaxial operating at 3 GHz is loaded with

ZL=50+j50 . Use the Smith chart to find the input impedance seen from the coaxial transmission line. Consider the dielectric constant of the line is 2.56. A

Graphically from the Smith Chart: Normalized load impedance (point A): z 1 j Transmission line electrical length:  9 l l l f  r 0.02 3·10 2.56    8  0.32 B v p f c 3·10 Normalized input impedance (point B): z  0.41 j0.10 0.32 from load Unnormalized input impedance: Zin  20.5  j5.0 to generator  360 Numerically from equations: l  l 115.20º  Z  jZ tanl Z Z L 0 19.3 4.9 j IN  0   Z0  jZ L tan l

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 59 EETAC-UPC TRANSMISSION LINES Impedance matching • Though the Smith chart is useful to avoid tedious computations involving complex numbers, nowadays this is not a problem thanks to the widespread use of calculators and computers. • The main advantage of using Smith chart is that it allows drawing conclusions without requiring complex calculations. • Smith chart is very useful when designing matching networks. Several alternatives are possible when trying to match a load (or a generator) to the reference impedance of a transmission line. Some of them are: . Impedance matching with lumped elements. . Single-stub matching. . Double-stub matching. . Triple-stub tuner. . Quarter-wave impedance transformer. • In a practical design of a matching network the technology to be used and the frequency bandwidth of the solution should be considered. • Conjugate matching can also be treated by using the Smith Chart.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 60 EETAC-UPC TRANSMISSION LINES Impedance matching

• When designing matching networks use to be helpful employing CAD tools. • An unlicensed tool (with limited functionalities) called Smith (designed by Prof. Fritz Dellsperger and Michael Baud from Bern University) is an example.

[Image from: http://www.fritz.dellsperger.net/] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 61 EETAC-UPC TRANSMISSION LINES Impedance matching

• Impedance matching using reactive elements is desirable due to the absence of losses.

• The matching solution depends on the technology to be used for its implementation.

L

Z-chart Y-chart B B -1/(L) L C

A A

C -1/(C)

Z-Smith chart D D Y-Smith chart

moving CW or CCW a load along a moving CCW or CW a load along a constant-R circle is equivalent to add constant-G circle is equivalent to add a a L or a C in series, respectively L or a C in shunt, respectively

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 62 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements • Impedance matching using two lumped elements.

. ZL inside the 1  j x circle can be matched using shunt-series elements.

C L Z Z Z Z 0 L L 0 C L

. YL inside the 1  j b circle can be matched using series-shunt elements.

L C Z Z Z Z 0 C L 0 L L

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 63 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements

. ZL outside the 1  j x circle and YL outside the 1  j b circle can be matched using series-shunt and shunt-series elements.

C1 C2 Z0 Z L Z0 Z L C2 C1

C1 L Z0 Z L Z0 Z L L C

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 64 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements • Impedance matching using three lumped elements networks (only some possible solutions are shown).

C C Z 2 1 Z 0 L L

C2 Z0 Z L C3 C1

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 65 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements • Impedance matching using four lumped elements networks (only some possible solutions are shown). • When designing matching networks, remember that shorter paths in the Smith chart provide a wider operational bandwidth.

C2 C1 Z0 Z L L2 L1

C3 C1 Z0 Z L C4 C2

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 66 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements • Lumped elements (should be smaller than /10) used to design the matching networks can be: . Capacitors: chip capacitors, MIM capacitors, interdigital capacitors, open-circuited stubs. ] [Images from [POZAR]] ] [Images from: [Images http://www.ad- mtech.com/products/thin_ film/index.html . Inductors: chip inductors, loop inductors, spiral inductors, short-circuited stubs.

. Resistor: chip resistors, planar resistors. [Images from: [Images http://www.hitachi- aic.com/english/products/ca pacitors/tantal/k_chip.htmll

• Lumped elements have parasitics in the microwave range. • The standard units used when describing the size of a lumped element is the mil: 1 mil=0.001 in=25 m=1/40 mm

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 67 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements

Example: Matching a dipole. Consider a dipole with input impedance 82+j45  and operating at 2.45 GHz. Consider all the possibilities of matching the dipole to the line using a two-lumped elements network when fed with a 50  transmission line. Solve the problem analytically and check the results using the application Smith.exe.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 68 EETAC-UPC TRANSMISSION LINES Impedance matching: lumped elements

Example: Matching a monopole. Consider a monopole with input impedance 30+j20  operating at 2.45 GHz. Consider all the possibilities of matching the monopole to the line using a two- lumped elements network when fed with a 50  transmission line. Solve the problem using the application Smith.exe. Make a frequency sweep from 2 to 3 GHz and decide which solution has a better bandwidth (assume that the monopole impedance does not change in the proposed frequency bandwidth).

worst RL is around 20 dB

worst RL is around 14 dB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 69 EETAC-UPC TRANSMISSION LINES Impedance matching: single-stub matching • Single-stub matching. . A stub is a short-circuited or open-circuited section of a transmission line. . There are two alternatives: single-shunt-stub and single-series-stub. . There are two parameters to adjust: distance l from load to stub and the shunt impedance.

Z0 Z0 Z Z0 0 ZL l Z0 Z0 ZL l

Z0

Z0 Z Z0 0 ZL Z0 Z0 ZL l Radiofrequency Engineering l C. Collado, J.M. González-Arbesú 70 EETAC-UPC TRANSMISSION LINES Impedance matching: double-stub matching • Double-stub matching. . Single-stub tuning can be a problem for a variable matching network due to the variable line length l between the load and the stub. . A double-stub with fixed separation d between stubs and variable stub lengths is used to solve this problem. Unfortunately, there is a region where impedances can’t be matched (it can be tuned out by adding a certain line length). . Although double-shunt-stub tuning is shown, double-series-stub tuning is also possible. d d Z Z 0 Z0 0 Z0 Z0 Z0 Z Z ZL 0 0 ZL

d d

Forbidden region where loads can’t be matched.

Radiofrequency Engineering d C. Collado, J.M. González-Arbesú d 71 EETAC-UPC TRANSMISSION LINES Impedance matching: triple-stub tuner • Triple-stub tuner. . A triple-stub tuning network does not have regions where loads can not can be matched. . This networks has more degrees of freedom (three variable stub lengths) in order to increase the bandwith of the tuner. d d

Z0 Z Z0 Z Z 0 Z 0 0 0 ZL

d d

Z0 Z Z0 Z Z 0 Z 0 0 0 ZL

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 72 EETAC-UPC TRANSMISSION LINES Impedance matching: quarter-wave transformer • Quarter-wave impedance transformer.

. A resistive load RL can be matched to a transmission line with reference ' impedance Z0 by means of a /4 section of a transmission load having a Z0  RL Z0 reference impedance of Z0’: . If the load is not purely resistive a series or paraler reactive element (lumped element or transmission line section) should be added to make it purely resistive before including the quarter-wave transformer.

Z0’ Z0 ZL /4 l

Z0’ -jXL ZL=RL +jXL

/4

Z0’ -jBL YL=GL +jBL

/4 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 73 EETAC-UPC TRANSMISSION LINES Impedance matching

Example: Matching a monopole with a microstrip single-stub network. Consider a monopole with input impedance 30+j20  operating at 2.45 GHz. Consider the possibility of matching the monopole using a single-stub network made with microstrip technology (avoid using vias) and a

Rogers Duroid 4003C substrate (r=3.55). Solve the problem using the application Smith.exe.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 74 EETAC-UPC Take your time…

High impedance matching. Which of the following options are possible to match an impedance of 188  to a 50  transmission line?

Matching Z =50  188  0 Network

a) b) c) d)

L C L Z ’=97  0 C L L

/4

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 75 EETAC-UPC TRANSMISSION LINE DESIGN Balanced and unbalanced lines

• A is a transmission line having two conductors with the same voltage magnitude but a phase shift of 180º with respect to . Impedance of both conductors is equal with respect to ground. • The balanced and unbalanced character of transmission lines has to be accounted for proper connection to circuits or devices. • An example of a balanced line is a twin-lead line, whereas an is a coaxial cable.

BALanced V UNbalanced V/2 -V/2 0 Volts 0 Volts

• The transition from a balanced to an unbalanced structure, or viceversa, requires a transformer. • are used to connect balanced to unbalanced lines or structures. They are required irrespective of transmission line technology. • Baluns are usually narrowband devices. It is difficult to design wide bandwidth baluns.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 76 EETAC-UPC TRANSMISSION LINE DESIGN Homogeneous and non-homogeneous lines

• Homogeneous dielectric media are uniform in all points and its physical properties are unchanged. . A transmission line in a homogeneous medium has a propagation velocity that

depends only on material properties (dielectric permittivity r and magnetic permeability r). . The principal wave existing in these kind of transmission lines could be a TEM (transversal electromagnetic) wave.

• Non-homogeneous media contain multiple materials with different dilectric constants. . Wave propagation velocity in non-homogeneous transmission lines depends on material properties and structure dimensions

. An effective r,eff dielectric constant is often used to represent an average dielectric constant. . These line do not propagate pure TEM modes.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 77 EETAC-UPC TRANSMISSION LINE DESIGN Homogeneous and non-homogeneous lines

• Cross sections of some common transmission lines:

centered stripline dual stripline

two-conductor

coaxial shielded two-conductor homogeneous

rectangular WG circular WG

coplanar strip microstrip coupled microstrip

non-homogeneous embedded microstrip coplanar

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 78 EETAC-UPC TRANSMISSION LINE DESIGN Coupled lines

• Coupled lines are balanced-type lines.

• These structures are analyzed by means of an odd and an even excitation mode that superpose. . In the even mode the currents in the strip conductors are equal in amplitude and flow in the same direction. . In the odd mode the currents in the strip conductors are equal in amplitude and flow in opposite directions. . Each strip conductor is characterized by its characteristic impedance (relative to ground) and its propagation constant. . Both parameters are different for the excited modes.

Z0e 0e even mode parameters

Z0o 0o odd mode parameters [Image from: [POZAR]] [Image

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 79 EETAC-UPC TRANSMISSION LINE DESIGN Line design

• Homogeneous lines propagate purely TEM modes and have analytical equations helping to match the characteristic impedance and propagation constants of the lines to given specific requirement.

• Non-homogeneous lines do not propagate purely TEM modes (but under certain circumstances they propagate quasi-TEM modes). The characteristic parameters of the line have to be numerically computed or other approximate techniques have to be used.

• Nowadays, CAD software helps the designer... (e.g. TX-Line Calculator of AWR, LineCalc part of ADS of Agilent Technologies).

http://web.awrcorp.com/Usa/Products/Optional-Products/TX-Line/

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 80 EETAC-UPC TRANSMISSION LINE DESIGN Line design

• TX-Line Calculator of AWR is a license free software for transmission line design.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 81 EETAC-UPC APPLICATION NOTES Coaxial cables • The coaxial cable was invented by Oliver Heaviside at the end of the 19th century for transmitting telegraphic signals. • The first propagating mode in a coaxial cable is TEM. It has a cut-off frequency of zero.

• Next mode is TE11. Its cut-off frequency depends on the mean radius between the conductors and the material filling this gap. • The characteristic impedance of the cable also depends on the geometry of the cable section and the materials. electric field distribution inside a coaxial ] idson.edu/stuhome/

TEM mode TE11

[Image from: [Image http://www.phy.dav phstewart/IL/speed/Cableinfo.html   TE11 1 1 r  D  fc  Z0   ln   rout  rin   2  r  d  2   r r  2  Radiofrequency Engineering C. Collado, J.M. González-Arbesú 82 EETAC-UPC APPLICATION NOTES Coaxial cables • Characteristic impedance, frequency bandwidth, attenuation, wave speed of propagation, and maximum power handling capability should be carefully considered when adquiring a cable for a given application.

[Table from SSi Cable Corp.] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 83 EETAC-UPC APPLICATION NOTES Connectors • Some 50  impedance connectors typically used in RF and microwave equipment. Others exist.

N BNC MCX MMCX TNC APC-7

2 GHz 6 GHz 6 GHz 11 (18) GHz 11 GHz 18 GHz

Warning! RP connectors BNC: Bayonet Neill-Concelman (RP: reverse polarity) 3.5 K MCX: Micro CoaXial SMA MMCX: Micro-Miniature CoaXial N: Neill connector TNC: Threaded Neill-Concelman APC-7: Amphenol Precission Connector with 7 mm diameter SMA: SubMiniature version A APC-3.5: Amphenol Precission 18 (26.5) GHz 26.5 (34) GHz 40 (45) GHz Connector with 3.5 mm diameter SMK: SubMiniature version K (also called 2.92 mm)

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 84 EETAC-UPC Take your time…

Maximum power transfer. Two complex impedances, ZS=(25-j15)  and ZL=(100-j25) , have to be matched by means of a LC network in order to have maximum power transfer between them. Calculate the values of the inductance and capacitance at 100 MHz using a reference impedance of 25 .

ZS ZL

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 85 EETAC-UPC Take your time…

Impedance matching. Use the Smith chart and an operating frequency of 1 GHz to find the capacitance and the transmission line length l that matches an impedance of

ZL=(100-j200)  to 50  by means of the network of the figure.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 86 EETAC-UPC Take your time…

Smith chart. A 50  lossless bifilar transmission line is loaded with an impedance of (100-j200) . Line length is 2.25 m and the frequency of the propagated harmonic wave is 100 MHz. a) Find the input impedance of the line using the Smith chart. b) Find the input impedance using the analytical equation that provides the input impedance of a transmission line as a function of its length, load and its electrical model parameters. c) Calculate the power dissipated at the load when the available power is 100 W.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 87 EETAC-UPC