RF Seminar The Smith Chart

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RF Seminar The Smith Chart
Jan H. Kuypers
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contents

• Part I Fundamentals
• imedance matrices
• transmission lines
• scattering parameters
• Part II The Smith Chart
• s-plane
• deriving the Smith Chart
• examples
• matching using the Smith Chart

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literature

• recommended RF literature
• Microwave Engineering, David M. Pozar
• RF Circuit Design, Chris Bowick

4

• Part I
• Fundamentals
• Things you should know before trying to
  understand the Smith Chart

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introduction

• Q What is displayed in the Smith Chart ?
• A Scattering Parameters Sij
• Review necessary of
• why use scattering parameters ?
• why not use impedance ?
• what units do scattering parameters have ?
• why are they complex ?

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review

• n-port device fully described by
• Impedance parameters Zij
• Admittance parameters Yij

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review

• The Impedance Parameters
• measured or computed as
• result in matrix Z

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• example

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example

• Impedance Parameters for simple 2 port
• 2-port circuit
• want to derive
• why ? ? we can fully describe the 2-port and
  simulate its frequency characteristics

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Z11

• Impedance Parameters for simple 2 port
• Z11 input impedance, forward impedance
• simplify circuit

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Z12

• Impedance Parameters for simple 2 port
• Z12 transfer impedance
• simplify circuit

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Z21

• Impedance Parameters for simple 2 port
• Z21 transfer impedance
• simplify circuit

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Z22

• Impedance Parameters for simple 2 port
• Z22 output impedance
• simplify circuit

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Zij

• We have the impedance matrix, so what ?
• R 30 W, C 100 pF

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Zij

• What else can we see ?
• assume the 2-port is used as a filter
• ? what is the transfer function/ frequency
  characteristics ?

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Zij

• 2-port frequency characteristic
• transfer function
• ? its a low pass filter!

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Zij

• Impedance Parameters for simple 2 port
• 2-port circuit and its impedance matrix
• lets try to measure the circuit exactly the same
  way!

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questions

• until now is everything clear ?
• you could do this by yourself ?
• Yes ? Lets do it!

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test yourself

• turn around todays slides
• prepare paper and pencil
• you have 5 min
• ok ?

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problem 1

• calculate the impedance matrix Z
• bonus
• could this 2-port be used as a filter ?
• what filter is it ?

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test results

• was it easy ?
• why was it easy ?
• what were the difficulties ?

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solution

• the sought-after impedance matrix
• it was easy because the problem is symmetric
• the 2-port can be used as high-pass filter
• at low frequencies L shorts, and incident voltage
  on port 1 can not be detected at port 2

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• circuit measurement

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example

• Impedance Parameters for simple 2 port
• 2-port circuit
• want to measure

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Z11

• measurement of Z11
• requirements
• measure voltage and current at port 1
• port 2 is open
• low frequencies no problem
• RF frequencies IMPOSSIBLE

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RF frequencies

• What makes RF frequencies so special ?
• ? Answer Nothing!
• Difference of circuit theory and transmission
  line theory depends only on size!
• circuit theory is valid for
• 100 MHz
• l lt 3 cm (vacuum)
• microstrip line ca. l lt 1.78 cm (FR4 er 4.7)
• 2.45 GHz
• l lt 1.2 mm
• microstrip line ca. l lt 0.71 mm (FR4 er 4.7)

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RF frequencies

• graphical explanation
• assume a wire cable used in combination with
  ordinary circuit boards
• lets examine the potential distribution on the
  line at the time t0
• at 50 Hz
• the potential on the whole line is constant
• the potential at port A and port B has the same
  amplitude
• that means the potential in A and B is in phase
• at 2.45 GHz
• we can measure a harmonic potential distribution
  along the line
• the waves in A and B do not have the same
  amplitude
• there is a phase difference between A and B

Port B
Port A
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RF frequencies

• RF frequency consequences
• voltage and current must be seen as waves
• dimensions become important
• circuit board design
• length and width of lines
• corners
• wave physics include
• reflection / transmission
• diffraction
• dispersion
• radiation
• ? the most important quantities for us will be
  reflection and transmission

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RF frequencies

• reflection and transmission
• when does reflection occur ?
• wave impedance changes? definition of reflection
  coefficient G
• Z0 is the source impedance
• Z is the impedance under measurement (both are
  complex)
• transmission line characteristic impedance (e.g.
  width)
• open
• short
• resistive element (R)
• reactance element (L, C)
• reflection means standing waves

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lossless transmission line

• wave propagation for a lossless transmission line
• voltage on the line
• current on the line
• propagation coefficient for lossless line

reflected wave
incident wave
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• transmission examples

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transmission line

• short-circuited transmission line

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transmission line

• short-circuited transmission line
• voltage on the line
• current on the line
• input impedance into line
• ? Zin is purely imaginary

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transmission line

• short-circuited transmission line
• short line
• ? line looks inductive
• length? impedance becomes infinite? line looks
  like open circuit
• long line? line looks capacitive
• impedance periodic in l with p l/2

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transmission line

• open-circuited transmission line

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transmission line

• open-circuited transmission line
• voltage on the line
• current on the line
• input impedance into line
• ? Zin is purely imaginary

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transmission line

• open-circuited transmission line
• short line
• ? line looks capacitive
• length? impedance becomes zero
• ? line looks like a short
• long line? line looks inductive
• impedance periodic in l with p l/2

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• loaded transmission lines
• general case

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terminated line

• general case of terminated line
• the input impedance into the line is

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terminated line

• general case of terminated line
• special cases
• length
• ? input impedance becomes? load impedance is
  not altered or transformed

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terminated line

• general case of terminated line
• special cases
• length ? input impedance becomes
• ? input impedance is transformed
• ? quarter-wave transformer
• ? we will study that later in the RF seminar

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• loaded transmission lines
• reflection and transmission

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transmission line

• change in impedance
• for z lt 0
• for z gt 0
• at z 0

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questions

• until now is everything clear ?
• you could do this by yourself ?
• Yes ? Lets do it!

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test yourself

• turn around todays slides
• prepare paper and pencil
• you have 5 min
• ok ?

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problem 2

• using these 2 formulas for reflection and
  transmission
• calculate G and T for
• a transmission line of Z0 50 W connected to a
  line of
• Z 25 W
• Z 250 W

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test results

• was it easy ?
• why was it easy ?
• what were the difficulties ?

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solution

• the sought-after G and T
• a transmission line of Z0 50 W connected to a
  line of Z 25 W
• a transmission line of Z0 50 W connected to a
  line of Z 250 W

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• S-parameters

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S-Parameters

• how to measure at RF frequencies ?
• reflection coefficient was introduced as
• definition for n-port devices

similar to impedance parameters
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S-Parameters

• properties of S-parameters
• complex
• magnitude
• phase q
• magnitude range of G
• short circuit
• open circuit

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S-Parameters

• properties of S-parameters range
• magnitude
• phase q
• magnitude range
• passive circuits
• active circuits (amplifier/transistor)
• recall
• we want amplification, so e.g. S21 gt 1!
• we often derive G from resistance values, but G
  is a reflection ratio!

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S-Parameters

• 2-port device
• generally (here 2-port)
• only if network is loss-less

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• measuring our circuit
• using S-parameters

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S-Parameters

• How to measure the S-Parameters for our simple
• 2-port device ?
• We had earlier derived the impedance matrix
• using
• How to find the S-parameters ?

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example

• S-Parameters for simple 2 port
• 2-port circuit
• want to derive
• by using

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S11

• S11 forward reflection coefficient
• meaning incident voltage wave on port-1, no
  incident waves
• from port-2. Requires port-2 to be terminated
  with matched
• loads, to prevent reflections.

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S11

• S11 forward reflection coefficient
• measure reflected incident voltage wave on port 1

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S11

• S11 forward reflection coefficient
• How to calculate S11 ?
• find the input impedance looking into port 1(not
  same as Z11, as Z0 termination has been
  inserted!)
• result

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S22

• S22 reverse reflection coefficient
• measure reflected voltage wave on port 2

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S22

• calculation of S22
• input impedance for port 2
• result

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S12

• S12 reverse transmissions coefficient
• measure

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S12

• calculating S12
• 1.) find the transmission T22 of V2 into the
  circuit
• where V2 sees Zin,22

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S12

• T22 using Zin,22
• we find T22 as

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S12

• 2.) port 1 only sees voltage across Z0
• S12 becomes
• finally using T22, S12 is found as

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S21

• S21 forward transmission coefficient
• measure

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S21

• calculating S21
• find the transmission T11 of V1 into the circuit
• where V1 sees Zin,11

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S21

• T11 using Zin,11
• we find T11 as

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S21

• port 2 only sees voltage across Z0
• S21 becomes
• finally using T11, S21 is found as

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Sij

• S-Parameters for example
• we have determined all S-parameters!!?
  measurement is easy!? but calculation is hard
  work!! (but this example was easy!!)? is there
  an easier way ?

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questions

• until now is everything clear ?
• you could do this by yourself ?
• Yes ? Lets do it!

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test yourself

• turn around todays slides
• prepare paper and pencil
• you have 5 min
• ok ?

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problem 3

• calculate S11 for the circuit at resonance
• given
• source impedance Z0 50 W
• C 24.31 pF
• L 4.7 nH
• R 50 W

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test results

• was it easy ?
• why was it easy ?
• what were the difficulties ?

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solution

• As L and C are at resonance we only deal with the
  resistive elements
• the impedance into the circuit is
• and thus

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• back to S-parameter calculation....

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Sij

• S-Parameters for example
• we have determined all S-parameters!!?
  measurement is easy!? but calculation is hard
  work!! (but this example was easy!!)? is there
  an easier way ?

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2-port relations

• finding S-parameters
• derive Zij or Yij
• derive network impedances
• use simulation software
• use own models for e.g. FBAR, SAW devices, RF
  Amps....
• use information from component data sheets
  (transistor Y-matrix)
• convert impedance or admittance to ABCD matrices
• cascade networks with ABCD parameters
• convert ABCD network to S-parameters
• alternative
• RF simulation software helps, but can be
  expensive...
• use Spice circuit simulation software and convert
  from impedance to S-parameters

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2-port relations
this is what we measure ? save experimental data
as S-data
converting from S-parameters to any format you
want