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RF Seminar The Smith Chart

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wave physics include. reflection / transmission. diffraction. dispersion. radiation ... using these 2 formulas for reflection and transmission. calculate G and T for ... – PowerPoint PPT presentation

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Title: RF Seminar The Smith Chart


1
RF Seminar The Smith Chart
Jan H. Kuypers
2
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

3
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

5
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 ?

6
review
  • n-port device fully described by
  • Impedance parameters Zij
  • Admittance parameters Yij

7
review
  • The Impedance Parameters
  • measured or computed as
  • result in matrix Z

8
  • example

9
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

10
Z11
  • Impedance Parameters for simple 2 port
  • Z11 input impedance, forward impedance
  • simplify circuit

11
Z12
  • Impedance Parameters for simple 2 port
  • Z12 transfer impedance
  • simplify circuit

12
Z21
  • Impedance Parameters for simple 2 port
  • Z21 transfer impedance
  • simplify circuit

13
Z22
  • Impedance Parameters for simple 2 port
  • Z22 output impedance
  • simplify circuit

14
Zij
  • We have the impedance matrix, so what ?
  • R 30 W, C 100 pF

15
Zij
  • What else can we see ?
  • assume the 2-port is used as a filter
  • ? what is the transfer function/ frequency
    characteristics ?

16
Zij
  • 2-port frequency characteristic
  • transfer function
  • ? its a low pass filter!

17
Zij
  • Impedance Parameters for simple 2 port
  • 2-port circuit and its impedance matrix
  • lets try to measure the circuit exactly the same
    way!

18
questions
  • until now is everything clear ?
  • you could do this by yourself ?
  • Yes ? Lets do it!

19
test yourself
  • turn around todays slides
  • prepare paper and pencil
  • you have 5 min
  • ok ?

20
problem 1
  • calculate the impedance matrix Z
  • bonus
  • could this 2-port be used as a filter ?
  • what filter is it ?

21
test results
  • was it easy ?
  • why was it easy ?
  • what were the difficulties ?

22
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

23
  • circuit measurement

24
example
  • Impedance Parameters for simple 2 port
  • 2-port circuit
  • want to measure

25
Z11
  • measurement of Z11
  • requirements
  • measure voltage and current at port 1
  • port 2 is open
  • low frequencies no problem
  • RF frequencies IMPOSSIBLE

26
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)

27
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
28
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

29
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

30
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
31
  • transmission examples

32
transmission line
  • short-circuited transmission line

33
transmission line
  • short-circuited transmission line
  • voltage on the line
  • current on the line
  • input impedance into line
  • ? Zin is purely imaginary

34
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

35
transmission line
  • open-circuited transmission line

36
transmission line
  • open-circuited transmission line
  • voltage on the line
  • current on the line
  • input impedance into line
  • ? Zin is purely imaginary

37
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

38
  • loaded transmission lines
  • general case

39
terminated line
  • general case of terminated line
  • the input impedance into the line is

40
terminated line
  • general case of terminated line
  • special cases
  • length
  • ? input impedance becomes ? load impedance is
    not altered or transformed

41
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

42
  • loaded transmission lines
  • reflection and transmission

43
transmission line
  • change in impedance
  • for z lt 0
  • for z gt 0
  • at z 0

44
questions
  • until now is everything clear ?
  • you could do this by yourself ?
  • Yes ? Lets do it!

45
test yourself
  • turn around todays slides
  • prepare paper and pencil
  • you have 5 min
  • ok ?

46
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

47
test results
  • was it easy ?
  • why was it easy ?
  • what were the difficulties ?

48
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

49
  • S-parameters

50
S-Parameters
  • how to measure at RF frequencies ?
  • reflection coefficient was introduced as
  • definition for n-port devices

similar to impedance parameters
51
S-Parameters
  • properties of S-parameters
  • complex
  • magnitude
  • phase q
  • magnitude range of G
  • short circuit
  • open circuit

52
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!

53
S-Parameters
  • 2-port device
  • generally (here 2-port)
  • only if network is loss-less

54
  • measuring our circuit
  • using S-parameters

55
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 ?

56
example
  • S-Parameters for simple 2 port
  • 2-port circuit
  • want to derive
  • by using

57
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.

58
S11
  • S11 forward reflection coefficient
  • measure reflected incident voltage wave on port 1

59
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

60
S22
  • S22 reverse reflection coefficient
  • measure reflected voltage wave on port 2

61
S22
  • calculation of S22
  • input impedance for port 2
  • result

62
S12
  • S12 reverse transmissions coefficient
  • measure

63
S12
  • calculating S12
  • 1.) find the transmission T22 of V2 into the
    circuit
  • where V2 sees Zin,22

64
S12
  • T22 using Zin,22
  • we find T22 as

65
S12
  • 2.) port 1 only sees voltage across Z0
  • S12 becomes
  • finally using T22, S12 is found as

66
S21
  • S21 forward transmission coefficient
  • measure

67
S21
  • calculating S21
  • find the transmission T11 of V1 into the circuit
  • where V1 sees Zin,11

68
S21
  • T11 using Zin,11
  • we find T11 as

69
S21
  • port 2 only sees voltage across Z0
  • S21 becomes
  • finally using T11, S21 is found as

70
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 ?

71
questions
  • until now is everything clear ?
  • you could do this by yourself ?
  • Yes ? Lets do it!

72
test yourself
  • turn around todays slides
  • prepare paper and pencil
  • you have 5 min
  • ok ?

73
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

74
test results
  • was it easy ?
  • why was it easy ?
  • what were the difficulties ?

75
solution
  • As L and C are at resonance we only deal with the
    resistive elements
  • the impedance into the circuit is
  • and thus

76
  • back to S-parameter calculation....

77
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 ?

78
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

79
2-port relations
this is what we measure ? save experimental data
as S-data
converting from S-parameters to any format you
want
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