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Electronic equations content :

 

S-parameter-equations :

S-parameter are very practical and have outset classical parameter  like VSWR and reflection. Here are the basic equations of S-parameter and it’s transformation formulas to impedance, reflection and standing wave ratio. S-parameter are widely used in electronic CAD programs in the popular touchstone format. We can type a touchstone format file from measured S parameter values.Using this files we can  compute Y to S-parameter values.>>>go to S-parameter example

 

S-parameter touchstone file format

S-parameter catalog or measurement values can be written in a touchstone cad-file to be used in CAD-programs. The green text must be written in a ASC2 DOS text-file.

  • !  filename.S2P
  • !  filename
  • #  HZ S MA R 50 Z
  • !  S-PARAMETER DATA
  • 1.200E+08 9.558E-01 -1.486E+02  2.632E-02 5.941E+01   2.632E-02SS 5.941E+01 9.544E-01 1.062E+02
  • NEXT FREQUENCY AND DATA .......AND SO ON

     frequency S11 S11angle S21 S21angle S12 S12angle S22 S22angle

 

Y-parameter touchstone file format

Y-parameter catalog or measurement values can be written in a touchstone cad-file to be used in CAD-programs. The blue text must be written in a ASC2 DOS text-file.

  • !  filename.Y2P
  • !  filename
  • #  HZ S MA R 50 Z
  • !  Y-PARAMETER DATA
  • 1.200E+08 9.558E-01 -1.486E+02  2.632E-02  5.941E+01   2.632E-02SS 5.941E+01 9.544E-01 1.062E+02
  • NEXT FREQUENCY AND DATA .......AND SO ON

     frequency S11 S11angle S21 S21angle S12 S12angle S22 S22angle

 

Y-parameter to S-parameter equations

Y-parameter can be expressed as S-parameter and vice versa. Some older component catalogues only show the y parameters. To avoid tedious computing , one can create a Y-block to be analyzed in a CAD-program and save the results as S-parameter file.

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Norton transformation of circuits

The circuit of a lumped filter can be simplified ,, using Norton’s transforming equations . That means a part of the circuit can be replaced by an other circuit and a ideal transformer. The transformer then must shifted to the input or output of the circuit and removed by changing the actual input or output impedance. An other way to free the changed circuit from the transformer, is to transform 2 circuit parts ore more having the opposite transforming ratio. The transformer than can be canceled if they are directly connected.. Go to  Norton’s example. At Fig.1-5 the CIR named circuit parts of a filter can be replaced by the transformer circuits 1...2...3...and / or transformed with the transformer ratio.There are other possibility’s of Norton’s transformation but they are not shown here.
                                              Fig.1 Transfomer with load and tapped coil

 

                                                                Fig.2 -5 Nortons transformation

Noise power and density

To measure the Noise produced in a communication system, the following formulas are valid:

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 Constants of RF-materials

Each Electronic book has an epsilon table of materials, but the constants of PCB-materials like FR4 or Pertinax or Araldid, you never will find. Here are the epsilon values of popular RF-materials:

 

Material

 Frequency/Mhz

tan delta

 Epsilon

Air

 

 

1

PCB-FR4

 

200

5,4

PCB-522

 

20

2,6

Polyguide

1-1Ghz

20

2,32

Duroid-Microw.

1-10GHz

15

2,2

Teflon

1

5

2

PVC

1

30

3,4

Polystrol

1

80

2,9

Makrolon

 

1,1

3,4

Styroflex

1

2

3,3

Polyäthylen

50Ghz

30

2,05

Paper

 

400

2,6

Mica

 

6

7

Ceramics

 

3-30

5-100

Eccosorb Emmerson

 

1,2

40

Silicon Wacker

5

30

3

Silicon Dow corning

1Ghz

400

3,5

Silicon Elastoul

5

280

3,7

Araltit

0,1

40

3,3

Scotchcast

0;1

 

3,0-5     20-80grad

Poleurethan

50Hz

230

2,6

Sylgard

1

10

2,8

Pertinax

 

 

4

The capacitance of a PCB-capacitor

Designing RF-circuits means to be aware of parasitics. Parasitic capacitors can be estimated using the basic formulas:

RF-transformer coupling:

We have two coupled RF-coils and want to know the coupling. Here is how we can measure it:

  • Measure the resonant frequency with a Q  or grid dip meter.
  • Measure the resonant frequency Fclosed with the secondary coil shorted.
  • Measure the resonant frequency Fopen with the secondary coil open
  • Compute the coupling using the following formula:

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Replace two resonators in lumped filters

Lumbed RF- Filter circuits from a catalogue, often seems to be very complicated. Here are the formulas to replace  a two resonator circuit into a one resonator circuit. ( only valid for narrowband circuits)

  Fig.1 Replacement of two parallel resonators

Fig.1 Replacement of two series resonators

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Basic Transformation of wave lines

Wave lines come out in different mechanical configurations , either as Coax-cables, lines on a PCB or ceramics  (Micro Strip) ,Parallel lines, Twisted wires and special Strip lines. Each type has its one impedance Z.

The impedance of all this configurations are different and depend on the material, and the mechanical size. The basic transformation formulas for all configurations are the same: A complex output resistor will be transformed to an other complex resistor at the input of the line. Fig.1  The values of this input resistor can be found, either  by means of the formulas, shown at Fig.2, or using a Smith Chart :

 

 

Fig.1 Transforming Wave line equations

Fig.2 Basic wave line transforming formulas

The shorted wave line resonator

As the length of the line is important, multiples of  the length to lambda factor, make the conditions, where the line will resonate. Either as parallel or series resonator. Using this length, we get new conditions especially if the line is shorted  on the end. Fig.3 shows the resonant conditions for the shorted line. The transformation equation is simplified to :                                                         

Fig. 3 Resonance condition  with short termination

     Fig.4 shorted line resonance                                                  

  • This means, a shorted Quarter Lambda resonator is a parallel resonator and will ring at other frequencies too, and a  shorted Half Lambda resonator works as a series resonator and will ring at other frequencies  too.
  • Theoretically the open wave line will resonate in the opposite manner. Series and parallel resonance will change. Practically this condition is unstable and is not used.

The capacitor input, wave line resonator

This resonator has an paralleled capacitor at the input. Fig.5 This is the typical filter resonator. The capacitor Cp is used to adjust the resonator frequency . The resonance formula shows ringing at frequencies different  from the  harmonics to the basic resonance.. The resonance frequency can be found using a little computer program on a pocket computer to solve following equation:   

  • Example : l = 30 cm; Cp = 10 pF ; Z = 60 Ohm; 
  • fres1 = 170 Mhz  >>>> basic resonance;
  • fres 2 = 580 MHz ;
  • fres 3 = 1030 MHz ;   Go to example of line resonator:

Fig.5 C-loaded Resonator                                        

The C or L loaded wave line Resonator

  • If  Ro at Fig.1 is a capacitance or inductance, we have an  other special condition, where  we get parallel resonance at shifted frequencies:
  • Wave line Parallel Resonance Condition using C- Termination :
  • Wave line Parallel Resonance Condition using L - Termination :
  • Important Knowledge : If at low frequencies the resonator is mechanically to long, an L-load is shorting  the length.

The lumped resonator values of  wave lines

  • For a given shorted wave line resonator, an equivalent lumped circuit can be computed. Fig.6
  • The equations : Fig. 7

 Fig.6 Equivalent lumped circuit

                                                                         Fig.7 The values of the lumped circuit    

  Link>>> Go to example of line resonator

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The Surface Resistance of metals at RF-Frequencies

RF- losses in wires, filters, resonators, PCB-connections a.s.o., depends on the RF-skineffect  at the surface of the used metals. The skin effect in the RF-range is defined as resistance at a certain area A. The skin effect factor Rho’( ) is  normalized to a length of 1cm. The following formula is valid:

For instance , R’ is the loss per cm of a coax cable having a an impedance of  Z = SQR(L’/C’). R’ must be used to calculate the Q-values of wave line resonators. The values of the specific surface resistance are shown in a diagram. Fig.1 shows the normalized factor Rho’ for different materials as silver, copper, aluminum, and brass in the RF-frequency range from 300 kHz to 30 Ghz
Fig.1 Specific surface resistance of metals versus wavelength

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Impedance’s of Wave lines

The impedance of a wave line depends on the mechanical dimension and the epsilon value of their insulation.

Fig.1 and 2 presents  the impedance equations of following wave lines:

  1. Single line above ground
  2. Coaxial line
  3. Parallel line
  4. Parallel line above ground
  5. Parallel line with round screen
  6. Parallel line with oval screen
  7. Strip line
  8. Micro strip
  9. Twisted wires

Link >>> Go to twisted wires

 Fig.1 Impedance equations

     Fig.3 Z of Micro-Strip  at epsilon air =1   (for other insulation materials Zx = Zair/SQR(Epsilonx)

 Link >>> Go to epsilon values

  Fig. 2 Wave lines

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Magnetics basic equation:

The basic magnetic equation often has been forgotten . Remember the induction in a magnetic core Fig.1, and go to an example of magnetic: >>> go to magnetic example  >>>> t.b.c.

Fig.1 Basic Magnetic formulas

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