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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:

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 :    Go to Wave line impedance’s >> coming

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 : Go to Smith chart >>coming

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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Explanation of the Smith Chart

For some electronic people the smith chart is a secret cobweb. But its easy to understand. Here is a careful explanation of the Smith Chart . This RF development diagram is the sum of four circle diagrams :

1. Rotation circles of the reflection pointer
2. Normalized Constant Resistance Circles
3. Normalized Constant Reactance Circles
4. Normalized wave line reflection versus line length

1. Reflection circles

Reflection is the result of mismatched electronic devices, cables, connectors and so on. It is a complex pointer

 This leads to a pointer 360 degree circle. The pointer is 0 for a matched system or 1 at totally reflection. The total reflection circle is the basis circle of the Smith diagram. Fig1. shows the reflection circles over the background of a Smith diagram. In the Smith diagram, only the outer total mismatch circle is obvius.The other reflection circles are invisible, and the designer has to know, that this circles exist. In the paper smith charts, outside the circles ,a measurement line of reflection are shown, either as S11/S22, VSWR, w or other reflection values. In smith chart software’s the reflection circles can be switched on and off.

2. 3. Normalized resistance and reactance circles.

As reflection is always the result of a complex unit input impedance ore load, we can write the reflection in the following equation: A simplified formula is:

Doing some algebraic work, the reflection comes out  as a quadratic equation including to circles. One for the resistances and an other for the reactances.

Fig.1 Reflection in the smith chart

The organization of this circles inside the maximum reflection circle leads then to the basic smith diagram. Fig.2 . The reactance circles are cut off outside  the reflection circle p =1. As it is shown, impedance’s are normalized to a the standard, Wave Line or Unit input Impedance Z.                                                    

4.Normalized Wavelength.    

Using Wave lines, the reflection is an other circle Pointer: The reflection here is dependent on the propagation . We get : The reflection circle for wave lines therefore is scaled in normalized values of the line length to Wave Length   . Fig. 3 shows this circles, one in the direction to the load and another from the load.This are only scales, the reflection is inside the maximum reflection circle. In a paper smith chart all 4 circles are combined in one chart Fig.4, to work with reflection . The main purpose of the Smith Chart is to match  RF. units inputs and outputs in a way to get a reflection S11/ S22  greater then 15 to 20 dB. Fig. 3  shows a typical input reflection (S11) versus frequency of a RF Unit. (black line). Matching lumped elements and wave lines can improve the reflection and move the curve near to the zero reflection value (brown line).

5. Application using the Smith Chart:

• Transform Resistances to Admittances and vice versa
• Matching of RF Units and cables
• Stability of RF-Amplifiers
• Noise figure optimization
• Improvement of Wave Line Resonators

 Fig.2 Resistance circles (above)

6.Examples working with the the Smith Chart, readings:

Dr. Henne Fachhochschule Augsburg : Die Smith Chart 1963

Microwaves, Dez.1975 page 58 ....

Norm Dye and Helge Grandberg :Radio Frequency Transistors / Butterwoth Heinemann 1993...

Z-Match for Windows / Number ONE System Limited 1994

To work with a paper Smith Chart, copy the readable high resolution smith chart of Fig.4 using the Browser

Fig.3 Wavelength circle

Fig.4 Paper Smith Chart ( Free to copy with the browser)

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