Basic Transformation of wave lines
Wave lines come out in different mechanical configurations , either as Coaxcables, 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 Cloaded 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 Lload 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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