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Quartz_Crystal_Filter_Designer_1.php   11083 Bytes    10-12-2016 12:31:59


Quartz Crystal Filter Design


Equivalent Circuit Determination of Quartz Crystals



Before we may design a crystal filter it is necessary to know the crystal's data. Unfortunately this information is not in the datasheet - or we have no datasheet because the crystals were bought on a market some decades ago ...




Quarz Quarzersatzschaltung

A quartz crystal and its equivalent circuit




As you may have guessed from the equivalent circuit above, a quartz crystal has a series resonance (Cs and L) as well as a parallel resonance. This may be observed very well, when doing a frequency sweep, using the quartz as a series element. In order to determine these four values (R, Cs, L, Cp) one measurement is not enough. We use a small testfix for the frequency sweep. Building such a thing allows for stable electronic and mechanical conditions, leading to reliable and reproduceable results.

Those four values may be obtained by four measurements :

• Holder Capacitance : Cp
• Series resonance frequency : fs
• Parallel resonance frequency : fp
• The dynamic Losses : R




Quartz Crystal Measurement Adapter

Measurement Adapter in Action, Details about this test fixture




Measure Cp




Quartz Crystal Measurement Adapter The easiest way to get this value is the use of a RLC-Meter. This has the ad- vantage, that no calibration is neces- sary. (Except zero).




Measure fs and fp




The Series / Paralle resonance frequency may be measured by a lot of setups. A vector network analyser is of course the luxury version, but you may as well use a generator and a scope, a spectrum analyser with a generator or any linear combination thereof. The quartz crystal is connected in series between the source and the load. We are intersted in the maximum amplitude (fs, where XL = XCs) and the minimum (fp, where XL and XCs, XCp form a block).




S21 measured by Rostig & Schwer

Measured Amplitude Response of a Quartz Crystal in the test fixture




Measure Insertion Loss at fs




This measurement targets the internal "Resistor". We measure exactly at the series resonance frequency, because there, XL and XCs sum up to zero. So what is left, is a (resistive) voltage divider, consisting of your source, the R from the quartz and the load. Whereas source and load are mostly 50 Ω. Please keep in mind to calibrate or to measure the difference (shorted / quartz inserted).




Quartz Crystal Equivalent Circuit #1


Series resonance frequency

  [MHz]

Parallel resonance frequency

  [MHz]

Holder Capacity Cp

  [pF]

Cs

  [fF]

Ls

  [mH]





Quartz Crystal Equivalent Circuit #2


System Impedance

  [Ω]

Insertion Loss

  [dB]

R

  [Ω]





PROCEED HERE TO LADDER FILTER CALCULATOR




The Mathematics behind




Series resonance frequency :      `fs = 1 / ( 2 pi sqrt(L Cs) `     (1)


Parallel resonance frequency :      ` fp = fs sqrt(1 + (Cs)/(Cp) ) `     (2)


Quality factor at fs :      ` Q = 1/R sqrt((Ls)/(Cs)) = ( 2 pi fs Ls) / R `     (3)


R as a function of Insertion Loss :      ` Rs = 2 Zo (10^(a/20)-1) `     (4)


Equation (2) may be rewritten as :      ` Cs = Cp ( (fp^2) / (fs^2) - 1) `


And from Equation (1) we get :      ` Ls = 1 / ( 4 pi^2 fs^2 Cs ) `



Credits :

The Visualisation uses https://www.mathjax.org/





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t1 = 3913 d

t2 = 560 ms

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