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Quartz Crystals
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Quartz Crystal Theory

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The Equivalent Circuit

L1 - Dynamic or Motional Inductance (mH)
C1 - Dynamic or Motional Capacitance (fF) (10-15 F)
R1 - Equivalent Series Resistance (ohms)
C0 - Parallel or Static Capacitance (pF)
The Equivalent Circuit

Note 1:  L1, C1 and R1 are not true electrical components. They are apparent values of inductance, capacitance and resistance which serve to model mechanical / piezoelectric of the vibrating crystal in the region of resonance.
Note 2:  In all cases the units shown in brackets are those normally used for AT-cut crystals.
Note 3:  It is necessary to use consistent units in the equations, i.e. Farads, Henries, etc.

Circuit Equations - Basic Crystals

Series Resonant Frequency:
Parallel Resonant Frequency:

Note:- In practice, an allowance usually has been made for 'strays'

Quality Factor:
Capacitance Ratio:
Figure of Merit:

Circuit Equations - with external load

Frequency Deviation - series to load:

Note 1: Series Capacitance (CL) increases the frequency.

Note 2: The effect of a trimmer may be readily assessed by calculating D and D' for CL and CL' and subtracting.

Note 3: Units are correct for C1, C0 and CL in Farads.

The Equivilent Circuit with load
Trim Sensitivity:

Load Capacitance v. Trim Sensitivity
The above graph shows the effect of external load capacitance on the trim sensitivity of a representative 20MHz fundamental mode crystal. Notice the large frequency shift with small load changes when fundamental mode crystals are operated at low load capacitance's. Overtone mode crystals are generally less affected by external loads. A rough approximation is to reduce the fundamental mode trim sensitivity by a factor of the overtone mode crystal.

Apparent Series Resistance (with CL):

Frequency v. Temperature Curve

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