Oscillation Circuit Evaluation Methods
What is "frequency deviation " (1) <Overview>
This time, we would like to present an overview of frequency deviation.
The fundamental of frequency deviation (symbol is "dF") is difference between the frequency to be measured and the reference frequency.
In order to clarify the degree of deviation, dF divided by the reference frequency is generally called " frequency deviation"(symbol is "dF/F" ).
General formula for "dF/F" is following equation 
(* DUT: device under test)
The concept of equation  is also used in crystal units specification sheets.
Frequency tolerance described in crystal units specification sheets are calculated by the following equation .
If you calculate the lowest and highest frequency expected for a crystal units, substitute the lowest and highest values of frequency tolerance into dF/F in formula (3) below, which is a variant of formula (2) above, to obtain them.
For example, the specification of a certain crystal unit states
Nominal frequency: 20.000MHz
Frequency tolerance: ±50 ppm max.
Load capacitance: 8 pF
The lowest and highest frequencies of the crystal unit are derived as follows.
Minimum frequency: at CL=8pF : 20.000MHz×(1-50×10-6 )＝19.999000MHz
Maximum frequency: at CL=8pF : 20.000MHz×(1+50×10-6 )＝20.001000MHz
２．Frequency deviation in circuit analysis
The frequency to be measured in the circuit analysis is the on-circuit frequency of the circuit under the test.
Generally, the nominal frequency is used as the reference frequency, but the nominal frequency is not used as the reference frequency in the circuit analysis.
The reason for this is that crystal units have frequency tolerance.
The following is an explanation based on a specific example.
For convenience of explanation, we will name minimum frequency product of a crystal oscillator with a nominal frequency of 20.000MHz described in "Basics" and maximum frequency product as follows.
Crystal A: minimum frequency product (19.999000MHz at CL=8pF)
Crystal B: maximum frequency product (20.001000MHz at CL=8pF)
Suppose that the circuit frequency is 19.999100MHz when crystal A is mounted on a circuit.
When crystal B is mounted on the exact same circuit, the frequency on the circuit is different from that when crystal A is mounted. The frequency by simple calculation is 20.001100MHz.
Assume that crystal B oscillates at 20.001100MHz as calculated above. If dF/F is calculated with the idea that the reference frequency is nominal frequency, the result is followings.
Case 1: using crystal A
Case 2: using crystal B
As just described above, the derived values are very different even though they are evaluated for exactly the same circuit.
To eliminate this difference, dF/F is derived by using the following equation  in circuit analysis.
|F nom||：Crystal nominal frequency|
|F CL※2||：Crystal frequency at CL= x pF (x : the value described in crystal units specification sheet)|
|F circuit※2||：Circuit frequency when the experimental crystal is mounted|
*1："Symbol of nearly equal in " holds because the difference between F nom and F CL is hundreds ppm, no matter how large.
*2：This symbol is not generic (this is provided for convenience in this report).
The results applying equation  to the previous example are the following.
Case : using crystal A
Case : using crystal B
The same value is derived.
Briefly summarized, equation  is used for dF/F in case of circuit analysis in order to exclude the influence of frequency tolerance any crystal unit has.
３．Frequency accuracy when frequency tolerance of crystal units are taken into consideration
When taking into account the frequency variation among products of crystal units, i.e., the frequency tolerance at room temperature (+25°C), the following equation can be used to determine the frequency tolerance of a crystal unit.
If you want to know what the oscillation frequency accuracy (the reference frequency is the nominal frequency) will be, add the frequency tolerance of the individual crystal unit to the value obtained in equation (4) to derive the value.
Frequency accuracy on circuit considering frequency tolerance of crystal unit is derived with the following equation .
·dF/F according to equation : +10ppm
·Frequency tolerance of crystal unit: within ±30 ppm (at +25°C).
In the above case, Frequency accuracy = +10 ppm ± 30 ppm = from -20 ppm to +40ppm
４．Frequency accuracy when frequency tolerance and frequency temperature characteristics of crystal units are taken into consideration
If you want to know what the oscillation frequency accuracy (the reference frequency is the nominal frequency) will be when taking into account frequency deviations between products and temperature fluctuations, the value can be derived by adding the frequency temperature characteristics of a single crystal unit to the value obtained in equation ⑤. In other words, the following equation (6) is obtained.
Oscillation frequency accuracy = Oscillation frequency accuracy according to formula (5) + Frequency-temperature characteristics of the single crystal unit ... ⑥. An example is shown below.
·dF/F according to equation : -10ppm
·Frequency tolerance of crystal unit: within ±40 ppm (at +25°C).
·Frequency temperature characteristic of crystal unit: within ±50 ppm (in range of operating temperature)
In the above case, oscillation frequency accuracy = -10 ppm ± 40 ppm ± 50 ppm = from -100 ppm to +80ppm
５．In case of changing the load capacitance specifications of the crystal unit
In the circuit analysis report, it may be proposed to change the load capacitance specifications of the crystal unit. How to derive dF/F in this case is described in the following.
The basic is the same as described in section 2.
The only difference is "which load capacitance is used to measure F CL."
Here is an example.
Nominal frequency of the experimental crystal unit: 10.000MHz (load capacitance specified in the specifications: 8pF)
F (CL=8pF)※3： F CL measured with load capacitance=8pF
F (CL=6pF)※3：F CL measured with load capacitance = 6pF
F circuit※3：Frequency on the circuit when the experimental crystal unit is mounted in the circuit to be measured.
※3：This symbol is not general (this is provided for convenience
In the above case, oscillation frequency deviation is derived as followings.
(A)：When the reference frequency is F (CL=8pF):
(B)： When the reference frequency is F (CL=6pF):