1. 負性抵抗（-R）の具体的算出方法

Negative resistance (-R) is the oscillation starting power of the circuit (IC), and is obtained by the following formula.

Re is the load equivalent resistance of the crystal oscillator, which is the resistance value when the crystal oscillator is mounted in the circuit. Re is given by the following formula:

Here, substituting Rs=18.7Ω, Co=1.1pF, and CL=9pF results in the following formula.

ESR = 1.26 times 18.7Ω.
(When the crystal oscillator is mounted on the board, the resistance value increases by 1.26 times.)

Next, Rv is the minimum resistance value at which the circuit starts to oscillate, and is obtained by actual measurement.
Gradually decrease the value from the state where Rv is maximized and oscillation is stopped, and find the Rv value when oscillation starts.
The measured value this time was Rv = 2430Ω.

Therefore, the value of the negative resistance is obtained by the following formula.

2. Specific calculation method for oscillation margin

Oscillation margin is a magnification that indicates how much the circuit (IC) has the power to oscillate with a margin when the ESR value of the crystal oscillator is the worst of the standard values. .
The magnification is expressed by the ratio of the resistance value (Re) when the crystal oscillator is mounted on the circuit and the negative resistance (-R).

The following is the formula for the oscillation margin.

This is the formula for Re, referring to the above.

When calculating the oscillation margin, Rs is calculated using the ESR standard value instead of the measured value of 18.7Ω.
This is because the ESR value of delivered crystal oscillators also varies, and it is possible that some of the delivered products may be just above the standard value, so it is safer to calculate the margin under the worst conditions. am.
This is an important point related to 5. Circuit analysis and pass/fail of the board, as described later.

In the case of Kyushu Dentsu SEG's 24.576MHz, the standard value is 90Ω.
Therefore, substituting Rs=90Ω, Co=1.1pF, and CL=9pF gives 112.9Ω.

If the ESR of the crystal is 90Ω, it will be 112.9Ω on the board.

Therefore, the oscillation margin is obtained by the following formula. (It means that there is 21.7 times more oscillation margin even when the ESR value is the worst 90Ω.)

3. Specific calculation method for drive level

The drive level value is calculated by the following formula.

Here, Re = Rs x ( 1 + Co / CL ) ², so substitute Rs = 18.7Ω, Co = 1.1pF, and CL = 9pF to find the value of Re.

The crystal current value (I) was actually measured to be 2.26mA, so substitute that to find the drive level.

A drive level of around 100µW is generally considered appropriate, so this value should be fine.

Negative resistance (-R), oscillation margin, drive level, etc. are obtained in this way.

4. Negative resistance (-R), method of increasing oscillation margin

Depending on the circuit received, the negative resistance and oscillation margin may not be sufficient. In such a case, you can increase the oscillation margin by increasing the negative resistance by the following method. If methods 1 and 2 are not sufficient, try method 3.

1. Decrease the value of damping resistor Rd
2. Decrease the value of the external capacitor
3. Tighten ESR standards

As mentioned in 2. How to specifically calculate the oscillation margin, the oscillation margin is calculated using the ESR standard value, so the margin can be increased by tightening the standard value. In this case, the value of the negative resistance does not change.

For example, if you halve the standard value instead of 90 in the formula below and substitute 45, for example, the value of Re will be halved, so it is theoretically possible to double the oscillation margin.

However, tightening the ESR standard may not be possible, and even if it is possible, the yield will be poor, so a separate estimate will be required.

5. Check by circuit matching (circuit analysis)

At the end of Kyushu Dentsu's circuit analysis report, it states, "The examination results are the evaluation results within the scope of the board and parts that we have received. Please check the accuracy design including the variation of each part by yourself." The wording is attached.

There is only one crystal oscillator mounted on the analysis board, but there will always be variations when it is delivered in mass production. The ESR value also always varies within the standard range.

1. When determining the negative resistance in the specific calculation method of the negative resistance (-R), the calculation is based on the actual measurement value of the crystal oscillator mounted on the board, but in the end, it is mainly the IC. Since the purpose is to determine the oscillation starting power (IC capability) of the circuit itself, the ESR value of the mounted crystal oscillator is used for calculation. Mounting crystals with different values of ESR does not change the value of the negative resistance.

However, the oscillation margin in 2. How to specifically calculate the oscillation margin must be considered based on the assumption of all ESR values that have variations, so the maximum value of variation = the standard value. It means that you have to calculate on the premise.

In addition, there are variations in the various parts mounted on the board, and there are also variations in the IC itself.

Even if there are variations in the frequency stability and ESR value of the crystal oscillator, it can be confirmed by circuit analysis that the crystal oscillator oscillates even during mass production, but the board itself does not pass the test. Please note that it is not intended to judge whether or not the final product is acceptable.

Even if the oscillation margin is 5 times or more, it may be returned due to oscillation failure.
There are various possible causes, but there are cases where oscillation problems may have occurred due to a change in the IC lot, and noise during the period after power-on until oscillation stabilizes. A case has been reported that it may have jumped in and prevented proper oscillation.

In any case, please keep in mind that ensuring an oscillation margin of at least 5 times through circuit analysis does not mean that future oscillation problems will never occur.