Open loop gain and open loop phase for an oscillator

Hi Pictou,

You are asking  a number of questions that are really outside the scope of this forum I think - but I will let Andrew make that judgement. The questions are really appropriate for a forum on quartz oscillator design. Nevertheless, I will provide my insights since I do quartz oscillator design,

I'm not sure if my results are correct or not.

I examined your simulation schematic and it appears you are following the methdology correctly - very good! However, your schematic suggests you are using a value of 750 fF for the quartz crystal unit C0. This is a very low value and I think it may not be correct, Most quartz crystal units in the 40 MHz range will have a C0 (which includes both the capacitance of the resonator and the packaging and board) in the range of 2 pf to 5 pf. The vendor will usually specify its value on the data sheet (without your board parasitic) - but it is usually specified as a value far in excess of the value for any speciifc frequency. Often, you need to contact the vendor to get the specific value for the frequency and package type of interest. The value of C0 will significantly impact the value of negative resistance you simulate at 40 MHz. Hence, I suspect the -880 ohm value you observed will be less in absolute value when the proper value of C0 is used.

My result is -881Ohm at Fosc (40MHz), my gain is then equal to : 880/R1 (which is 21.58), 880/21.58 = 41.

My question is the following, I found on the internet that the result should be around 4-8 times the motional resistance R1, am I not a little too high?

How should I assess this result? What can I say about it?

I think that this result means that if I wanted my oscillator to oscillate at 40MHz, I could do it because I have enough gain. Am I correct?

 Your basic calculation methodology for oscillator open loop gain is correct for the specific simulation you performed. 

First, I do not know if you have considered what the minimum absolute value of negative resistance is over all process, supply voltage, and temperature conditions. I also noted that you appear to bias one transistor with an ideal voltage source. If you are trying to determine the minimum absolute value of negative resistance, you must examine all cases and exclude ideal bias elements. As mentioned above, I think if you use a more realistic value of C0, this will also significantly reduce the magnitude of your measured negative resistance at 40 MHz.

Second, there are many other considerations that one must weigh when choosing the proper value of negative resistance. A rule of thumb is not appropriate. For example, how does the negative resistance saturate as the amplitude of oscillation increases? How temperature and voltage sensitive is the negative resistance? How much quartz crystal resonator drive level sensitivity exists? How much frequency trim capacitance is required and how does that impact negative resistance? Hence, you need to answer these questions to determine how much margin is required for your circuit. 

EDIT : My colleage asked me a question that I couldn't answer to. Why is the gain the ratio between the negative resistance and the motional resistance? Thank you.

 I think my answer to this is outside the scope of this forum. Andrew, if you feel otherwise, I will respond.

Shawn

 

Hi Pictou,

You are asking  a number of questions that are really outside the scope of this forum I think - but I will let Andrew make that judgement. The questions are really appropriate for a forum on quartz oscillator design. Nevertheless, I will provide my insights since I do quartz oscillator design,

I'm not sure if my results are correct or not.

I examined your simulation schematic and it appears you are following the methdology correctly - very good! However, your schematic suggests you are using a value of 750 fF for the quartz crystal unit C0. This is a very low value and I think it may not be correct, Most quartz crystal units in the 40 MHz range will have a C0 (which includes both the capacitance of the resonator and the packaging and board) in the range of 2 pf to 5 pf. The vendor will usually specify its value on the data sheet (without your board parasitic) - but it is usually specified as a value far in excess of the value for any speciifc frequency. Often, you need to contact the vendor to get the specific value for the frequency and package type of interest. The value of C0 will significantly impact the value of negative resistance you simulate at 40 MHz. Hence, I suspect the -880 ohm value you observed will be less in absolute value when the proper value of C0 is used.

My result is -881Ohm at Fosc (40MHz), my gain is then equal to : 880/R1 (which is 21.58), 880/21.58 = 41.

My question is the following, I found on the internet that the result should be around 4-8 times the motional resistance R1, am I not a little too high?

How should I assess this result? What can I say about it?

I think that this result means that if I wanted my oscillator to oscillate at 40MHz, I could do it because I have enough gain. Am I correct?

 Your basic calculation methodology for oscillator open loop gain is correct for the specific simulation you performed. 

First, I do not know if you have considered what the minimum absolute value of negative resistance is over all process, supply voltage, and temperature conditions. I also noted that you appear to bias one transistor with an ideal voltage source. If you are trying to determine the minimum absolute value of negative resistance, you must examine all cases and exclude ideal bias elements. As mentioned above, I think if you use a more realistic value of C0, this will also significantly reduce the magnitude of your measured negative resistance at 40 MHz.

Second, there are many other considerations that one must weigh when choosing the proper value of negative resistance. A rule of thumb is not appropriate. For example, how does the negative resistance saturate as the amplitude of oscillation increases? How temperature and voltage sensitive is the negative resistance? How much quartz crystal resonator drive level sensitivity exists? How much frequency trim capacitance is required and how does that impact negative resistance? Hence, you need to answer these questions to determine how much margin is required for your circuit. 

EDIT : My colleage asked me a question that I couldn't answer to. Why is the gain the ratio between the negative resistance and the motional resistance? Thank you.

 I think my answer to this is outside the scope of this forum. Andrew, if you feel otherwise, I will respond.

Shawn