oscillation frequency estimate using harmonic balance

Dear fishbed,

I have experience designing quartz based designs and I would recommend that a different approach will provide the level of accuracy you desire and reduce the simulation accuracy requirements significantly. I will outline the methodology and hope it is both clear and useful to you.

For very high Q resonator based circuits such as a bulkwave quart crystal unit, the use of a "negative resistance" approach is one methodology that very efficiently and accurately will estimate the temperature dependence of the oscillating amplifier.

The methodology requires that one simulate the negative impedance of the oscillator sustaining amplifier by breaking the loop at the quartz resonator nodes. I've illustrated the concept in the attached figure. The input impedance of the parallel combination of the sustaining amplifier input node(s) with the C0 of the quartz crystal unit (including any board parasitics of the one or two quartz crystal unit node(s)) is simulated by applying a sinusoidal current source at or near the desired oscillation frequency in lieu of the series impedance arm of the quartz crystal unit model. A second, independent, current source is connected to the series arm of the quartz crystal unit model to simulate its impedance. The input impedance of the parallel combination of the oscillating amplifier input nodes with the C0 of the quartz crystal unit is determined by a transient simulation with the amplitude of the sinusoid set to a value that provides a negative real impedance that is equal and opposite in sign to the resistance of the series arm of the quartz crystal unit model. This is found by a series of iterative transient simulations of the sustaining amplifier where the amplitude of the current source is varied. This process can be automated. Note that since the high Q element is not in the sustaining amplifier simulation, the simulation time is minimal as the settling time of the sustaining amplifier will be orders of magnitude less than the time constant of the high Q element. The sustaining amplifier input impedance is found by taking the Fourier transform of the impedance at the applied sinusoidal input current frequency oscillation.

To determine the temperature coefficient of the sustaining amplifier, one needs to determine the imaginary part of the sustaining amplifier input impedance as the junction temperature is changed that produces a real part that is equal and opposite to the series resistance of the series arm of the quartz crystal unit model. In this fashion, if you know the temperature characteristics of the imaginary part of the quart crystal unit series arm, you can combine this with the temperature characteristics of the imaginary part of the sustaining amplifier to compute the resulting output frequency. This methodology is used to design temperature compensated oscillators.

I am sure this will take some thought to fully understand I will apologize if my explanation is either too brief or not clear. However, the process drastically reduces the simulation accuracy and simulation time for high Q resonator based circuits. I have responded to similar questions in the designer guide forums and these responses may be helpful.

Shawn

Dear fishbed,

I have experience designing quartz based designs and I would recommend that a different approach will provide the level of accuracy you desire and reduce the simulation accuracy requirements significantly. I will outline the methodology and hope it is both clear and useful to you.

For very high Q resonator based circuits such as a bulkwave quart crystal unit, the use of a "negative resistance" approach is one methodology that very efficiently and accurately will estimate the temperature dependence of the oscillating amplifier.

The methodology requires that one simulate the negative impedance of the oscillator sustaining amplifier by breaking the loop at the quartz resonator nodes. I've illustrated the concept in the attached figure. The input impedance of the parallel combination of the sustaining amplifier input node(s) with the C0 of the quartz crystal unit (including any board parasitics of the one or two quartz crystal unit node(s)) is simulated by applying a sinusoidal current source at or near the desired oscillation frequency in lieu of the series impedance arm of the quartz crystal unit model. A second, independent, current source is connected to the series arm of the quartz crystal unit model to simulate its impedance. The input impedance of the parallel combination of the oscillating amplifier input nodes with the C0 of the quartz crystal unit is determined by a transient simulation with the amplitude of the sinusoid set to a value that provides a negative real impedance that is equal and opposite in sign to the resistance of the series arm of the quartz crystal unit model. This is found by a series of iterative transient simulations of the sustaining amplifier where the amplitude of the current source is varied. This process can be automated. Note that since the high Q element is not in the sustaining amplifier simulation, the simulation time is minimal as the settling time of the sustaining amplifier will be orders of magnitude less than the time constant of the high Q element. The sustaining amplifier input impedance is found by taking the Fourier transform of the impedance at the applied sinusoidal input current frequency oscillation.

To determine the temperature coefficient of the sustaining amplifier, one needs to determine the imaginary part of the sustaining amplifier input impedance as the junction temperature is changed that produces a real part that is equal and opposite to the series resistance of the series arm of the quartz crystal unit model. In this fashion, if you know the temperature characteristics of the imaginary part of the quart crystal unit series arm, you can combine this with the temperature characteristics of the imaginary part of the sustaining amplifier to compute the resulting output frequency. This methodology is used to design temperature compensated oscillators.

I am sure this will take some thought to fully understand I will apologize if my explanation is either too brief or not clear. However, the process drastically reduces the simulation accuracy and simulation time for high Q resonator based circuits. I have responded to similar questions in the designer guide forums and these responses may be helpful.

Shawn