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DATA COMMUNICATIONS APPLICATION NOTE

DAN108

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5. Achieving Barkhausen’s Criteria

Satisfying both Barkhausen’s Criteria for Oscillation is not easily achieved when one wants to

design a sinusoidal oscillator with a well-defined frequency of oscillation. Fortunately digital

circuits work off square wave clock, which can be produced from an amplifier whose gain is

high and whose input is overdriven.

An amplifier’s output voltage is limited by the supply voltage – which means that the amplifier’s

gain is not constant. The gain approaches zero as the input sinusoidal signal’s

amplitude

increases. This means that the output amplitude of the amplifier/inverter is independent of both

frequency and input amplitude, assuming a high gain amplifier. This makes achieve

Barkhausen’s Amplitude Criteria feasible.

In order to satisfy Barkhausen’s Phase Condition for oscillation, a suitable feedback element,

F(

ω), needs to be selected. Since the inverter creates 180° phase delay (within its pass-band), an

additional 180

° is needed. This can be achieved with passive R, L, C elements. However

frequency selectivity is limited by the accuracy of the LC components. Enter the piezo-electric

quartz crystal.

6. Quartz Crystals

Electrically, a quartz crystal can be modeled by the equivalent circuit given in Figure 5.

Figure 5: Crystal Equivalent Circuit

Cp represents the capacitance of the crystal electrodes plus the holder and the leads. Rs, Cs and L

model the motional parameters of the crystal. The motional inductance, L, represents the

vibrating mass of the crystal. The motional capacitance, Cs, represents the elasticity of the

quartz. Rs represents the bulk losses that occur within the quartz. The relative magnitudes of the

shunt capacitance, Cp, series capacitance, Cs, inductance, L, and series resistance, R, results in a

typical crystal reactance curve shown in Figure 6.

L

Rs

Cp

Cs