Frequency reference standards are essential to achieving frequency accuracy and phase stability in electronic systems. Such sources require the chief characteristics of low phase noise and good frequency stability.^1-13 The best oscillator performance can be expensive, however. Fortunately, a patented approach has been developed to design and optimize the performance of voltage-controlled crystal oscillators (VCXOs), even those with relative low quality-factor (Q) resonators, to achieve excellent phase noise and frequency stability.

A typical oscillator consists of a tuned circuit and an active device such as a transistor. Ideally, the tuned circuit provides a high loaded Q, generally from less than 100 for simple circuits to more than 1 million for crystal-resonator-based circuits. Noise arises from the active device as well as from resonator losses. Noise from a bipolar transistor, for example, stems from base and collector contributions and from device parasitic elements, such as the base-spreading resistor. The filtering effect of the resonator tends to remove the device noise, with higher Qs delivering greater filtering effects. The Leeson equation relates these noise effects.¹ The formula was modified by Rohde for use with VCOs.²

The equation is linear, with many unknowns. Among the more difficult oscillator performance parameters to predict are output power, noise figure, operating Q, and flicker corner frequency. The parameters can not be derived for linear conditions but require large-signal (nonlinear) analysis.³ But by combining Leeson's formula with the contributions of the tuning diode,² Eq. 3 results, making it possible to calculate oscillator noise based on a linear approach:

where:

£(f_m) = the ratio of sideband power in a 1-Hz bandwidth to the total power (in dB) at the frequency offset (f_m);
f₀ = the center frequency;
f_c = the flicker frequency;
Q_L = the loaded quality factor (Q) of the tuned circuit;
F = the noise factor;
kT = 4.1 10^–21 at 300°K (room temperature);
P_sav = average power at oscillator output;

R = the equivalent noise resistance of tuning diode (typically 50 Ω to 10 kΩ); and
K_o = the oscillator voltage gain.

Equation 1 is limited by the fact that loaded Q typically must be estimated; the same applies to the noise factor. The following equations, based on this equivalent circuit, are the exact values for P_sav, Q_L, and F, which are required for the Leeson equation. Figure 1 shows the typical simplified Colpitts oscillator giving some insights into the novel noise calculation approach.⁴

From ref. 3, the noise factor can be calculated by:

After some small approximation,

Figure 2 (left) illustrates the dependency of the noise factor on feedback capacitors C₁ and C₂. From Eq. 1, the phase noise of the oscillator circuit can be enhanced by optimizing the noise factor terms as given in Eq. 3 with respect to feedback capacitors C₁ and C₂.

Equation 4 can be found by substituting 1/r_e for Y₂₁⁺ (+ sign denotes the large-signal Y-parameter).

When an isolating amplifier is added, the noise of an LC oscillator is determined by Eq. 5.

where:

G = the compressed power gain of the loop amplifier;
F = the noise factor of the loop amplifier;
k = Boltzmann's constant;
T = the temperature (in degrees K);
P₀ = the carrier power level (in W) at the output of the loop amplifier;
F₀ = the carrier frequency (in Hz); f_m = carrier offset frequency (in Hz);
Q_L = (πF₀τ_g) = the loaded Q of the resonator in the feedback loop; and a_R and a_E = the flicker noise constants for the resonator and loop amplifier, respectively.