Phase noise is a differentiating performance specification when comparing voltage-controlled oscillators (VCOs). it can be estimated for VCOs operating at different frequencies by means of Leeson's expressions, although a VCO parameter other than frequencyits tuning bandwidthalso plays a key role in the phase-noise performance of the oscillator. What follows is the development of an expression that suggests that the phase noise performance of two VCOs implemented using the same approach and technology are related by the ratio of the VCO tuning bandwidths. Te validity of the expression will be put to the test against the known measured phase-noise performance of several commercial VCOs.

Leeson's equation1 relates the phase noise of an oscillator to various factors:

L(ΔΩ) = 10log{(2FkT/P_{sig})0/2QΔΩ)^{2}> 1/f3)/|ΔΩ?|>} (1)

This empirical expression has been found to capture the key features of oscillator phase noise for a wide range of oscillator designs. as shown in Fig. 1, the phase noise associated with this expression can be broken into three distinct regions:

- A -30-dB/decade slope for close-in offset frequencies, due to upconversion of 1/f noise;
- A -20-dB/decade slope for intermediate offset frequencies due to limited resonator quality factor (Q); and
- Flat phase noise for large offsets where performance is dominated by the noise characteristics of the oscillator's active device(s).

Of course, these distinct regions are approximations; a real VCO has a phase-noise characteristic that is smooth and continuous. For offset frequencies of interest here, a slope in the range of 20 to 30 dB/decade will apply. A slope of "(20 + x)" will be used to represent a general case, where x is in the range of 0 to 10.

There is a wealth of literature on oscillator circuit design which relates phase-noise performance to key circuit parameters. A phase-noise figure of merit (FOM) is often used to assess a given oscillator design in a normalized sense, and can also be used to compare different oscillator solutions. This FOM takes account of phase noise, oscillation frequency, offset frequency, and DC power consumption. The FOM is given by the following expression: FOM = L(ΔΩ) -20log(Ω_{0}/ΔΩ) + 10log(P_{DC})

This FOM is generally applied to offsets in the 20-dB/decade region.

In a conventional VCO, the resonator incorporates at least one varactor diode. By adjusting the varactor bias voltage, its capacitance and, hence, the VCO's frequency of oscillation, is tuned.

When a VCO is oscillating at a given varactor-diode tuning voltage, the tank AC voltage waveform associated with the oscillation imposes a swing on the varactor voltage around its DC tuning value. This AC varactor bias results in a varying instantaneous frequency of oscillation which manifests itself as phase-noise skirts around an ideal single-tone spectral response.

The greater the AC tank voltage impact on the instantaneous frequency of oscillation, the worse the associated phase noise will be. Circuit designers typically go to great lengths to optimize VCO phase noise. One method that can be employed to achieve this involves the use of back-to-back varactor pairs.^{2} In the presence of AC tank swing, as one of these varactor diodes shifts to a higher instantaneous capacitance value, the other varactor shifts to a lower instantaneous capacitance value. The net effect is that the combined series capacitance shifts much less than the change in capacitance for either individual varactor diode. It is this combined series capacitance which plays a key role in determining the instantaneous frequency of oscillation. A more stable frequency of oscillation and improved phase noise result.

In a similar fashion, any noise or spurious signal on the applied varactor tuning voltage modulates the varactor capacitance and instantaneous frequency of oscillation, and also degrades the oscillator phase noise. Good decoupling of the tuning port minimizes this effect.

Generally, fixed-frequency oscillators do not contain varactor diodes, so the phase-noise degradation associated with the above mechanism does not arise. Therefore, fixed-frequency oscillators tend to exhibit better phase-noise performance than tunable VCOs using the same design approach and technology.

For VCO designs based on the same circuit topology and technology, it is important to understand how phase noise is related to tuning bandwidth. Experience and common sense suggest that the greater the tuning range, the worse the phase noise. However, a more quantitative understanding would be helpful in predicting tradeoffs between phase noise and VCO tuning bandwidth.

Consider the general case of a baseline VCO with tuning sensitivity S0 (MHz/V). Assume for simplicity's sake that the tuning characteristic is linear over the full tuning range (Fig. 2). This assumption is not strictly true for real VCOs but makes it possible to draw conclusions that are approximately applicable to real VCO designs.

It follows that:

f_{osc0} = f_{00} + S_{0}V_{tune} where f_{00} is the frequency of oscillation at zero tuning voltage.

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Assume that the VCO's tuning voltage is set to V_{tune_DC}. Now, consider the phase-noise characteristic. In the presence of AC tank swing, assume that the instantaneous varactor voltage traverses the range V_{tune_min} to V_{tune_max}.

The corresponding instantaneous frequency range is f_{osc0_min} to f_{osc0_max}, where:

f_{osc0_min} = f_{00} + S_{0} V_{tune_min} f_{osc0_max} = f_{00} + S_{0} V_{tune_max}

Note that V_{tune_DC} ~ 0.5 (V_{tune_min} + V_{tune_max}). The range of instantaneous frequencies of oscillation is given by: Δf_{osc0} = S_{0}(V_{tune_max} V_{tune_min}) = S_{0}ΔV_{tune} The greater the value of Δf_{osc0}, the higher and wider the skirts on the spectral response and the worse the associated phase noise.

Now, consider a second VCO, similar to the baseline design, based on the same circuit topology and technology, operating at the same center frequency, and delivering similar output power so that the tank swing (ΔV_{tune}) is also the same as for the baseline design. Assume, however, that this new VCO has sensitivity S_{1}. Its idealized tuning characteristic is also shown in Fig. 2, where it has been arbitrarily assumed that S_{1} > S_{0}.

Since the same ΔV_{tune} is assumed, it follows that:

Δf_{osc1} = S_{1} ΔV_{tune} = S_{1}/ S_{0} Δf_{osc0} = nΔf_{sc0}

where n represents the ratio of the tuning sensitivities.

If one assumes minimum and maximum VCO DC tuning voltage settings of V_{DC_min} and V_{DC_max}, respectively, it also follows from Fig. 2 that the corresponding frequency tuning ranges for the two VCOs are S_{0}ΔV_{DC} and S_{1}ΔV_{DC}, where: ΔV_{DC} = V_{DC_max} - V_{DC_min}

Hence, n is also equal to the ratio of the frequency tuning ranges or bandwidths of the two VCOs.

Assuming that the baseline VCO has the idealized phase-noise response shown in Fig. 3, one can immediately infer that the spectral response of the second VCO is similar to that of the baseline, but is scaled in terms of offset frequency by n. The idealized phase-noise response of the second VCO is also shown in Fig. 3, for two scenarios (n > 1 and n

Now, consider frequency offset Δf_{x} in Fig 3, where the baseline phase noise is PN_{0} (dBc/Hz). The same phase noise is obtained at nΔf_{x} for the second VCO with sensitivity S_{1}. Assuming local slope of the phase-noise characteristic is (20 + χ) dB/decade, then it follows that the phase noise of the second VCO at Δf_{x} is: PN_{1} = PN_{0} + (20 + x) log(nΔf_{x}/Δf_{x}) = PN_{0} + (20 + x) log(n) (2)

For example, if the tuning bandwidth of an oscillator is doubled (n = 2) while maintaining the same center frequency, then the phase noise at a given offset is degraded by 6 dB in 20 dB/decade zone, or by 9 dB in the 30 dB/decade zone.

Thus far, constant VCO center frequency has been assumed. Next, consider the following, more generalized question: Given a baseline VCO, VCO1, with parameters: center frequency f_{01}01/(2π)>, tuning bandwidth = BW_{01} and phase noise at offset Δf_{REF} = PN_{01}, what is the anticipated phase-noise performance PN_{02} at the same offset Δf_{REF} of another VCO, VCO2 with parameters: center frequency f_{02}02/(2π)> and tuning bandwidth = BW02 assuming the same technology and circuit topology? Note that the tuning bandwidths alluded to are absolute values with dimension Hz, as distinct from percent tuning range values relative to the center frequencies.

An estimate of PN_{02} can be developed using a two-step operation. In the first step, VCO1 is scaled to the same center frequency as VCO2 by means of an ideal frequency multiplication. The scaled VCO is designated VCO1'. Assuming the same output power, Q etc., it is clear that the associated phase noise of VCO1' at Δf_{REF} is given by:

PN'_{01} = PN_{0}1 + (20 + x) log (Ω_{02} /π_{01})

Note that after this first step, the tuning bandwidth of VCO1' has scaled to: BW'_{01} = BW_{01} (Ω_{02} /Ω_{01})

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Now that the center frequency of VCO1' is the same as the target VCO2, the next step involves adjusting the tuning bandwidth of VCO1' using Eq. 2 above so that it equals the tuning bandwidth of target VCO2. In other words, the tuning bandwidth is changed from BW'_{01} to BW_{02}. This further modified VCO is designated as VCO1''. From Eq. 2, the associated phase noise of VCO1'' is given by:

PN''_{01} = PN'_{01} + (20 + x) log (n)

where n = ratio of tuning bandwidth = BW_{02} / BW'_{01} = (BW_{02} / BW_{01})(ω_{01} / ω_{02}).

Hence:

PN''_{01}= PN_{01} + (20 + x) log (ω_{02} / ω_{01}) + (20 + x) log 02 / BW_{01}) (ω_{01} / ω_{02})> = PN_{01} + (20 + x) log 02 / BW_{01}>

Given that VCO'' is now consistent with VCO2 in terms of both its center frequency and its tuning bandwidth, PN''_{01} can be viewed as an estimate of the phase-noise performance of VCO2 at offset Δf_{REF}. This result suggests that the phase-noise performance is independent of f_{01} and f_{02} and only depends on the ratio of BW01 and BW02: PN (f_{02}, BW_{02}) = PN (f_{01}, BW_{01}) + (20 + x) log 02 / BW_{01}> (3)

Given a baseline VCO design with a known level of phase-noise performance, it should be possible to estimate the phase-noise performance of a similar VCO based on the same design approach and technology, with the only information required being the ratio of the tuning bandwidths of the two VCOs. The operating frequencies do not need to be known. Of course, this conclusion is an approximation and should be applied with caution, although it is a useful relationship and can be used to assess and compare the performance of different VCOs.

To demonstrate the effectiveness of Eq. 3, consider a case study using commercial VCO products for comparison. A family of VCOs based on the same circuit topology and fabricated with the same GaAs heterojunction-bipolar-transistor (HBT) process are shown in the table.

A conventional way of analyzing these data would be to consider the relationship between phase noise and VCO center frequency. Even though it does not take tuning bandwidth into account, this is an obvious trend to investigate in the context of Eq. 1. To be precise, the relationship of interest is phase noise versus the logarithm of a VCO's center frequency. Data in this format are plotted in Fig. 4. There appears to be an underlying approximately linear trend, and this is confirmed by the R2 value of 83%. This shows a strong correlation between phase noise and the logarithm of a VCO's center frequency. Based on this, one may be inclined to conclude that this is a fundamental relationship.

However, further analysis is required. Figure 5 plots VCO tuning bandwidth versus center frequency. Again, it is clear that there is a strong correlation between these two parameters. There is nothing fundamental about this relationship, of course; but rather, for this VCO family, it is a result of the product and market requirements which have dictated that the tuning bandwidth generally increases with increasing center frequency (though there are exceptions to this general trend, as will be discussed presently). Given the more or less linear relationship between tuning bandwidth and center frequency, this introduces the prospect that the correlation shown in Fig. 4 may not be indicative of a fundamental relationship at all, but rather, may be simply a byproduct of a fundamental relationship between phase noise and tuning bandwidth (which would be consistent with Eq. 3) in conjunction with this particular VCO family's strong correlation between tuning bandwidth and center frequency.

With this in mind, a plot of phase noise versus the log of the tuning bandwidth is presented in Fig. 6. Again, these parameters are strongly correlated, and this time the correlation factor (88%) exceeds that in Fig. 4. There is stronger correlation between these parameters than shown in Fig. 4 for phase noise vs. the logarithm of center frequency.

This is further illustrated by examining two VCOs in the family, models MAOC-009265 and MAOC-009266. The general trend of center frequency versus tuning bandwidth is violated for these two VCOs, and the corresponding two points are highlighted within the red rings superimposed on the curves in Fig. 4, Fig. 5, and Fig. 6. It is clear that while these two points lie significantly off the trend lines in both Fig. 4 and Fig. 5and more importantly, the lines drawn between the two points deviate a great deal from those same trend linesthe points lie much closer to the trend line in Fig. 6, and a line connecting the two points there does not deviate so much from that trend line. For these two VCOs which are not well modeled by the trend line in Fig. 4, the trend line in Fig. 6 and hence Eq. 3 provide a significantly better fit.

Thus, the VCO family data are reasonably consistent with a linear relationship between phase noise and the logarithm of the tuning bandwidth. Also shown in Fig. 6, in addition to the R2 value, is the best-fit linear expression: phase noise at an offset of 100 kHz = -112.97 + 30.07log_{10}(tuning bandwidth). This is consistent with Eq. 3 and suggests that the log multiplier is about 30. This in turn suggests that the 100-kHz offset phase noise is closer to the 30-dB/decade region rather than the 20-dB/decade region of the phase-noise spectral response (i.e. x ~ 10).

It should be noted that the tuning bandwidths for the VCOs in the table are the target ranges needed to meet typical customer requirements, and are not necessarily the same as the actual ranges achieved by the designs. In fact, the designs exceed the ranges listed in the table with some margin; the degree of margin is not necessarily the same for all 12 VCOs. As a consequence, it would be unreasonable to expect a perfect R2 of unity in Fig. 6. The fact that an R2 value of 88% is achieved provides confidence that using Eq. 3 as an approximate guide will yield meaningful phase-noise predictions that should prove helpful in future VCO developments. Care should be taken to ensure that a suitable value for x is assumed in the (20 + x) dB/decade local slope term.

**References**

- D. B. Leeson, "A simple model of feedback oscillator noise spectrum," Proceedings of the IEEE, Vol. 54, February 1966, pp. 329-330.
- A. Bonfanti, S. Levantino, C. Samori, and A. L. Lacaita, "A Varactor Configuration Minimizing the Amplitude-to-Phase Noise Conversion in VCOs," IEEE Transactions on Circuits and SystemsI, Regular Papers, Vol. 53, No. 3, March 2006, pp. 481-488.