1. Introduction

LC-based voltage controlled oscillators (VCOs) have been widely employed in various wideband and multi-band communication systems^[1-4]. To achieve a good phase noise performance, the VCO frequency band is usually divided into multiple sub-bands by adopting a binary weighted capacitor array^[5]. However, with respect to this type of VCO, the gain ($K_{\rm VCO})$ is proportional to the cubic oscillation frequency ($f_{\rm VCO})$^[6], and the loop bandwidth of the frequency synthesizer (FS) changes with the square of $f_{\rm VCO}$^[7], which leads to huge variation in FS performance, such as phase noise and locking time^[8]. An effective method to keep the loop bandwidth less dependent on the oscillation frequency is therefore highly desired.

To achieve this, several methods have been previously proposed, and can be categorized into two types: the calibration method^[9-12] and the compensation method^{[6, 13-15]}. The calibration method measures the$K_{\rm VCO}$ and then calibrates the current of the charge pump to cancel the variation in loop bandwidth. The compensation method uses a specially designed charge pump with the current changeable with $f_{\rm VCO}$ to compensate the variation in loop bandwidth. However, whether using the calibration or compensation method, the bandwidth problem is solved by adjusting the other function blocks or adding new blocks, and this will increase the complexity and reduce the robustness of the whole FS system.

This paper proposes a VCO with the gain ($K_{\rm VCO})$ proportional to the oscillation frequency ($f_{\rm VCO})$. By adopting this kind of VCO, the bandwidth variation problem of the wideband/multiband FS is alleviated. To obtain the proposed VCO character, a switched varactor array, as well as a switched capacitor array, is adopted. The tuning technique of changing the values of the capacitors and varactors at the same ratio is also derived. As an example, a 2.5 G/3.5 G dualband QVCO for WiMAX applications is fabricated and measured.

2. Analysis and derivation of the proposed VCO character

Figure 1 shows a block diagram of a classical frequency synthesizer, which consists of the VCO, a low pass filter (LPF), a divider, a phase frequency detector (PFD) and a charge pump (CP). The open-loop transfer function of Fig. 1 can be expressed as

$\begin{equation} \label{eq1} H_{\rm OPEN} (s)=\frac{I_{\rm CP} K_{\rm VCO} }{2\pi Ns} \frac{1+R_1 C_1 s}{s(R_1 C_1 C_2 s+C_1 +C_2 )}, \end{equation}$

(1)

where $I_{\rm CP}$ represents the sink or source current of the charge pump. $K_{\rm VCO}$ is the gain of VCO and $N$ is the division ratio. The FS transfer function has one zero and three poles.

In a locked frequency synthesizer, it can be easily derived that:

$\begin{equation} \label{eq2} N=\frac{f_{\rm VCO} }{f_{\rm ref}}. \end{equation}$

(2)

Then, Equation (1) can be rewritten as:

$\begin{equation} \label{eq3} H_{\rm OPEN} (s)=\frac{I_{\rm CP} f_{\rm ref} }{2\pi s} \frac{K_{\rm VCO} }{f_{\rm VCO} } \frac{1+R_1 C_1 s}{s(R_1 C_1 C_2 s+C_1 +C_2 )}. \end{equation}$

(3)

From Eq. (3), if:

$\begin{equation} \label{eq4} \frac{K_{\rm VCO} }{f_{\rm VCO} }={\rm constant}, \end{equation}$

(4)

then:

$\begin{equation} \label{eq5} H_{\rm OPEN} (s)=\frac{I_{\rm CP} f_{\rm ref} }{2\pi s} \gamma \frac{1+R_1 C_1 s}{s(R_1 C_1 C_2 s+C_1 +C_2 )}, \end{equation}$

(5)

where $\gamma$ represents the value of $K_{\rm VCO}/f_{\rm VCO}$. It can be seen from Eq. (5) that the loop transfer function, as well as the distribution of zeros and poles, is independent of $f_{\rm VCO}$, and the bandwidth can be approximately given^[12] by

$\begin{equation} \label{eq6} \omega _{\rm BW} =R_1 f_{\rm ref} \frac{I_{\rm cp}}{2\pi } \gamma. \end{equation}$

(6)

All the parameters in the right hand of Eq. (6) are fixed values, and the bandwidth will remain constant over the whole frequency range. This demonstrates that, by adopting the VCO with the gain proportional to the oscillation frequency, the bandwidth variation problem of FS in wideband/multiband applications can be greatly alleviated. To implement the VCO with constant $K_{\rm VCO}/f_{\rm VCO}$, an extra switched varactor array is adopted. The VCO tuning technique is derived in the next section.

3. VCO tuning technique

3.1 The LC-VCO tuning scheme

In LC-VCO, the oscillation frequency $f_{\rm VCO}$ can be written as:

$\begin{equation} \label{eq7} f_{\rm VCO} =\frac{1}{2\pi \sqrt {LC_{\rm tot}}}=\frac{1}{2\pi \sqrt {L(C_{\rm V} +C_{\rm F} )} }, \end{equation}$

(7)

where $L$ is the inductance and $C_{\rm tot}$ the total capacitance. $C_{\rm tot}$ consists of two parts: the fixed capacitor part $C_{\rm F}$, which does not change with the control voltage ($V_{\rm C})$ and the changeable varactor part $C_{\rm V}$, which changes linearly with $V_{\rm C}$. In Eq. (7), $C_{\rm V}$ is equal to 0 when $V_{\rm C}$ is 0.

Assuming at the highest frequency:

$\begin{equation} \label{eq8} f_{\rm VCO, max} =\frac{1}{2\pi \sqrt {L(C_{\rm F0} +C_{\rm V0} )} }, \end{equation}$

(8)

where $C_{\rm F0}$ and $C_{\rm V0}$ represent the specific value of $C_{\rm F}$ and $C_{\rm V}$ when VCO resonates at the highest frequency. Equation (5) can be rewritten as:

$\begin{equation} \label{eq9} f_{\rm VCO} =\frac{1}{2\pi \sqrt {L(C_{\rm F} +C_{\rm V} )} }=\frac{1}{2\pi \sqrt {L(\alpha C_{\rm F0} +\beta C_{\rm V0} )} }, \end{equation}$

(9)

where $\alpha$ and $\beta$ represent the normalized capacitance of $C_{\rm F}$ and $C_{\rm V}$, respectively. The VCO gain $K_{\rm VCO}$ can be derived as:

$\begin{equation} \begin{split} K_{\rm VCO} {}& =\frac{\partial f_{\rm VCO} }{\partial V_{\rm C}} \\& =\frac{f_{\rm VCO} }{4\pi \sqrt L (\alpha C_{\rm F0} +\beta C_{\rm V0} )} \left(\frac{C_{\rm F0} }{C_{\rm V0} }\frac{\partial \alpha }{\partial \beta }+1\right) \beta \frac{\partial C_{\rm V0} }{\partial V_{\rm C}}. \end{split} \end{equation}$

(10)

From Eq. (10), it can be derived that if

$\begin{equation} \label{eq10} \alpha =\beta, \end{equation}$

(11)

then Equation (10) can be rewritten as:

$\begin{equation} \label{eq11} K_{\rm VCO} =\frac{1}{4\pi \sqrt L C_{\rm V0} } \frac{\partial C_{\rm V0} }{\partial V_{\rm C}} f_{\rm VCO} ={\rm Constant} \cdot f_{\rm VCO} . \end{equation}$

(12)

In Eq. (12), $C_{\rm V0}$ and $\partial C_{\rm V0} /\partial V_{\rm C}$ are all constants in the linear region of VCO. From Eqs. (11) and (12), if $C_{\rm F}$ and $C_{\rm V}$ changes with the same ratio, the VCO gain $K_{\rm VCO}$ will be proportional to the oscillation frequency $f_{\rm VCO}$.

3.2 Switched capacitor and varactor array calculation

In general, a switched capacitor array is adopted in LC-VCO to divide the frequency band into several sub-bands. The in-sub-band frequency tuning is controlled continuously by the analog control voltage $V_{\rm C}$ and the sub-band selection is controlled digitally and discretely by switches.

To implement the VCO with the proposed performance, both $C_{\rm F}$ and $C_{\rm V}$ are made to change with the same ratio. So a switched varactor array, as well as a switched capacitor array, is needed, as shown in Fig. 2.

In a single-band VCO, for a given number of control bits, $m$, the available analog range $V_{\rm RAN}$ of the control voltage $V_{\rm C}$, as well as the frequency range $f_{\rm min}$ and $f_{\rm max}$, the procedure to calculate the switched capacitor and varactor arrays is as follows.

Choose a proper value of inductance $L$, and then the initial value of capacitance $C_{\rm F0}$ can be written as:

$\begin{equation} \label{eq12} C_{\rm F0} =\frac{1}{4\pi ^2L} \frac{1}{f_{\rm max}^2}. \end{equation}$

(13)

For a VCO with a binary weighted switched array, the number of sub-bands is $2^m$. The $i$-th switched capacitance part value $C_{{\rm F}i}$ can be derived as:

$\begin{equation} \label{eq13} C_{{\rm F}i} =\frac{1}{4\pi ^2L}\left(\frac{1}{f_{\rm min} ^2}-\frac{1}{f_{\rm max } ^2}\right) \frac{2^{i-1}}{2^m}, \quad i=1, 2, \ldots, m. \end{equation}$

(14)

So the total value of the switched capacitance is:

$\begin{equation} \label{eq14} \Delta C_{\rm F} =\sum\limits_{i=1}^m {C_{{\rm F}i}} =\frac{1}{4\pi ^2L}\left(\frac{1}{f_{\rm min } ^2}-\frac{1}{f_{\rm max} ^2}\right) \frac{2^m-1}{2^m}. \end{equation}$

(15)

The initial value of the varactor part $C_{\rm V0}$ should make the frequency cover the top sub-band when $V_{\rm C}$ changes from 0 to $V_{\rm RAN}$. It equals $C_{\rm Fm}$, and thus

$\begin{equation} \label{eq15} C_{\rm V0} =\frac{1}{4\pi ^2L} \left(\frac{1}{f_{\rm min } ^2}-\frac{1}{f_{\rm max } ^2}\right) \frac{1}{2^m}. \end{equation}$

(16)

As $C_{\rm F}$ and $C_{\rm V}$ change with the same ratio

$\begin{equation} \label{eq16} \frac{C_{\rm F0} }{\Delta C_{\rm F}}=\frac{C_{\rm V0} }{\Delta C_{\rm V}}, \end{equation}$

(17)

the total value of the switched capacitance $\Delta C_{\rm V}$ can be written as

$\begin{equation} \label{eq17} \Delta C_{\rm V}=\frac{\Delta C_{\rm F}}{C_{\rm F0}} C_{\rm V0} =\frac{f_{\rm max}^2}{4\pi ^2L} \left(\frac{1}{f_{\rm min}^2}-\frac{1}{f_{\rm max}^2}\right)^2 \frac{2^m-1}{2^{2m}}. \end{equation}$

(18)

And the $i$-th switched capacitance part value $C_{{\rm V}i}$ can be derived as

$\begin{equation} \label{eq18} C_{{\rm V}i} =\frac{f_{\rm max} ^2}{4\pi ^2L}\left(\frac{1}{f_{\rm min } ^2}-\frac{1}{f_{\rm max } ^2}\right)^2 \frac{2^{i-1}}{2^{2m}}, \quad i=1, 2, \ldots, m. \end{equation}$

(19)

Since $C_{\rm F0}$, $\Delta C_{\rm F}$, $C_{\rm V0}$ and $C_{{\rm V}i}$ are all derived, the next step is to calculate the size of each capacitor and varactor in the switched capacitor and varactor arrays. It is worth mentioning that in CMOS technology, for IC designers, the changeable size of both the varactor and capacitor is their effective area. The capacitance of varactor $C_{\rm VAR}$ contains two parts: the basic capacitance and the changed capacitance. In the linear area, $C_{\rm VAR}$ can be written as

$\begin{equation} \label{eq19} \begin{split} C_{\rm VAR} {}&=S_{\rm VAR} (C_{\rm VAR} ^\prime \, +\, k_{\rm VAR} ^\prime V_{\rm C})\\[2mm] &=S_{\rm VAR} C_{\rm VAR} ^\prime \, +\, S_{\rm VAR} k_{\rm VAR} ^\prime V_{\rm C}, \end{split} \end{equation}$

(20)

where $C_{\rm VAR} ^\prime$ and $k_{\rm VAR}^\prime$ represent the basic capacitance and the slope of the varactor in unit area, respectively. $S_{\rm VAR}$ represents the effective area of the varactor. In Eq. (20), the first and second term of the right-hand side should be counted as the fixed capacitor part $C_{\rm F}$ and the changeable varactor part $C_{\rm V}$, respectively. So the area of the switched varactor array can be written as:

$\begin{equation} \label{eq20} \begin{split} {}& S_{{\rm V}i} =\frac{C_{{\rm V}i} }{k_{\rm VAR} ^\prime V_{\rm RAN} }, \quad i=1, 2, \ldots, m, \\[2mm]& S_{\rm V0} =\frac{C_{\rm V0} }{k_{\rm VAR} ^\prime V_{\rm RAN}}. \\ \end{split} \end{equation}$

(21)

In CMOS technology, the capacitance of the capacitor can be written as:

$\begin{equation} \label{eq21} C_{\rm CAP} =S_{\rm CAP} C_{\rm C}^\prime, \end{equation}$

(22)

where $C_{\rm C}^\prime$ represents the capacitance of the capacitor in unit area. $S_{\rm CAP}$ represents the effective area of the capacitor. The basic capacitance of the varactors contributes a part to the fixed capacitance $C_{\rm F0}$ and $\Delta C_{\rm F}$, so the needed capacitance of the switched capacitor array is decreased, and can be written as:

$\begin{equation} \label{eq22} \begin{split} {}& C_{{\rm C}i} =C_{{\rm F}i} -S_{{\rm V}i} C_{\rm VAR} ^\prime, \quad i=1, 2, \ldots, m, \\[2mm]& C_{\rm C0} =C_{\rm F0} -S_{\rm V0} C_{\rm VAR} ^\prime. \\ \end{split} \end{equation}$

(23)

The area of the switched capacitor array can be written as:

$\begin{equation} \label{eq23} \begin{split} {}& S_{{\rm C}i} =\frac{C_{{\rm C}i} }{C_{\rm C}^\prime }=\frac{k_{\rm VAR} ^\prime V_{\rm RAN} C_{{\rm F}i} -C_{\rm VAR} ^\prime C_{{\rm V}i} }{k_{\rm VAR} ^\prime V_{\rm RAN} C_{\rm C} ^\prime }, \quad i=1, 2, \ldots, m, \\[3mm] & S_{\rm C0} =\frac{C_{\rm C0} }{C_{\rm C} ^\prime }=\frac{k_{\rm VAR} ^\prime V_{\rm RAN} C_{\rm F0} -C_{\rm VAR} ^\prime C_{\rm V0} }{k_{\rm VAR} ^\prime V_{\rm RAN} C_{\rm C}^\prime}. \\ \end{split} \end{equation}$

(24)

3.3 The tuning scheme of multi-band VCO

From the above analysis, both the switched capacitor and varactor arrays in single wideband VCOs are calculated. For multi-band VCOs, a band-selected capacitor and varactor pair is needed, respectively, for one more frequency band. There are three steps to calculate the multi-band VCOs. Firstly, the band with the highest ratio of bandwidth to center frequency is chosen. Then the sub-band-selected switched capacitor and varactor arrays are calculated, as discussed above. Secondly, the band-selected capacitors and varactors of each band are calculated. Note that the band-selected $C_{\rm F}$ and $C_{\rm V}$ also change with the same ratio. Lastly, ensure that the VCO can cover all the frequency ranges of each band. If this is not satisfied, then the capacitor and varactor array must be adjusted and re-calculated from the first step.

4. VCO circuit implementation

A dualband QVCO with the $K_{\rm VCO}$ proportional to $f_{\rm VCO}$ is implemented for the WiMAX application, as shown in Fig. 3. The quadrature signals are generated by the anti-phase cross coupling of two symmetric differential LC oscillators. The detailed circuit schematic of the single-side oscillator is shown in Fig. 4. This is a PMOS-tailed current splitting differential LC-VCO. Both switched capacitor and varactor arrays are adopted in this work. Each frequency band is divided into four sub-bands, which are controlled by en1 and en2 signals. The band-select switch is controlled by the en3 signal.

5. Measurement results

The proposed LC QVCO is implemented in a 0.13 $\mu$m 1P8M RF CMOS technology. A micrograph of the fabricated QVCO is shown in Fig. 5. The sizes of the core circuit are 1.2 $\times$ 0.55 mm$^{2}$, and the output of the QVCO is driven out by QVCO buffers with separated power supplies.

The QVCO tuning characteristics are shown in Fig. 6, and are tunable from 2.44 to 2.63 GHz for the lower band and 3.36 to 3.60 GHz for the higher band. The available tuning range of the control voltage is 0.55-0.95 V in the lower band and 0.45-0.85 V in the higher band.

In the available tuning range, the QVCO covers the entire frequency range. The gain ($K_{\rm VCO})$ grows with oscillation frequency from sub-band to sub-band. Figure 7 shows the average gain versus the centre frequency of each sub-band. The markers represent the measured average $K_{\rm VCO}$ in each sub-band and the line represents the fitting curve of $K_{\rm VCO}$ versus frequency $f_{\rm VCO}$. From Fig. 7, the average $K_{\rm VCO}$ is closely proportional to the oscillation frequency. The fitting value of $K_{\rm VCO}/f_{\rm VCO}$ is approximately 0.06 (unit: 1/V). The variation is less than 5.22%, and considering the whole frequency range, the variation is $\pm$16%.

The supply voltage of the proposed QVCO is 1.2 V, and the power consumption is 12 mW. The measured phase noise is -138.15 dBc/Hz at 10 MHz from the 2.5 GHz carrier, and -137.44 dBc/Hz at 10 MHz from the 3.5 GHz carrier, as shown in Figs. 8 and 9, respectively.

The QVCO figure of merit (FoM) is calculated according to:

$\begin{equation} \label{eq24} {\rm FoM}_{\rm QVCO} =20\lg \frac{f_{\rm o}}{\Delta f}-10\lg \frac{P_{\rm DC} }{2}/1\, {\rm mW}-{\rm PN}, \end{equation}$

(25)

where $f_{\rm VCO}$ is the oscillation frequency, PN is the phase noise in dBc/Hz, $\Delta f$ is the offset frequency of the phase noise, and $P_{\rm DC}$ is the power dissipation. For QVCO, the power is twice that of VCO. So when calculating the FoM of the QVCO, the power is divided by 2, as shown in the second term of the right-hand side of Eq. (25). Table 1 shows a comparison of the proposed QVCO with previous works.

6. Conclusion

This paper proposed a quadrature VCO with gain proportional to oscillation frequency. By adopting this kind of VCO, the bandwidth variation problem of wideband/multiband FS can be greatly alleviated. To implement the proposed VCO, the technique of changing the capacitance of the capacitors and varactors at the same ratio is derived. The procedure for calculating the switching capacitor and varactor sizes at each frequency band is also described. A 2.5 G/3.5 G dual band QVCO by the proposed method is fabricated in 0.13 $\mu$m CMOS technology. This achieves a tuning range of 2.44-2.63 GHz in the lower band and 3.36-3.60 GHz in the higher band. The measured phase noise is -138.15 dBc/Hz at 10 MHz from a 2.5 GHz carrier and -137.44 dBc/Hz at 10 MHz from a 3.5 GHz carrier. The $K_{\rm VCO}$ is closely proportional to the oscillation frequency with a $\pm$16% variation over the entire frequency range.