APPLICATION NOTE 530

# VCO Tank Design for the MAX2310

Abstract: This application note presents various voltage-controlled oscillator (VCO) designs for popular IF frequencies of 85MHz, 190MHz, and 210MHz. These designs reduce the number of iterations required for optimized results. Analysis can be accomplished with a simple spreadsheet program.

## Introduction

This application note presents various voltage-controlled oscillator (VCO) designs for popular IF frequencies of 85MHz, 190MHz, and 210MHz. These designs reduce the number of iterations required for optimized results. Analysis can be accomplished with a simple spreadsheet program.

## VCO Design

Figure 2 shows the differential tank circuit used for the MAX2310 IF VCO. For analysis purposes, the tank circuit must be reduced to an equivalent simplified model. Figure 1 depicts the basic VCO model. The frequency of oscillation can be characterized by EQN1:

 EQN1

fosc = frequency of oscillation
L = inductance of the coil in the tank circuit
Cint = internal capacitance of the MAX2310 tank port
Ct = total equivalent capacitance of the tank circuit

Figure 1. Basic VCO model.

Rn = equivalent negative resistance of the MAX2310 tank port
Cint = internal capacitance of the MAX2310 tank port
Ct = total equivalent capacitance of the tank circuit
L = inductance of the coil in the tank circuit

Figure 2. The MAX2310 tank circuit.

Inductor L resonates with the total equivalent capacitance of the tank and the internal capacitance of the oscillator (Ct+Cint) (see Figure 1). Ccoup provides DC block and couples the variable capacitance of the varactor diodes to the tank circuit. Ccent is used to center the tank's oscillation frequency to a nominal value. It is not required but adds a degree of freedom by allowing one to fine-tune resonance between inductor values. Resistors (R) provide reverse-bias voltage to the varactor diodes via the tune voltage line (Vtune). Their value should be chosen large enough so as not to affect loaded-tank Q but small enough so that 4kTBR noise is negligible. The resistors' noise voltage gets modulated by KVCO, producing phase noise. Capacitance Cv is the variable tuning component in the tank. The capacitance of varactor diode (Cv) is a function of reverse-bias voltage (see Appendix A for the varactor model). Vtune is the tuning voltage from a phase-locked loop (PLL).

Figure 3 shows the lumped Cstray VCO model. Parasitic capacitance and inductance plague every RF circuit. In order to predict the frequency of oscillation, the parasitic elements must be taken into account. The circuit in Figure 3 lumps the parasitic elements in one capacitor called Cstray. The frequency of oscillation can be characterized by EQN2:

 EQN2

L = inductance of the coil in the tank circuit
Cint = internal capacitance of the MAX2310 tank port
Ccent = tank capacitor used to center oscillation frequency
Cstray = lumped stray capacitance
Ccoup = tank capacitor used to couple the varactor to the tank
Cv = net variable capacitance of the varactor diode (including series inductance)

Figure 3. Lumped Cstray model.

Figure 4 depicts the detailed VCO model. It takes into account the capacitance of the pads but does not include the effects of series inductance for simplicity. Cstray is defined as:

 EQN3

CL = capacitance of the inductor
CLP = capacitance of the inductor pads
CDIFF = capacitance due to parallel traces

Figure 4. Detailed VCO model.

Rn = equivalent negative resistance of the MAX2310 tank port
Cint = internal capacitance of the MAX2310 tank port
LT = inductance of series trace to the inductor tank circuit
CDIFF = capacitance due to parallel traces
L = inductance of the coil in the tank circuit
CL = capacitance of the inductor
CLP = capacitance of inductor pads
Ccent = tank capacitor used to center oscillation frequency
Ccoup = tank capacitor used to couple the varactor to the tank
Cvar = variable capacitance of the varactor diode
LS = series inductance of the varactor
R = resistance of the varactor reverse-bias resistors

To simplify analysis, inductance LT is ignored in this design. The effects of LT are more pronounced at higher frequencies. To mathematically model the shift in frequency due to LT with the spreadsheets that follow, the value of CDIFF can be increased appropriately. Minimize inductance LT to prevent undesired series resonance. This can be accomplished by making the traces short.

## Tuning Gain

Tuning gain (Kvco) must be minimized for best closed-loop phase noise. Resistors in the loop filter as well as the resistors "R" (Figure 2) will produce broadband noise. Broadband thermal noise ( ) will modulate the VCO by Kvco, which is measured in MHz/V. There are two ways to minimize Kvco. One is to minimize the frequency range over which the VCO must tune. The second way is to maximize the tuning voltage available. To minimize the frequency range over which the VCO must tune, tight tolerance components must be used, as will be shown. To maximize tuning voltage, a charge pump with a large compliance range is needed. This is usually accomplished by using a larger Vcc. The compliance range for the MAX2310 is 0.5V to Vcc-0.5V. In battery-powered applications, the compliance range is usually fixed by battery voltage or a regulator.

## Basic Concept for Trimless Design

VCO design for manufacturability with real-world components will require an error budget analysis. In order to design a VCO to oscillate at a fixed frequency (fosc), the tolerance of the components must be taken into consideration. Tuning gain (Kvco) must be designed into the VCO to account for these component tolerances. The tighter the component tolerance, the smaller the possible tuning gain, and the lower the closed-loop phase noise. For worst-case error budget design, we will look at three VCO models:
1. Maximum-value components (EQN5)
2. Nominal tank, all components perfect (EQN2)
3. Minimum-value components (EQN4)
All three VCO models must cover the desired nominal frequency. Figure 5 shows visually how the three designs must converge to provide a manufacturable design solution. Observation of EQN1 and Figure 5 reveal that minimum-value components will shift the oscillation frequency higher and that maximum-value components will shift the oscillation frequency lower.

Figure 5. Worst-case and nominal-tank centering.

Minimum tuning range must be used in order to design a tank with the best closed-loop phase noise. Therefore, the nominal tank should be designed to cover the center frequency with overlap to take into account device tolerance. The worst-case high-tune tank and worst-case low-tune tank should tune just to the edge of the desired oscillation frequency. EQN2 can be modified by component tolerance to produce a worst-case high-tune tank EQN4 and a worst-case low-tune tank EQN5.

 EQN4

 EQN5

TL = % tolerance of the inductor (L)
TCINT = % tolerance of the capacitor (CINT)
TCCENT = % tolerance of the capacitor (CCENT)
TCCOUP = % tolerance of the capacitor (CCOUP)
TCV = % tolerance of the varactor capacitance (CV)

EQN4 and EQN5 assume that the strays do not have a tolerance.

## General Design Procedure

### Step 1

Estimate or measure pad capacitance and other strays. The stray capacitance on the MAX2310 Rev C EV kit has been measured with a Boonton Model 72BD capacitance meter. CLP = 1.13pF, CVP = 0.82pF, CDIFF = 0.036pF.

### Step 2

Determine the value for capacitance Cint. This can be found in the MAX2310/MAX2312/MAX2314/MAX2316 data sheet on Page 5. Typical operating characteristic TANKH PORT 1/S11 vs. FREQUENCY shows the equivalent parallel RC values for several popular LO frequencies. Appendix B includes tables of Cint versus frequency for the high- and low-band tank ports. Keep in mind that the LO frequency is twice the IF frequency.

### Example:

For an IF frequency of 210MHz (high-band tank), the LO will operate at 420MHz. From Appendix B, Table 5, Cint = 0.959pF.

### Step 3

Choose an inductor. A good starting point is using the geometric mean. This will be an iterative process.
 EQN6

This equation assumes L in (nH) and C in (pF) (1x10-9 x 1x10-12 = 1x10-21). L = 11.98nH for a fosc = 420MHz. This implies a total tank capacitance C = 11.98pF. An appropriate initial choice for an inductor would be 12nH Coilcraft 0805CS-12NXGBC 2% tolerance.

When choosing an inductor with finite step sizes, the following formula EQN6.1 will be useful. The total product LC should be constant for a fixed oscillation frequency fosc.

 EQN6.1

LC = 143.5 for a fosc = 420MHz. The trial-and-error process with the spreadsheet in Table 3 yielded an inductor value of 18nH 2% with a total tank capacitance of 7.9221pF. The LC product for the tank in Figure 8 is 142.59, close enough to the desired LC product of 143.5. One can see this is a useful relationship to have on hand. For best phase noise, choose a high-Q inductor like the Coilcraft 0805CS series. Alternatively, a micro-strip inductor can be used if the tolerance and Q can be controlled reasonably.

### Step 4

Determine the PLL compliance range. This is the range over which the VCO tuning voltage (Vtune) will be designed to work. For the MAX2310, the compliance range is 0.5V to Vcc-0.5V. For a Vcc = 2.7V, this would set the compliance range to 0.5 to 2.2V. The charge-pump output will set this limit. The voltage swing on the tank is 1Vp-p centered at 1.6VDC. Even with large values for Ccoup, the varactor diodes will not be forward-biased. This is a condition to be avoided, as the diode will rectify the AC signal on the tank pins, producing undesirable spurious response and loss of lock in a closed-loop PLL.

### Step 5

Choose a varactor. Look for a varactor with good tolerance over your specified compliance range. Keep the series resistance small. For a figure of merit, check that the self-resonant frequency of the varactor is above the desired operating point. Look at the Cv(2.5V)/Cv(0.5V) ratio at your compliance-range voltage. If the coupling capacitors Ccoup were chosen large, then the maximum tuning range can be calculated using EQN2. Smaller values of capacitor Ccoup will reduce this effective frequency tuning range. When choosing a varactor, it should have a tolerance specified at your given compliance-range mid and end points. Select a hyperabrupt varactor such as the Alpha SMV1763-079 for linear tuning response. Take the value for total tank capacitance, and use that for Cjo of the varactor. Remember, Ccoup will reduce the net capacitance coupled to the tank.

### Step 6

Pick a value for Ccoup. Large values of Ccoup will increase tuning range by coupling more of the varactor into the tank at the expense of decreasing tank-loaded Q. Smaller values of Ccoup will increase the effective Q of the coupled varactor and loaded Q of the tank at the expense of reducing tuning range. Typically this will be chosen as small as possible, while still getting the desired tuning range. Another benefit of choosing Ccoup small is that it reduces the voltage swing across the varactor diode. This will help thwart forward-biasing the varactor.

### Step 7

Pick a value for Ccent, usually around 2pF or greater for tolerance purposes. Use Ccent to center the VCO's nominal frequency.

## MAX2310 VCO Tank Designs for IF Frequencies of 85MHz, 190MHz, and 210MHz

The following spreadsheets show designs for several popular IF frequencies for the MAX2310. Keep in mind that the LO oscillates at twice the desired IF frequency.

Figure 6. 85MHz low-band IF tank schematic.

Table 1. 85MHz Low-Band IF Tank Design
 Light grey indicates calculated values.

 Darker grey indicates user input.

 MAX2310 Low-Band Tank Design and Tuning Range Total Tank Capacitance vs. V tune V tune Total C Ct (Nominal) Ct (Low) Ct (High) 0.5V Ct high 14.1766pF 13.3590pF 14.9459pF 1.375V Ct mid 12.8267pF 11.7445pF 13.7620pF 2.2V Ct low 11.4646pF 10.3049pF 12.4534pF Tank Components Tolerance C coup 18pF 0.9pF 5% C cent 5.6pF 0.1pF 2% C stray 0.70pF L 68nH 2.00% C int 0.902pF 10.00% Parasitics and Pads (C stray) Due to Q C L 0.1pF Ind. pad C Lp 1.13pF Due to || C diff 0.036pF Var. pad C vp 0.82pF Varactor Specs Alpha SMV1255-003 Cjo 82pF Varactor Tolerance Vj 17V 0.5V 19.00% M 14 1.5V 29.00% Cp 0pF 2.5V 35.00% Rs 1Ω Reactance Ls 1.7nH X Ls 1.82 Freq 170.00MHz Nominal Varactor X c Net Cap Cv high 54.64697pF -17.1319 61.12581pF Cv mid 27.60043pF -33.92 29.16154pF Cv low 14.92387pF -62.7321 15.36874pF Negative Tol Varactor (Low Capacitance) Cv high 44.26404pF -21.1505 48.42117pF Cv mid 19.59631pF -47.7746 20.37056pF Cv low 9.700518pF -96.5109 9.886531pF Positive Tol Varactor (High Capacitance) Cv high 65.02989pF -14.3965 74.41601pF Cv mid 35.60456pF -26.2945 38.24572pF Cv low 20.14723pF -46.4682 20.96654pF Nominal LO (Nom) Range Low Tol IF (High) Range Nominal IF (Nom) Range High Tol IF (Low) Range F low 162.10MHz 84.34MHz 81.05MHz 78.16MHz F mid 170.42MHz 89.95MHz 85.21MHz 81.45MHz F high 180.25MHz 96.03MHz 90.13MHz 85.62MHz BW 18.16MHz 11.69MHz 9.08MHz 7.46MHz % BW 10.65% 12.99% 10.65% 9.16% Nominal IF Frequency 85.00MHz Design Constraints Condition for bold number IF Delta 0.66 -0.21 0.62 Test pass pass pass Raise or lower cent freq by -0.21 MHz Inc or dec BW -1.28 MHz Cent adj for min BW 84.98 MHz K vco 10.68MHz/V

Figure 7. 190MHz high-band IF tank schematic.

Table 2. 190MHz High-Band IF Tank Design
 Light grey indicates calculated values.

 Darker grey indicates user input.

 MAX2310 High-Band Tank Design and Tuning Range Total Tank Capacitance vs. V tune V tune Total C Ct (Nominal) Ct (Low) Ct (High) 0.5V Ct high 10.4968pF 10.0249pF 10.9126pF 1.375V Ct mid 9.6292pF 8.8913pF 10.2124pF 2.2V Ct low 8.6762pF 7.7872pF 9.3717pF Tank Components Tolerance C coup 12pF 0.1pF 1% C cent 3.4pF 0.1pF 3% C stray 0.70pF L 18nH 2.00% C int 0.954pF 10.00% Parasitics and Pads (C stray) Due to Q C L 0.01pF Ind. pad C Lp 1.13pF Due to || C diff 0.036pF Var. pad C vp 0.82pF Varactor Specs Alpha SMV1255-003 Cjo 82pF Varactor Tolerance Vj 17V 0.5V 19.00% M 14 1.5V 29.00% Cp 0pF 2.5V 35.00% Rs 1Ω Reactance Ls 1.7nH X Ls 4.06 Freq 380.00MHz Nominal Varactor X c Net Cap Cv high 54.64697pF -7.66426 116.1695pF Cv mid 27.60043pF -15.1747 37.67876pF Cv low 14.92387pF -28.0643 17.44727pF Negative Tol Varactor (Low Capacitance) Cv high 44.26404pF -9.46205 77.51615pF Cv mid 19.59631pF -21.3728 24.19031pF Cv low 9.700518pF -43.1759 10.70708pF Positive Tol Varactor (High Capacitance) Cv high 65.02989pF -6.44056 175.8588pF Cv mid 35.60456pF -11.7633 54.36221pF Cv low 20.14723pF -20.7884 25.03539pF Nominal LO (Nom) Range Low Tol IF (High) Range Nominal IF (Nom) Range High Tol IF (Low) Range F low 366.15MHz 189.23MHz 183.07MHz 177.78MHz F mid 382.29MHz 200.94MHz 191.14MHz 183.78MHz F high 402.74MHz 214.71MHz 201.37MHz 191.84MHz BW 36.59MHz 25.47MHz 18.29MHz 14.06MHz % BW 9.57% 12.68% 9.57% 7.65% Nominal IF Frequency 190MHz Design Constraints Condition for bold number < IF = IF > IF Delta 0.77 -1.14 1.84 Test pass pass pass Raise or lower cent freq by -1.14 MHz Inc or dec BW -2.61 MHz Cent adj for min BW 190.54 MHz K vco 21.52MHz/V

Figure 8. 210MHz high-band IF tank schematic.

Table 3. 210MHz High-Band IF Tank Design
 Light grey indicates calculated values.

 Darker grey indicates user input.

 MAX2310 High-Band Tank Design and Tuning Range Total Tank Capacitance vs. V tune V tune Total C Ct (Nominal) Ct (Low) Ct (High) 0.5V Ct high 8.8304pF 8.1465pF 9.4877pF 1.35V Ct mid 7.9221pF 7.0421pF 8.6970pF 2.2V Ct low 6.9334pF 5.9607pF 7.7653pF Tank Components Tolerance C coup 12pF 0.6pF 5% C cent 1.6pF 0.1pF 6% C stray 0.70pF L 18nH 2.00% C int 0.959pF 10.00% Parasitics and Pads (C stray) Due to Q C L 0.1pF Ind. pad C Lp 1.13pF Due to || C diff 0.036pF Var. pad C vp 0.82pF Varactor Specs Alpha SMV1255-003 Cjo 82pF Varactor Tolerance Vj 17V 0.5V 19.00% M 14 1.5V 29.00% Cp 0pF 2.5V 35.00% Rs 1Ω Reactance Ls 1.7nH X Ls 4.49 Freq 420.00MHz Nominal Varactor X c Net Cap Cv high 54.64697pF -6.93433 154.787pF Cv mid 27.60043pF -13.7295 40.99616pF Cv low 14.92387pF -25.3916 18.12647pF Negative Tol Varactor (Low Capacitance) Cv high 44.26404pF -8.56091 92.99806pF Cv mid 19.59631pF -19.3373 25.51591pF Cv low 9.700518pF -39.0639 10.95908pF Positive Tol Varactor (High Capacitance) Cv high 65.02989pF -5.82717 282.5852pF Cv mid 35.60456pF -10.643 61.54791pF Cv low 20.14723pF -18.8086 26.45795pF Nominal LO (Nom) Range Low Tol IF (High) Range Nominal IF (Nom) Range High Tol IF (Low) Range F low 399.20MHz 209.92MHz 199.60MHz 190.67MHz F mid 421.47MHz 225.78MHz 210.73MHz 199.14MHz F high 450.52MHz 245.41MHz 225.26MHz 210.75MHz BW 51.31MHz 35.49MHz 25.66MHz 20.09MHz % BW 12.18% 15.72% 12.18% 10.09% Nominal IF Frequency 210MHz Design Constraints condition for bold number < IF = IF > IF Delta 0.08 -0.73 0.75 Test pass pass pass Raise or lower cent freq by -0.73 MHz Inc or dec BW -0.83 MHz Cent adj for min BW 210.34 MHz K vco 30.18MHz/V

Figure 9. High-Q 210MHz high-band IF tank schematic.

Table 4. High-Q 210MHz High-Band IF Tank Design
 Light grey indicates calculated values.

 Darker grey indicates user input.

 MAX2310 High-Band Tank Design and Tuning Range Total Tank Capacitance vs. V tune V tune Total C Ct (Nominal) Ct (Low) Ct (High) 0.5V Ct high 5.8856 5.5289 6.2425 1.375V Ct mid 5.2487 4.9113 5.5858 2.2V Ct low 4.8371 4.5156 5.1581 Tank Components C coup 15pF 0.75pF 5% C cent 1.6pF 0.1pF 6% C stray 0.77pF L 27 2.00% C int 0.959 10.00% Parasitics and Pads (C stray) Due to Q C L 0.17pF Ind. pad C Lp 1.13pF Due to || C diff 0.036pF Var. pad C vp 0.82pF Varactor Specs Alpha SMV1763-079 Cjo 8.2pF Varactor Tolerance Vj 15V 0.5V 7.50% M 9.5 1.5V 9.50% Cp 0.67pF 2.5V 11.50% Rs 0.5Ω Reactance Ls 0.8nH X Ls 2.11 Freq 420.00MHz Nominal Varactor X c Net Cap Cv high 6.67523pF -56.7681 6.933064pF Cv mid 4.23417pF -89.4958 4.336464pF Cv low 2.904398pF -130.471 2.952167pF Negative Tol Varactor (Low Capacitance) Cv high 6.174588pF -61.3709 6.39456pF Cv mid 3.831924pF -98.8904 3.915514pF Cv low 2.570392pF -147.425 2.607736pF Positive Tol Varactor (High Capacitance) Cv high 7.175873pF -52.8076 7.474698pF Cv mid 4.636416pF -81.7313 4.759352pF Cv low 3.238404pF -117.015 3.297904pF Nominal LO (Nom) Range Low Tol IF (High) Range Nominal IF (Nom) Range High Tol IF (Low) Range F low 399.25MHz 208.05MHz 199.62MHz 191.92MHz F mid 422.78MHz 220.75MHz 211.39MHz 202.89MHz F high 440.40MHz 230.22MHz 220.20MHz 211.14MHz BW 41.15MHz 22.16MHz 20.58MHz 19.21MHz % BW 9.73% 10.04% 9.73% 9.47% Nominal IF Frequency 210MHz Design Constraints Condition for bold number < IF = IF > IF Delta 1.95 -1.39 1.14 Test pass pass pass Raise or lower cent freq by -1.39 MHz Inc or dec BW -3.08 MHz Cent adj for min BW 209.60 MHz K vco 24.21MHz/V

## Appendix A

Figure 10. Varactor model.

Alpha Application Note AN1004 has additional information on varactor models. Varactor capacitance is defined in EQN7:

 EQN7

 Alpha SMV1255-003 Alpha SMV1763-079 Cjo = 82 pF Cjo = 8.2 pF Vj =17 V Vj =15 V M = 14 M = 9.5 Cp = 0 Cp = 0.67 Rs = 1Ω Rs = 0.5Ω Ls = 1.7 nH Ls = 0.8 nH

The series inductance of the varactor is taken into account by backing out the inductive reactance and calculating a new effective capacitance Cv:
 EQN8

## Appendix B

Table 5. Cint vs. Frequency for the MAX2310 High-Band Tank
 Frequency (MHz) Cint (pF) Frequency (MHz) (cont.) Cint (pF) (cont.) 100 0.708 360 0.949 110 0.759 370 0.955 120 0.800 380 0.954 130 0.809 390 0.954 140 0.839 400 0.954 150 0.822 410 0.955 160 0.860 420 0.959 170 0.869 430 0.956 180 0.880 440 0.959 190 0.905 450 0.964 200 0.917 460 0.962 210 0.920 470 0.963 220 0.926 480 0.963 230 0.924 490 0.960 240 0.928 500 0.964 250 0.935 510 0.965 260 0.932 520 0.968 270 0.931 530 0.966 280 0.933 540 0.968 290 0.927 550 0.967 300 0.930 560 0.974 310 0.933 570 0.977 320 0.943 580 0.976 330 0.944 590 0.984 340 0.945 600 0.982 350 0.956 - -

Figure 11. Cint vs. frequency for the MAX2310 high-band tank (sixth-order polynomial curve fit)

Table 6. Cint vs. Frequency for the MAX2310 Low-Band Tank
 Frequency (MHz) Cint (pF) Frequency (MHz) (cont.) Cint (pF) (cont.) 100 0.550 360 1.001 110 0.649 370 0.982 120 0.701 380 0.992 130 0.764 390 1.001 140 0.762 400 0.985 150 0.851 410 0.980 160 0.838 420 0.986 170 0.902 430 0.992 180 0.876 440 0.994 190 0.907 450 1.001 200 0.913 460 1.003 210 0.919 470 1.007 220 0.945 480 0.992 230 0.952 490 1.010 240 0.965 500 1.004 250 0.951 510 1.011 260 0.954 520 1.022 270 0.974 530 1.019 280 0.980 540 1.044 290 0.973 550 1.026 300 0.982 560 1.041 310 0.970 570 1.038 320 0.982 580 1.032 330 0.991 590 1.036 340 0.993 600 1.025 350 0.991 - -

Figure 12. Cint vs. frequency for the MAX2310 low-band tank (sixth-order polynomial curve fit).

### References

1. Chris O'Connor, Develop Trimless Voltage-Controlled Oscillators, Microwaves and RF, July1999.
2. Wes Hayward, Radio Frequency Design, Chapter 7.
3. Krauss, Bostian, Raab, Solid State Radio Engineering, Chapters 2, 3, 5.
4. Alpha Industries Application Note AN1004.
5. Coilcraft, RF Inductors Catalog, March 1998, p.131.
6. Maxim, MAX2310/MAX2312/MAX2314/MAX2316 Data Sheet, Rev 0.
7. Maxim, MAX2310/MAX2314 Evaluation Kit Data Sheet, Rev 0.
8. Maxim, MAX2312/MAX2316 Evaluation Kit Data Sheet, Rev 0.