Lecture 100 Voltage-Controlled Oscillators (09/01/03) Page 100-1

Transcription

1 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 00 ETURE 00 VOTAGEONTROED OSIATORS INTRODUTION Objective The objective of this presentation is examine and characterize the types of voltagecontrolled oscillators compatible with both discrete and integrated technologies. Outline haracterization of VO s Oscillators R Relaxation oscillators Ring oscillators Direct digital synthesis (DDS) Varactors Summary ecture 00 Voltageontrolled Oscillators (09/0/03) Page 002 HARATERIZATION OF VOTAGEONTROED OSIATORS Introduction to Voltageontrolled Oscillators What is an oscillator? An oscillator is a circuit capable of maintaining electric oscillations. An oscillator is a periodic function, i.e. f(x) = f(xnk) for all x and for all integers, n, and k is a constant. All oscillators use positive feedback of one form or another. lassification of oscillators: Oscillators Tuned Oscillators Untuned Oscillators R Oscillators S Oscillators Oscillators rystal Oscillators Relaxation Oscillators Ring Oscillators Fig. Osc0 What are tuned oscillators? A tuned oscillator uses a frequencyselective or tunedcircuit in the feedback path and is generally sinusoidal. An untuned or oscillator uses nonlinear feedback and is generally nonsinusoidal

2 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 003 Types of Oscillators Ring Oscillator: ascade of inverters Frequency of oscillation = Σ of stage delays ontrolled by current or power supply Higher power Oscillator: Frequency of oscillation = ontrolled by voltage dependent capacitance (varactor) Medium power Relaxation Oscillator: Frequency determined by circuit time constants ontrolled by current Medium power R Oscillators: Don t require inductors Operate at lower frequencies (00MHz) ecture 00 Voltageontrolled Oscillators (09/0/03) Page 004 haracteristics of Oscillators Frequency of oscillation Frequency tuning range as a function of the controlling variable (either voltage or current) Frequency stability phase noise and jitter Amplitude stability (adjustable?) Purity (harmonics)

3 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 005 inear Feedback Oscillator System Simplified block diagram: V f F(jw) V in A(jw) The loop gain of this diagram is, G(jω) = A(jω)F(jω) When the loop gain is equal to, oscillation occurs. Re[G(jω)] Im[G(jω)] = j0 The frequency of oscillation is found from, Im[G(jω)] = 0 and the gain necessary for oscillation is found from, Re[G(jω)] = Vout Fig. Osc02 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 006 inear Oscillator Amplitude Stabilization What determines the amplitude of the oscillator? Good question. A(jω) and/or F(jω) must have an outputinput characteristic that looks like an s shape. For small amplitudes, the magnitude of the loop gain is greater than one and the oscillation grows. As the amplitude grows, the effective gain decreases and stabilizes at just the right amplitude to give an effective loop gain of unity. Illustration: v out jω Peaktopeak amplitude Effective oop Gain = v in σ Fig. Osc03 Pole ocations as a function of amplitude

4 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 007 Van der Pol Equations for Oscillators Basic R oscillator and negative resistance circuit: i i i R R p i v = di dt di dt = v i = dv dt di dt = d2 v dt 2 i R = v R p di Rp dt = dv R p dt di dt di dt di Rp dt di dt = 0 d2 v dt 2 R p a 3a 3 v 2 dv d 2 v dt 2 R p a 3a 3v 2 dv R v i v VanderPol i i i R i = 0 and i = f(v) = a v a 3 v 3 di dv = a 3a 3 v 2 di dv v=0 = a = R n di dt = a dv dt 3v 2dv dt v d2 v dt 2 dv R p dt dv a dt 3a 3v 2dv dt = 0 dt v = 0 dt v = 0 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 008 Van der Pol Equations At start up, v is very small so that 3a 3 v 2 0 s 2 V(s) R a p sv(s) V(s) = 0 s2 ms = 0 Poles = 0.5m ±0.5 m 2 4 = 0.5m ± j0.5 4 m2 For jω axis poles, m = 0. In steadystate, the following relationship must hold. m = R a p 3a 3v 2 = 0 We see that the amplitude of oscillation (ω osc = V = For V = V, a R p 3a 3 = R n = R p 3a 3 R n R p 3a 3 ) will be,

5 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 009 Openoop oncept of an Oscillator Basic closedloop oscillator: Frequency shaping network & amplifier X(jω) H(jω) Y(jω) Fig Oscillator oscillates when H(jω) = j0 Openloop Q: The openloop Q is a measure of how much the closed loop system opposes variations in the oscillation frequency. The higher the Q, the lower the phase noise. Definitions of Q:.) Q = ω o ω where ω o is the frequency of oscillation. 2π Energy Stored 2.) Q = Energy Dissipated per ycle 3.) Q = ω o d 2 dω Arg[H(jω)] ω ω o ω Fig ecture 00 Voltageontrolled Oscillators (09/0/03) Page 000 Voltage ontrolled Oscillators Tuning A voltage controlled oscillator (VO) is an oscillator whose frequency can be varied by a voltage (or current). In local oscillator applications, the VO frequency must be able to be varied over the Rx or Tx range (quickly). Frequency Nonlinear tuning inear tuning Tx or Rx Range Voltage tuning range Fig Tuning variables: apacitance (varactor) urrent Power supply Speed of tuning will be determined by the bandwidth of the phase lock loop.

6 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 00 OSIATORS R Oscillators WienBridge Oscillator ircuit: R A K B R 2 V out Openoop Gain: For simplicity, let R = R 2 = R and = =. K s R G(s) = s2 3 R s G(jω) = (R)2 Equating the loop gain to j0 gives K jω ο R (R)2 ω ο 2 3 R jω ο = j0 The only way this equation can be satisfied is if ω o 2 = R Fig. Osc04 K jω R (R)2 ω2 3 R jω and K =3. ecture 00 Voltageontrolled Oscillators (09/0/03) Page 002 WienBridge Oscillator ontinued How do you realize the amplifier of K = 3? V in V out V out V in = R 2 R R R 2 Fig. Osc05 How does the amplitude stabilize? Thermistor (a resistor whose resistance decreases with increasing temperature) Nonlinear transfer function Example: v in D V R 4 R KR v out v out Slope = R 4 R 4 R 3 2R 2 R R 3 R R 2 R 2 D2 2V v in 2V 2 Slope = R 4 R 4 R 4 2R 2R R 3 V 2 Slope = R 4 R 4 R 3 2R Fig. Osc06

7 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 003 Other R Oscillators R PhaseShift Oscillator: R R 2 3 R 3 R 4 v out Fig. Osc07 If R = R 2 = R 3 = R and = = 3 =, then (R 4 R 3 )(jωr)(ωr)2 G(jω) = [6(ωR)2] jωr [5 (ωr)2] ω osc = 6 R and K = R 4 R 3 = 29 Quadrature Oscillator: R R R R 3 4 V out Fig. Osc08 G(jω) = (jω)2r R 3 4 ω osc = R R 3 4 Other R oscillators: Twintee R oscillator, SallenKey bandpass filter with Q =, Infinite gain, bandpass filter with Q = How do you tune the R oscillator? Must vary either R or or both. ecture 00 Voltageontrolled Oscillators (09/0/03) Page 004 Example The circuit shown is a R oscillator. Find the frequency of oscillation in Hertz and the voltage gain, K, of the voltage amplifiers necessary for oscillation. The voltage amplifiers have infinite input resistance and zero output resistance. R = 0kΩ Solution The loop gain can be found from the schematic R = 0kΩ = nf Vr V shown: x K K T(s) = V = R = r V x = K 2 sr nf 0kΩ S03FES sr sr K = 2 sr s 2 R 2 2sR T(jω) = K 2 jωr ω 2 R 2 jω 2R = j0 We see from this equation that for oscillation to occur, the following conditions must be satisfied: ω 2 R 2 = 0 and K 2 = 2 or ω osc = R = = 0 5 radians/sec. f osc = 5.9kHz and K = 2 =.44 K = nf R = 0kΩ = nf K V out S03FEP

8 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 005 Gm Oscillators Same the quadrature oscillator only implemented in a more I friendly manner. g m V o g m2 V o2 V o V o2 t Fig t Open oop Gain = (s) = s s etting s = jω and setting (jω) = gives, g m g m2 = g mg m2 s 2 g m g m2 ω osc = = g m = g m2 if g m = g m2 and = This circuit is much easier to tune. If the transconductors are MOS transistors, then 2K'I D W g m = Varying the bias current will vary g m and tune the frequency. ecture 00 Voltageontrolled Oscillators (09/0/03) Page 006 Switched apacitor Oscillators oncept Theoretically, the R s of any R oscillator can be replaced by switches and capacitors to create an S oscillator. Quadrature S Oscillator φ φ2 α e V (z) φ φ 2 α 2 φ φ 2 φ 2 φ e V out (z) Fig. Osc09 α α 2 ω osc = T2 = α α 2 f clock (really a frequency translator) The output is a sinusoid at frequency of α α 2 f clock.

9 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 007 Oscillator All oscillators require feedback for oscillation to occur..) Hartley or olpitts oscillators G m R 2 v out G m R 2 v out ω o = 2.) Negative resistance tank ω o = olpitts and R Rp = g m R ω o = Hartley Fig ( 2 ) and 2 = g m R G m R Freq. ontrol v out Fig ecture 00 Voltageontrolled Oscillators (09/0/03) Page 008 Example Hartley Oscillator Find the oscillation frequency, ω osc and g m r ds necessary to oscillate in terms of,, and for the oscillator shown. Solution The smallsignal model for solving g m v gs g m v gs ' this problem is shown. Note that the current from the r ds r ds independent source has two paths. One is through the parallel combination of r ds and, and the v gs v gs other is through and. The openloop gain, V gs /V gs can be found as, SU03E2S4 V gs (s) V gs (s) = g m [r ds (/ s )] s s (/ s ) r ds (/ s ) = s s r ds s2 r ds s s r ds G(s) = V gs(s) V gs (s) = g m r ds (s 2 )(s r ds ) s r ds g m r ds = s 3 r ds s 2 sr ds ( ) g m V Bias r ds I Bias v out SU03E2P4

10 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 009 Example ontinued g m r ds G(jω) = ω 2 jω[ r ds ( ) ω 2 r ds ] = j0 ω osc 2 = ω osc = At the oscillation frequency, we can write that, g m r ds = ω osc 2 = = = g m r ds = ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0020 Example 2 lapp Oscillator A lapp oscillator which is a version of the olpitt s oscillator is shown. Find an expression for the frequency of oscillation and the value of g m R necessary for oscillation. Assume that the output resistance of the FET, r ds, and R arge can be neglected (approach infinity). Solution The smallsignal model for this problem is shown below. The loop gain will be defined as V gs /V gs. Therefore, g m V gs ' R (/s 3 ) V gs = R (/s 3 ) s s s s 2 R (/s 3 ) g m V gs ' R (/s 3 ) s = R (/s 3 ) R (/s 3 ) s s s T(s) = V gs V gs = g m R sr 3 s R sr 3 s s s = g m V gs ' R arge V Bias g m R s R (sr 3 ) s s s R 3 3 R I Bias F02FEP5 V gs F02FES5

11 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 002 Example 2 ontinued g m R T(s) = s R (sr 3 )(s 2 ) T(s) = T(jω) = g m R s R s 3 R 3 sr 3 s 3 R s 2 g m R [ ω 2 ] jω[r ( 3 ) R 3 ω 2 R 3 ] 3 3 = ω osc 2 3 ω osc = 3 = j0 Also, g m R = ω osc 2 = 2 3 = 3 g m R = 3 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0022 Oscillators ircuits: v o v o2 v o v o2 M v o M2 v o2 M M2 M 2nd Harmonic trap 2 M2 PMOS Oscillator onditions for oscillation: H(s) = g m s 2 s2 s R H(jω) = NMOS Oscillator g m jω ω2 jω R Improved NMOS Oscillator Fig. 2.40A 2 = j0 ω osc 2 = & g m = Output swing of the improved circuit is twice that of the other circuits plus the second harmonic is removed.

12 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0023 Example 3 Oscillator An oscillator is shown. The value of the inductors,, are 5nH and the capacitor,, is 2.5pF. If the Q of each inductor is 5, find (a.) the value of negative resistance that should be available from the crosscoupled, sourcecoupled pair (M and M2) for oscillation and (b.) design the W/ ratios of M and M2 to realize this negative resistance (if you can t find the negative resistance of part (a.) M assume that the desired negative resistance is 00Ω). Solution (a.) The equivalent circuit seen by the negative resistance circuit is: The frequency of oscillation is given as / 2 or ω o = 2πx09 radians/sec. Therefore the series resistance, R s, is found as R s = ω Q = 2πx0 9 5x09 5 = 2π Ω onverting the series impedance of 2 and 2R s into a parallel impedance gives, Y = 2R s jω2 = 0.5 R s jω R sjω R s jω = 0.5R s R s 2ω22 j 0.5ω s R s 2ω22 2 M2 2mA F00E2P2 2R s F00E2S2A ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0024 Example 3 ontinued The reciprocal of the conductance is the parallel resistance, R p, given as R p = R s 2 ω22 0.5R s = 4π 2 4π2 25 π = 4π(26) = 326.7Ω R neg = 04π Ω = 326.7Ω (b.) The negative resistance seen by the R circuit is found as follows. i in = g m v gs = g m2 v gs2 R in = v in i in = v gs2v gs i in = g m2 g m = 2 g m Assuming the 2mA splits evenly between M and M2 for the negative resistance calculation gives, Thus, g m = g m = g m2 = 2 2mA 0x06 (W/) = W/ = 52π = 84 W/ = 84 W/ 508 = 2 04π M i in v in v gs v gs2 2mA M2 F00E2S2B

13 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0025 Oscillator VO with PMOS pair: I SS Design: v2 V Inductors = 2 = 7.nH, Q = 8.5 at ontrol 90MHz (Metal 3 with spacing of 2.µm and width of 6.µm and 5 turns) R, and R 2, form ac coupling filters Buffers A and B isolate the VO from the next stages to avoid the pulling effect of the center frequency due to injection from the external load or the prescaler fed by the VO. Buffer A provides a matched 50Ω output impedance and buffer B drives the large capacitance of the prescaler. I SS =.5mA M v ontrol Voltage (V) Fig VO Frequency (MHz) M R R 2 Buffer A Buffer B Fig. 2.4 Output Prescaler ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0026 Relaxation Oscillators ircuit: R R R R I SS M M2 ISS I SS M A I SS M2 A Open oop: losedoop Openoop Fig Assume that in the closed loop circuit is really two 2 capacitors in series and ground the midpoint to obtain the openloop circuit. and are capacitances to ground from the drains of M and M2. H(s) = g mr A s (g m A s)(r D s ) 2 where g m = transconductance of each transistor and D = =. It can be shown that, g m ω o = R A D and Q = 4 D D A A Q max = S oθ (f m ) = f 2 o 4 f m S θ (f m )

14 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0027 Relaxation Oscillator Experimental Results R R W/ = 00µm/0.5µm g m = /84µS I SS M M2 ISS R = 275Ω = 0.6pF I SS = 3mA Fig f o = 920 MHz Phase noise = 05 dbc/hz for f m = MHz and 5dBc/Hz for f m = 5MHz ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0028 Ring Oscillator VO Threestage ring oscillator: Output M3 M4 Output Input M M2 omparison of a ThreeStage and FourStage Ring Oscillator: Frequency ontrol haracteristic 3Stage Ring Oscillator 4Stage Ring Oscillator Min. Required Gain 2 2 Noise Shaping f Function 27 o 2 f o 2 f m f m 6 Openoop Q Power Dissipation.8mW 3.6mW Frequency ontrol M5 M6 Fig. 2.44

15 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0029 Example 4 Differential Ring Oscillator A differential ring oscillator is shown. (a.) Find the frequency of oscillation in Hz if R = kω and = pf. R R R R R R (b.) What value of g m is required for oscillation assuming all stages are identical? (c.) What is the maximum positive and maximum negative voltage swing at the drains if I SS = ma and = 2V? Solution I SS I SS I SS SU03FEP6 (a.) The voltage transfer function of a single stage is, V out (s) V in (s) = g mr sr = g mr sτ where τ = R = 09 secs. When the phase shift of each stage is equal to 60, the phase shift around the loop will be 360 or 0. Therefore, the oscillation frequency can be found as, tan(ω osc τ ) = 60 f osc =.732 2π R =.732 2π 0 9 = MHz f osc = MHz ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0030 Example 4 ontinued (b.) The magnitude of the loop gain at the oscillation frequency is given as g m R 3 = (ω osc R) 2 gm R = = 3 = 2 g m = 2 R = 2mS g m = 2mS (c.) v max = = 2V and v min = I SS R = 2 = V v max = 2V and v min = V

16 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 003 Ring Oscillator Experimental Results M3 M Input Frequency ontrol M4 M2 M5 Output M and M2: W/ = 97µm/0.5µm gm = /24µS M3 and M4: W/ = 3.4µm/0.5µm gm = /630µS M5: W/ = 3.4µm/0.5µm ID = 790µA gm = /530µS Fig f o = 2.2GHz Phase noise at MHz = 99.2 dbc/hz General omments: Phase noise is proportional to power dissipation Wide tuning range (2 to ) Poor phase noise performance ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0032 Direct Digital Frequency Synthesizer DDFS A sinusoid is digitized and stored in a ROM. These stored values are applied to a DA at regular time intervals by a reference clock. The frequency is increased by taking fewer, but further separated samples from the ROM lookup table. Block Diagram: Reference lock Frequency Setting Data Phase Accumulator Sine lookup table in ROM bits M bits N bits DA PF Output Fig Operation:.) Phase accumulator adds the frequency setting data to the previous contents once every clock cycle. The most significant bits of the results are used to address the ROM lookup table. 2.) The address decoding circuitry of the ROM selects the corresponding N bit sample and feeds it to the DA. 3.) The DA converts this digital data to an analog signal. 4.) The analog signal is passed through the lowpass filter to smooth the waveform and remove outofband high frequency noise from the signal. Advantages: Disadvantages: Very high frequency resolution (f clock /2N) High power consumption Fast switching time Restricted to low frequencies

17 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0033 Quadrature VO s Polyphase Oscillator: Oscillator A Oscillator B Fig Two crosscoupled oscillators synchronize in exact quadrature Phase inaccuracy insensitive to mismatch in resonators arge amplitudes available, balanced oscillation available to drive mixer FETs Performance: 830 MHz Unwanted sideband 46 db below wanted sideband eakage 49 db below wanted sideband ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0034 Quadrature Oscillators Frequency Division Approach: K sin(ßt) O Frequency Doubler BPF I Q 2 O 90 K cos(ßt) Fig omments: Start with a single phase local oscillator at twice the desired frequency Divide by 2 is done by positive and negative edge triggered flipflops Phase accuracy depends on timing skews between the flipflop channels (typically 2 )

18 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0035 VARATORS VARIABE APAITORS One of the components that can be used to vary the frequency is the capacitor. Types of apacitors onsidered pn junction capacitors Standard MOS capacitors Accumulation mode MOS capacitors Polypoly capacitors Metalmetal capacitors haracterization of apacitors Assume is the desired capacitance:.) Dissipation (quality factor) of a capacitor is Q = ωr p where R p is the equivalent resistance in parallel with the capacitor,. 2.) max / min ratio is the ratio of the largest value of capacitance to the smallest when the capacitor is used as a variable capacitor. 3.) Variation of capacitance with the control voltage. 4.) Parasitic capacitors from both terminal of the desired capacitor to ac ground. ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0036 Desirable haracteristics of Varactors.) A high quality factor 2.) A control voltage range compatible with supply voltage 3.) Good tunability over the available control voltage range 4.) Small silicon area (reduces cost) 5.) Reasonably uniform capacitance variation over the available control voltage range 6.) A high max / min ratio Some References for Further Information.) P. Andreani and S. Mattisson, On the Use of MOS Varactors in RF VO s, IEEE J. of SolidState ircuits, vol. 35, no. 6, June 2000, pp ) AS Porret, T. Melly,. Enz, and E. Vittoz, Design of HighQ Varactors for ow Power Wireless Applications Using a Standard MOS Process, IEEE J. of SolidState ircuits, vol. 35, no. 3, March 2000, pp ) E. Pedersen, RF MOS Varactors for 2GHz Applications, Analog Integrated ircuits and Signal Processing, vol. 26, pp. 2736, Jan. 200

19 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0037 Digitally Varied apacitances In a digital process, highquality capacitors are very difficult to achieve. A highquality variable capacitor is even more difficult to realize. Therefore, digitally controlled capacitances are becoming more popular. V Fine Tune S 4 V Fine Tune S S2 S3 S4 8 6 S2 S3 8 6 SingleEnded S4 Fig oncerns: Switch parasitics Switch ON resistance (will lower Q) Differential ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0038 K vo of Digitally Tuned VOs The value of the VO gain constant can be made smaller which is desirable in many applications of the VO. VO Frequency Digitally Adjustable apacitors V min ontolling Voltage(Fine tune) V max Fig. 3.45

20 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0039 apacitor Errors.) Oxide gradients 2.) Edge effects 3.) Parasitics 4.) Voltage dependence 5.) Temperature dependence ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0040 apacitor Errors Oxide Gradients Error due to a variation in oxide thickness across the wafer. A A 2 B No common centroid layout A B A 2 ommon centroid layout y x x2 x Only good for onedimensional errors. An alternate approach is to layout numerous repetitions and connect them randomly to achieve a statistical error balanced over the entire area of interest. A B A B A B A B A B A B B A B A B A 0.2% matching of poly resistors was achieved using an array of 50 unit resistors.

21 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 004 apacitor Errors Edge Effects There will always be a randomness on the definition of the edge. However, etching can be influenced by the presence of adjacent structures. For example, Matching of A and B are disturbed by the presence of. A B Improved matching achieve by matching the surroundings of A and B. A B ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0042 apacitor Errors Area/Periphery Ratio The best match between two structures occurs when their areatoperiphery ratios are identical. et = ± and 2 = ± where = the actual capacitance = the desired capacitance (which is proportional to area) = edge uncertainty (which is proportional to the periphery) Solve for the ratio of 2 /, 2 = ± ± = ± ± ± ± If = 2, then 2 = Therefore, the best matching results are obtained when the area/periphery ratio of is equal to the area/periphery ratio of.

22 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0043 apacitor Errors Relative Accuracy apacitor relative accuracy is proportional to the area of the capacitors and inversely proportional to the difference in values between the two capacitors. For example, 0.04 Relative Accuracy Unit apacitance = pf Unit apacitance = 0.5pF Unit apacitance = 4pF Ratio of apacitors ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0044 apacitor Errors Parasitics Parasitics are normally from the top and bottom plate to ac ground which is typically the substrate. Top Plate Top plate parasitic Desired apacitor Bottom Plate Bottom plate parasitic Top plate parasitic is 0.0 to 0.00 of desired Bottom plate parasitic is 0.05 to 0.2 desired

23 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0045 Other onsiderations on apacitor Accuracy Decreasing Sensitivity to Edge Variation: A A' A A' B B' Sensitive to edge variation in both upper andlower plates B B' Sensitive to edge varation in upper plate only. Fig A structure that minimizes the ratio of perimeter to area (circle is best). Top Plate of apacitor Bottom plate of capacitor Fig ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0046 Definition of Temperature and Voltage oefficients In general a variable y which is a function of x, y = f(x), can be expressed as a Taylor series, y(x = x 0 ) y(x 0 ) a (x x 0 ) a 2 (x x 0 ) 2 a (x x 0 ) 3 where the coefficients, a i, are defined as, df(x) a = dx x=x 0, a 2 = d2 f(x) 2 dx 2 x=x 0,. The coefficients, a i, are called the firstorder, secondorder,. temperature or voltage coefficients depending on whether x is temperature or voltage. Generally, only the firstorder coefficients are of interest. In the characterization of temperature dependence, it is common practice to use a term called fractional temperature coefficient, T F, which is defined as, T F (T=T 0 ) = df(t) f(t=t 0 ) dt T=T 0 parts per million/ (ppm/ ) or more simply, T F = df(t) f(t) dt parts per million/ (ppm/ ) A similar definition holds for fractional voltage coefficient.

24 ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0047 apacitor Errors Temperature and Voltage Dependence PolysiliconOxideSemiconductor apacitors Absolute accuracy ±0% Relative accuracy ±0.2% Temperature coefficient 25 ppm/ Voltage coefficient 50ppm/V PolysiliconOxidePolysilicon apacitors Absolute accuracy ±0% Relative accuracy ±0.2% Temperature coefficient 25 ppm/ Voltage coefficient 20ppm/V Accuracies depend upon the size of the capacitors. ecture 00 Voltageontrolled Oscillators (09/0/03) Page 0048 SUMMARY haracterization of VO s Frequency, Frequency tuning range Frequency stability Amplitude stability Spectral purity Oscillators R Relaxation oscillators Ring oscillators Direct digital synthesis (DDS) Varactors Used to vary the frequency of R and oscillators