Thesis Organization

The thesis is organized into five chapters including the introduction. Chapter 2 deals with the basic concepts of VCO design, its metrics and some popular voltage-controlled oscillator (VCO) topologies. In chapter 3, some advanced popular VCO topologies are reviewed. In chapter 4, we design the low phase noise low power consumption dual-band VCO with the simulated and measured results. Chapter 5 conclusion is drawn.

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Chapter 2 Basics of CMOS VCO

2.1 General Consideration

A simple oscillator produces a periodic output, usually in the form of voltage. As such, the circuit has no input while sustaining the output indefinitely. How can a circuit oscillate?

Consider the unity gain negative feedback circuit shown in Fig. 2.1, where

( ) ( )

1 ( )

out in

V s H s

V = H

+ s (1)

If the amplifier itself experiences so much phase shift at high frequencies that the overall feedback becomes positive, then oscillation may occur. More accurately, if for s=jω0, H(jω)=‐1, then the closed loop gain approach infinity at ω0 indefinitely. In fact, as conceptually illustrated in Fig. 2.2, a noise component at ω0 experiences a total gain of unity and a phase shift of 180°, returning to the subtractor as a negative replica of the input. Upon subtraction, the input and the feedback signals give a lager difference. Thus, the circuit continues to “regenerate,” allowing the component at ω0 to grow.

Figure 2.1 Feedback system

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Figure 2.2 Evolution of oscillatory system with time

For the oscillation to begin, a loop gain of unity or greater is necessary. This can be seen by following the signal around the loop over many cycles and expressing the amplitude of the subtractor’s output in Fig. 2.2 as geometric series (if ∠H(jω)=180°):

2 3

In summary, if a negative feedback circuit has a loop gain that satisfies two conditions:

( 0)

H jω ≥ 1 (4)

( 0) 180 H jω

∠ = ° (5)

then the circuit may oscillate at ω0, which is Called “Barkhausen criteria.” These conditions are necessary but not sufficient. In order to ensure oscillation in the presence of temperature and process variations, we typically choose the loop gain to be at least twice or three times the required value.

We may state the second Barkhausen critertion as ∠H jω( ₀) 180= °or a total phase shift of 360°. This should not be confusing: if the system is designed to have a low frequency negative feedback, it already produces 180° of phase shift in the signal traveling around the loop in Fig 2.1, and ∠H jω( ₀) 180= ° denotes an additional frequency dependent phase shift that, as illustrated in Fig 2.2, ensures the feedback signal enhances the original signal. Thus, three illustrated in Fig 2.3 are equivalent in terms of the second criterion. We say the system of Fig 2.3 exhibits a frequency dependent phase shift of 180°. The difference between Figs.

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2.3(b) and (c) is that the open loop amplifier in the former contains enough stages with proper polarities a total phase shift of 360° at ω0 whereas that in the latter produces no phase shift at ω0.

Figure 2.3 Various views of oscillatory feedback system

2.2 One port oscillator

An alternative view that provides more insight into the oscillation phenomenon employs the concept of “negative resistance.” To arrive at this view, let us first consider a simple tank that is stimulated by a circuit impulse Fig. 2.4(a). The tank responds with a decaying

oscillatory behavior because, in every cycle, some of the energy that reciprocates between the capacitor and the inductor is lost in the form of heat in the resistor. Now suppose a resistor equal to –Rp is placed in parallel with Rp and experiment is repeated Fig. 2.4(b). Since Rp∥(-Rp)=∞, the tank oscillates indefinitely. Thus, if a one port circuit exhibiting a negative resistance is placed in parallel with a tank Fig. 2.4(c), the combination may oscillate. Such a topology is called a one port oscillator.

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Figure 2.4 (a) Decaying impulse response of a tank, (b) addition of negative resistance to cancel loss in Rp, (c) use of an active circuit to provide negative resistance.

How can a circuit provide a negative resistance? Recall that feedback multiplies or divides the input and output impedances of circuits by a factor equal to one plus the loop gain. Thus, if the loop gain is sufficiently negative, (i.e., the feedback is sufficiently positive), a negative resistance is achieved.

Figure 2.5 (a) Source follower with positive feedback to create negative input impedance, (b) equivalent circuit if (a) to calculate the input impedance

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2.3 Negative-R LC Oscillator

With a negative resistance available, we can now construct an oscillator as illustrated in Fig. 2.6. Here, Rp denotes the equivalent parallel resistance of the tank and, for oscillation build-up, Rp-2/gm ≧ 0.

Figure 2.6 Oscillator using negative input resistance of a source follower with positive feedback

More interestingly, the circuit can be redrawn as in Fig. 2.7(a), bearing a resemblance to Fig. A.6(b). in fact, if the drain current of M1 flows through a tank and resulting voltage is applied to the gate of M2, the topology of Fig. 2.7(b) is obtained.

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Figure 2.7 (a) Redrawing of the topology shown in Fig. 2.6, (b) differential version of (a), (c) Equivalent circuit

of Fig. 2.7(b)

For oscillation build-up 2Rp-2/gm ≧ 0, Rp≧1/gm.

2.4 Voltage-Controlled Oscillators

Most applications require that oscillators be “tunable,” i.e., their output frequency be a function of a control input, usually a voltage. An ideal voltage-controlled oscillator is a circuit whose output frequency is a linear function of its control voltage (Fig. 2.8):

0

out kVCO contV

ω =ω + (9)

Here, ω0 represents the intercept corresponding to Vcont=0 and KVCO denotes the “gain” or

“sensitivity” of the circuit (expressed in rad/s/V). The achievable range, ω2-ω1, is called the

“tuning range.”

Figure 2.8 Definition of a VCO

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2.5 Center Frequency

The center frequency ( i.e., the midrange value in Fig. 2.8) is determined by the environment in which the VCO used.

2.6 Tuning Range

The required tuning range is dictated by two parameters: (1) the variation of the VCO center frequency with process and temperature and (2) the frequency range necessary for the application. The center frequency of some CMOS oscillator may vary by a factor of two at the extreme of process and temperature, thus mandating a sufficiently wide (≧2) tuning range to guarantee that the VCO output frequency can be driven to the desired value.

An important concern in the design of VCOs is the variation of the output phase and frequency as a result of noise on the control line. For a given noise amplitude, the noise in the output frequency is proportional to KVCO because ωout=ω0+KVCOVcont. Thus, to minimize the effect of noise in Vcont, the VCO gain must be minimized, a constraint in direct conflict with the required tuning range. In fact, if, as shown in Fig. 2.8, the allowable range of Vcont is from V1 to V2 (e.g., from 0 to VDD) and the tuning range must span at least ω1 to ω2, then KVCO

must satisfy the following requirement:

2 1

Note that, for a given tuning range, KVCO increase as the supply voltage decreases, making the oscillator more sensitive to noise on the control line.

Tuning Linearity

As exemplified by Eq. (A.16), the tuning characteristics of VCOs exhibit nonlinearity, i.e., their gain, KVCO, is not constant. Nonlinearity degrades the settling behavior of phase-locked loops. For this reason, it is desirable to minimize the variation of KVCO across the tuning range.

Actual oscillator characteristics typically exhibit a high gain region in the middle of the range and a low gain at the two extremes (Fig. 2.9). Compared to a linear characteristic (the gray

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line), the actual behavior displays a maximum gain greater than that predicted by (10), implying that, for a given tuning range, nonlinearity inevitably leads to higher sensitivity for some region of the characteristic.

Figure 2.9 Nonlinear VCO characteristic

2.7 Output Power

In general, it is not easy to predict the output power of the realistic VCO, but we can know that the maximum output power of VCO is not larger than the output power of the transistor in the VCO through large-signal analysis. The output power must be maximized in order to make the waveform less sensitive to noise or to lower phase noise. It trades with power consumption, supply voltage, and tuning range. The designer can choose the active devices whose parameter is known. Therefore, when the VCO is designed, we also can predict the output power of the VCO.

2.8 Harmonic Rejection

The VCO has a good harmonic rejection performance that means it is closed to a sinusoidal output waveform. In wireless communication systems, harmonic rejection is specified how much smaller the harmonics of the output signal are compared with the fundamental output power.

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2.9 Power Consumption

With fast growth in the radio-frequency (RF) wireless communications market, the demand for low-power and high-performance but low-cost RF solutions is rising. Low–power operation can extend the lifetime of the battery and save money for consumers.

2.10 Phase Noise

Noise injected into an oscillator by its constituent devices or by external means may influence both the frequency and the amplitude of the output signal. In most cases, the disturbance in the amplitude is negligible or unimportant, and only the random deviation of the frequency is considered.

For a nominally periodic sinusoidal signal, we can write x(t)=Acos[fct+φn(t)], where φn(t) is a small random excess phase representing variations in the period. The function φn(t) is called “phase noise”. Note that for∣φn(t)∣<<1 rad, we have x(t)≒Acosfct-Aφn(t) sinfct; that is, the spectrum of φn(t) is translated to ±fc.

In RF applications, phase noise is usually characterized in the frequency domain. For an ideal sinusoidal oscillator operating at fc, the spectrum assumes the shape of an impulse, whereas for an actual oscillator, the spectrum exhibits “skits” around the carrier frequency (Fig. 2.10). The frequency fluctuations correspond to jitter in the time domain, which is a random perturbation of zero crossings of a periodic signal (Fig. 2.11).

Figure 2.10 Frequency spectrum of ideal and real oscillators

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Figure 2.11 Jitter in the time domain relates to phase noise in the frequency domain

Frequency fluctuations are usually characterized by the single sideband noise spectral density normalized to the carrier signal power (Fig. 2.10). It is defined as

( ,1

( ,_c ) 10 log ^{sideband c}

carrier

P f f Hz

L f f

P

⎡ + Δ )⎤

Δ = ⎢ ⎥

⎣ ⎦ (11)

and has units of decibels below the carrier per hertz (dBc/Hz). Pcarrier is the carrier signal power at the carrier frequency fC and Psideband(fc+Δf, 1 Hz) denotes the single sideband power at the offset Δf from the carrier fC at a measurement bandwidth of 1 Hz.

Figure 2.12 Oscillator output power spectrum

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The typical oscillator output power spectrum is shown in Fig. 2.12. The noise distribution on each side of the oscillator signal is subdivided into a larger number of strips of width Δf located at the distance fm away from the single. It should be noted that, generally, the spectrum of the output single consists of the phase noise components. Hence, to measure the phase noise close to the carrier frequency, one needs to make sure that any contributions of parasitic amplitude modulation to the oscillator output noise spectrum are negligible compared with those from frequency modulation. The single sideband phase noise L(fm) usually given logarithmically is defined as the ratio of signal power PssΔf in one phase modulation sideband per bandwidth Δf=1 Hz, at an offset fm away from the carrier, to the total signal power Ps.

Time invariant model

In this section, phase noise analysis is described by using time invariant model. Time invariant means whenever noise sources injection, the phase noise in VCO is the same. In the other words, phase shift of VCO caused by noise is the same in any time. Therefore, it’s no need to consider when the noise is coming. Suppose oscillator is consists of amplifier and resonator. The transfer function of a band-pass resonator is written as

( )

2

The transfer function of a common band-pass is written as

( )

Compare equation (12) with (13). Thus,

0

1

ω = LC and Q=ω₀RC (14)

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The frequency ω ω= ₀+ Δω which is near oscillator output frequency. If ω₀ Δω, we can use Taylor expansion for only first and second terms. Hence

( )

The close-loop response of oscillator is expressed by

( ) ( )

The above equation is double sideband noise. The phase noise faraway center frequency ω

Where F is empirical parameter (“often called the device excess noise number”), k is Boltzman’ s constant, T is the absolute temperature, PS is the average power dissipated in the resistive part of the tank, ω₀ is the oscillation frequency, and Q is the effective quality factor of the tank with all the loading in place(also known as loaded Q). From equation (18), increasing power consumption and higher Q factor can get better phase noise. Increasing power consumption means increasing the power of amplifier. This method will decrease noise figure (NF) and improve phase noise.

From, equation (18), we can briefly understand phase noise. But the equation and actual measured results are different. The VCO spectrum is shown as Fig. 2.12. The phase noise equation can be modified as the same as equation (31) that is called Lesson’s model [5].

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These modifications, due to Leeson, consist of a factor to account for the increased noise in the 1/(Δω)² region, an additive factor of unity (inside the braces) to account for the noise floor, and a multiplicative factor (the term in the second set of parentheses) to provide a 1/ Δω3 behavior at sufficiently small offset frequencies. With these modifications, the phase-noise spectrum appears as in Fig. 2.13.

It is important to note that the factor F is an empirical fitting parameter and therefore must be determined from measurements, diminishing the predictive power of the phase-noise equation. Furthermore, the model asserts that Δω1/ f3, the boundary between the 1/(Δω)²

and 1/ Δω³ regions, is precisely equal to the 1/f corner of device noise. However, measurements frequently show no such equality, and thus one must generally treat Δω1/ f3 as an empirical fitting parameter as well. Also, it is not clear what the corner frequency will be in the presence of more than one noise source with 1/f noise contribution. Last, the frequency at which the noise flattens out is not always equal to half the resonator bandwidth, ω₀/ 2Q. Both the ideal oscillator model and the Leeson model suggest that increasing resonator Q and signal amplitude are ways to reduce phase noise. The Leeson model additionally introduces the factor F, but without knowing precisely what it depends on, it is difficult to identify specific ways to reduce it. The same problem exists with Δω1/ f3 as well. Last, blind application of these models has periodically led to earnest but misguided attempts to use active circuits to boost Q. Sadly, increases in Q through such means are necessarily accompanied by increases in F as well, preventing the anticipated improvements in phase noise. Again, the lack of analytical expressions for F can obscure this conclusion, and one

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continues to encounter various doomed oscillator designs based on the notion of active Q boosting.

Figure 2.13 Phase noise: Leeson versus (18).

Time Variant

In the general case, multiple noise sources affect the phase and amplitude of an oscillator.

This chapter begins by investigating the effect of a single noise source on the amplitude and phase of the oscillator.

Figure 2.14 Equivalent systems for phase and amplitude

Since each input source generally affects both amplitude and phase, a pair of equivalent systems, one each for amplitude and phase, can be defined. Each system can be viewed as a single-input, single-output system as shown in Fig. 2.14. The input of each system in Fig.

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2.14 is a perturbation current (or voltage) and the outputs are the excess phase, φ(t), and amplitude, A(t). Both systems shown in Fig. 2.14 are time-variant as shown by the following examples.

The first example is an ideal parallel LC tank oscillating with voltage amplitude, as shown in Fig. 2.15. If one injects an impulse of current at the voltage maximum, only the voltage across the capacitor changes; there is no effect on the current through the inductor. Therefore, the tank voltage changes instantaneously, as shown in Fig. 2.15. Assuming a voltage- and time-invariant capacitor, the instantaneous voltage change ΔV is given by

total

V q C

Δ = Δ (19)

where Δq is the total charge injected by the current impulse and Ctotal is the total capacitance in parallel with the current source. It can be seen from Fig. 2.15 that the resultant change in A(t) and φ(t) is time dependent. In particular, if the impulse is applied at the peak of the voltage across the capacitor, there will be no phase shift and only an amplitude change will result, as shown in Fig. 2.15(a). On the other hand, if this impulse is applied at the zero crossing, it has the maximum effect on the excess phase, φ(t), and the minimum effect on the amplitude, as depicted in Fig. 2.15(b).

Figure 2.15 Impulse response of an ideal LC oscillator

To emphasize the generality of this time-variance, consider two more examples. The

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relaxation oscillator known as the Bose oscillator is shown in Fig. 2.16. It consists of a Schmitt-trigger inverter and an RC circuit. The hysteresis in the transfer function of the inverter and the RC time constant determine the frequency of oscillation. The resulting capacitor voltage waveform is shown with a solid line in Fig. 2.17.

As before, imagine an impulsive current source in parallel with the capacitor, injecting charge at t=τ, as shown in Fig. 2.16. All of the injected charge goes into the capacitor and changes the voltage across it instantaneously. This voltage change, ΔV, results in a phase shift, Δφ, as shown in Fig. 2.17. As can be seen from Fig. 2.17, for a small area of the current impulse (injected charge), the resultant phase shift is proportional to the voltage change, ΔV, and hence to the injected charge, Δq. Therefore, Δφ can be written as

0 0

max max

( ) V ( ) q

V q

φ ω τ Δ ω τ Δ

Δ = Γ = Γ Δ q q_max (20)

Figure 2.16 Bose oscillator with parallel perturbation current source

Figure 2.17 The waveform of the Bose oscillator shown in Figure 3

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where Vmax is the voltage swing across the capacitor and qmax=CnodeVmax is the maximum charge swing. The function, Γ(x) is the time-varying “proportionality factor”. It is called the impulse sensitivity function (ISF), since it determines the sensitivity of the oscillator to an impulsive input. It is a dimensionless, frequency- and amplitude-independent function periodic in 2π that describes how much phase shift results from applying a unit impulse at any point in time.

In any event, to develop a feel for typical shapes of ISF’s, consider two representative examples, first for an LC and a ring oscillator in Fig. 2.18(a) and (b).

Figure 2.18 Example ISF for (a) LC oscillator and (b) ring oscillator.

It is critical to note that the current-to-phase transfer function is linear for small injected charge, even though the active elements may have strongly nonlinear voltage current behavior.

It should also be noted that the linearity and time-variance of a system depends on both the characteristics of the system and its input and output variables. The linearization of the current-to-phase system of Fig. 2.14 does not imply linearization of the nonlinearity of the voltage-current characteristics of the active devices. In fact, this nonlinearity affects the shape of the ISF and therefore has an important influence on phase noise, as will be seen shortly.

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Noting that the introduced phase shift persists indefinitely, the unity phase impulse response can be easily calculated from (20) to be

0

Thanks to linearity, the output excess phase, φ(t), can be calculated for small charge injections using the superposition integral

0

where i(t) represents the input noise current injected into the node of interest. Equation (22) is one of the most important results of this section and will be referred to frequently.

The output voltage, V(t), is related to the phase, φ(t), through a phase modulation process.

Thus the complete process by which a noise input becomes an output perturbation in V(t) can be summarized in the block diagram of Fig. 2.19. The essential features of the block diagram of Fig. 2.19 are a modulation by a periodic function, an ideal integration and a nonlinear phase modulation. The complete process thus can be viewed as a cascade of an LTV system that converts current (or voltage) to phase, with a nonlinear system that converts phase to voltage.

Figure 2.19 The equivalent block diagram of the process.

Since the ISF is periodic, it can be expanded in a Fourier series

0 0 0

23 

where the coefficients cn are real-valued, and θn is the phase of the nth harmonic. As will be seen later, θn is not important for random input noise and is thus neglected here. Using the expansion in (23) for Γ(ω0τ) in the superposition integral and exchanging the order of summation and integration, the following is obtained:

0

Equation (24) identifies individual contributions to the total φ(t) for an arbitrary input current

Equation (24) identifies individual contributions to the total φ(t) for an arbitrary input current

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