Thermal noise

The random motion of electrons in a conductor introduces fluctuations in the voltage measured across the conductor even if the average current is zero. Thus, the spectrum of thermal noise is proportional to the absolute temperature.

Figure 2.4 Thermal noise model of a resistor Noiseless

As shown in Figure 2.4, the thermal noise of a resistor R can be modeled by a series voltage source, with the one-sided spectral density

Where Κis the Boltzmann constant. Thus, we write where the overlie indicates an average. to emphasize that 4KTR is the noise power per unit bandwidth. To simplify the notation, we assume △f =1Hz, unless otherwise stated. The thermal noise of a resistor can be represented by a parallel current source as well. For the representations of Figure 2.4 and Figure 2.5 to be equivalent, we have , that is

. Note that is expressed in .

Noiseless Resistor

R

Figure 2.5 Representation of resistor thermal noise by a current source Consider a resistor R in parallel with a capacitor C. As a result of the random thermal agitation of the electrons in the resistor, the capacitor will be charged and discharged at random. The average energy stored in the capacitor will be:

Figure 2.6 Noise generated in a low pass filter

(2.2)

Where V 2 is the mean-square value of the voltage fluctuation impressed across the capacitor. This equation can be proved by the theorem as follows：

Solving for the spectral density Sv(0) we get

Equation (2.5) is called Nyquist's Theorem and the symbol Sv(0) for the spectral density means that there is no frequency dependence. Noise with such a spectrum is called white.

2.3 MOSFETS thermal noise

This approach becomes more attractive with the observation that the mobility of charge carries in MOS devices increases at low temperature. MOS transistors also exhibit thermal noise. The most significant source is the noise generated in the channel. It can be proved that for long channel MOS devices operating in saturation, the channel noise can be modeled by a current source connected between the source and drain terminals (Figure 2.7) with a spectral density.

The actual equation reads, where gds is the drain-source conductance with

V^ds =0. The coefficient γ is derived to be equal to 2/3 for long channel transistors and may need to be replaced by a large value for submicron MOSFETs. It also varies to some extent with the drain-source voltage. The theoretical determination of γ is still under active research.

The maximum output noise occurs if the transistor sees only its own output impedance as the load. The output noise voltage is then given by

So the noise current of a MOS transistor decreases if the transconductance drop. For example, if the transistor operates as a constant current source, it is desirable to minimize its transconductance. SO, the Figure 2.8 may be a common source or a common gate stage, exhibiting the same output noise.

2.4 Flicker noise

The interface between the gate oxide and the silicon substrate in a MOSFET entails an interesting phenomenon. Since the silicon crystal reaches an end at this interface, many

“dangling＂ bonds appear, giving rise to extra energy states. As charge carriers move at the interface, some are randomly trapped and later released by such energy states, introducing “Flicker＂ noise in the drain current. In addition to trapping, several other mechanisms are believed to general flicker noise. The flicker noise is more easily modeled as a voltage source in series with the gate and roughly given by

Where K is a process-dependent constant on the order of. In Figure 2.9 the noise spectral density is inversely proportional to the frequency. For this reason, flicker noise is also called 1/f noise. Note that Equation (2.9) does not depend on the bias current or the temperature. The inverse dependence of Equation (2.9) on WL suggests that to decrease 1/f noise, the device area must be increased. It is therefore not surprising to see devices having areas of several thousand square microns in low-noise applications. It is also believed that PMOS devices exhibit less 1/f noise than NMOS transistors because the former carry the holes in a “buried channel＂, at some distance form the oxide-silicon interface. Nonetheless, this difference between PMOS and NMOS transistors is not consistently observed.

We plot both spectral densities on the same axes (Figure 2.9). Called the 1/f noise “corne frequency＂, the intersection point serves as a measure of what part of the band is mostly corrupted by flicker noise.

that is

This result implies that ^c f generally depends on device dimensions and bias current.

Nonetheless, since for a given L, the dependence is relatively weak, the 1/f noise corner is relatively constant, falling in the vicinity of 500KHz to 1MHz for submicron transistors.

2.5 Phase noise

As other analog circuits, oscillators are susceptible to noise. Noise injected into an oscillator by its constituent devices or by external may influence both the frequency and the amplitude of the output signal. In most cases, the disturbance in amplitude is negligible or unimportant, and only the random deviation of the frequency is considered. The latter can be viewed as random variation in the period or deviation of zero crossing points from their ideal position along time axis. For a nominally periodic sinusoidal signal we can write , where is small random excess phase representing variations in period. The function ( ) ⁿ f t is called phase noise. Note that for < 1 rad, we have

; that is, the spectrum of is translated to .

In RF applications, phase noise is usually characterized in the frequency domain. For an ideal sinusoidal oscillator operating at , the spectrum assumes the shape of an impulse, whereas an actual oscillator, the spectrum exhibits skirts around the carrier frequency (Figure 2.10). To quantify phase noise, we consider a unit bandwidth at an offset Δw with respect to , calculate the noise power in this bandwidth, and divide the result by the carrier (average) power.

It is conventionally given the units of decibels below the carrier per Hertz (dBc/Hz) and is define as shown in Equation (2.12) where represents the single sideband power at a frequency offset, Δw , from the carrier in a measurement bandwidth of 1Hz as shown in Figure 2.10, and is the total power under the power spectrum.

The advantage of in Equation (2.12) is its ease of measurement. Its disadvantage is that is shows the sum of both amplitude and phase variations; it does not show them separately in a circuit. It is often important to known the amplitude and phase noise separately because they behave differently in a circuit. For instance, the effect of amplitude noise can be reduced by amplitude limiting, while the phase noise cannot be reduced in an analog manner. Therefore, in most practical oscillators, is dominated by its

phase portion, , known as phase noise, which will be simply denoted as , unless specified otherwise. For example, if the carrier power is –2 dBm and the noise power measured in a 1kHz bandwidth at an offset of 1 MHz is equal to –70 dBm then the phase noise is specified as –70dBm+2dBm-30dB=-98dBc/Hz, where dBc means “ in db with respect to carrier＂ [1].

Phase noise is measured in the frequency domain, and is expressed as a ratio of signal power to noise power measured in a 1 Hz bandwidth at a given offset from the desired signal. A plot of responses at various offsets from the desired signal is usually comprised of three distinct slopes corresponding to three primary noise generating mechanisms in the oscillator, as shown in Figure 2.11. Noise relatively close to the carrier (Region A) is called Flicker FM noise; its magnitude is determined primarily by the quality of the crystal.

Noise in Region B of Figure 2.11, called "1/f" noise, is caused by semiconductor activity.

Design techniques employed in low noise crystal oscillators limit this to a very low, often insignificant value. Region C of Figure 2.11 is called white noise or broadband noise.

It is important to determine from measurements, diminishing the predictive power of the phase-noise equation. Furthermore, the model asserts that , the boundary between

the and 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 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, . The ideal oscillator model suggest that increasing resonator Q and signal amplitude are ways to reduce phase noise. Referring to the ideal case depicted in Figure 2.12(a), we note that the signal of interest is convolved with an impulse and thus translated to a lower (and a higher) frequency with no change in its shape. In reality, however, the wanted signals may be accompanied by a large interferer in adjacent channel,

and the local oscillator exhibits finite phase noise [Figure 2.12(b)]. When the two signals are mixed with the LO output, the down converted band consists of two overlapping spectra, with the wanted signal suffering from significant noise due to the tail of the interferer. This effect is called “reciprocal mixing＂. Shown in Figure 2.12(c), the effect of phase noise on transmit path is slightly different. Suppose a noiseless receiver is to detect a weak

2.6 Varactors

2.6.1 Diode varactor

The varactor diode symbol is shown below with a diagram representation.

When a reverse voltage is applied to a PN junction, the holes in the p-region are attracted to the anode terminal and electrons in the n-region are attracted to the cathode terminal creating a region where there is a little current. This region, the depletion region, is essentially devoid of carriers and behaves as the dielectric of a capacitor. The depletion region increases as reverse voltage across it increases; and since capacitance varies inversely as dielectric thickness, the junction capacitance will decrease as the voltage across the PN junction increases. So by varying the reverse voltage across a PN junction the junction capacitance can be varied. This is shown in the typical varactor

voltage-capacitance curve below. The capacitance is then a function of the width of the depletion region, which is controlled by the reverse voltage (VR). If a small VR is applied across the junction, the width is correspondingly small, and hence a large capacitance results. If VR is increased, the junction width also increases, and a smaller capacitance is obtained. Hence, by changing the reverse voltage, the junction capacitance is also changed, and the wanted functionality is obtained.

Major varactor considerations are:

(a) Capacitance value (b) Voltage

(c) Variation in capacitance with voltage (d) Maximum working voltage

(e) Leakage current

2.6.2 MOS varactor

A MOS transistor with drain, source, and bulk (D, S, B) connected together realizes a MOS capacitor with capacitance value dependent on the voltage Vgs between B and gate (G). The cross sectional view of the NMOS varactor is shown in Figure 2.14. In the case of an NMOS capacitor, an inversion channel with mobile holes builds up for , where is the threshold voltage of the transistor. The condition guarantees that the MOS capacitor works in the strong inversion region, the region where the MOS device shows a transistors behavior. On the other hand, for some voltage , the MOS device enters the accumulation region, where the voltage at the interface between gate oxide and semiconductor is positive and high enough to allow electrons to move freely. Thus, in both strong inversion and accumulation region the value of the MOS

capacitance Cgs is equal to , where S and tox are the transistor channel area and the oxide thickness, respectively [3].

A very attractive realization of an MOS varactor is given by the pMOS device working in the depletion and accumulation regions only. This solution allows for the implementation of a MOS varactor with a large tuning range (Cmos does not climb back to Cox, since

accumulation-mode MOS capacitor, we must make sure that the formation of the strong, moderate, and weak inversion regions is inhibited, which requires the suppression of any injection of holes in the MOS channel. This, in turn, can be accomplished with the removal of the D/S diffusions (p+-doped) from the MOS device. At the same time, we can implement the bulk contacts (n+) in the place left by D/S, as shown in Figure 2-15, which minimizes the parasitic N-well resistance of the device [4].

2.7 Voltage control oscillator

An ideal voltage control oscillator is a circuit whose output frequency is a linear function of its control voltage

represents the intercept corresponding to and denotes the “gain＂ or

“sensitivity＂ of the circuit . The achievable range, , is called the “tuning range＂.

Figure 2.16

ω

Center frequency ： The center frequency is determined by the environment in which the VCO is used. Today CMOS VCO＇s achieve center frequencies as high as 1.8GHz, 2.4GHz and 5.2~5.8GHz.

The center frequency of some CMOS oscillators may vary by a factor of the two at the extremes 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 VCO＇s 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 because Thus, to minimize the effect of noise in , the VCO gain must be minimized, a constraint in direct conflict with the require tuning range. In fact, if, as shown in Figure 2.16, the allowable range of is from V1 to V2 and the tuning range must span at least ω1 to ω2, then

must satisfy the following requirement：

Output amplitude ： It is desired to achieve large output oscillation amplitude, thus making the waveform less sensitive to noise. The amplitude trades with power dissipation, supply voltage, and tuning range. Also, the amplitude may vary across the tuning range, an undesirable effect.

Power dissipation： In analog circuit, oscillators suffer from trade-offs between speed, power dissipation, and noise.

Supply and common mode rejection ： Oscillators are quite sensitive to noise, especially if they are realized in single-ended form. The design of oscillators for high noise immunity is a difficult challenge. So that noise may be coupled to the control line of a VCO as well. For these reasons, it is preferable to employ differential paths for both the oscillation signal and the control line.

Output signal purity ： Even with a constant control voltage, the output waveform of a VCO is not perfectly periodic. The electronic noise of the devices in the oscillator and supply noise lead to noise in the output phase and frequency. These effects are quantified by “Jitter＂ and “Phase Noise＂ and determined by the requirements of each application [2].

2.8 Inductor

In integrated RF works in silicon, inductors are normally implemented as a planar spiral-shaped metal. Figure 2.17 shows the top view of an example spiral inductor in silicon, realized using the top metal layer while the metal layer below the top metal layer is used for an interconnection for terminal 2. Figure 2.18 shows an equivalent electrical circuit model for the spiral inductor, obtained using an electromagnetic simulation. In this model, L, Rs, Rp, and Cp represent inductance, metal loss due to the skin effect, substrate loss, and metal-substrate capacitance, respectively. Cs accounts for the metal overlap capacitance between the top metal and the metal below. Using Cadence, measures the quality factor, Q of the spiral inductor (See the note below) over the frequency range.

Figure 2.18 Spiral inductor equivalent electrical circuit model

is discussed in class, the quality factor, Q is originally defined for a

“ resonator ＂ as：

where is the resonance frequency. For a given resonator whose resonance frequency is fixed, Q is not a function of frequency. Since inductors are not resonators, the original Q definition above cannot be used for inductors. However, the following frequency -dependent Q definition may be used instead as the quality factor for inductors:

It can be easily shown that Q (ω) in (2.16) is on the same order as, but not exactly the same as,

where Z (ω) is the frequency-dependent input impedance of a given inductor. RF engineers traditionally choose to use (2.15) over (2.16) due to the simplicity of (2.17). In

the problem above, we can evaluate the Q of the spiral inductor using (2.17) while the input impedance Z (ω) shown in Figure 2.18 can be measured using Cadence.

CHAPTER 3

LC TANK OSCILLATOR THEORY

3.1 LC tank oscillator architecture

Figure 3.1 (a) Ideal, (b) realistic LC tank

As shown in Figure 3.1(a), an inductor L1 placed in parallel with a capacitor C1 resonates

at a frequency . We say the circuit has an infinite quality factor, Q. In practice, inductors (and capacitors) suffer from resistive components. When a charged capacitor is connected to an inductor, the conventional analysis is to equate the voltage across the capacitor with the voltage across the inductor.

(3.1) Differentiating, we get

(3.2) This is then recognised as having as a solution of simple harmonic motion (SHM),

(3.3) The traditional analysis assumes that when current is switched into the inductor, it appears instantaneously at all points in the inductor; the use of the single, lumped quantity L implies this. Similarly, it is assumed that the electric charge density at all points in the capacitor is the same; that there are no transient effects such that the charge density is greater in certain regions of the capacitor plates.

For this circuit reader can show that the equivalent impedance is given by

(3.4)

(3.5) that is, the impendence does not go to infinite at any s=jω. We say the circuit has a finite

Q. The magnitude of Zeq in (3.5) reaches a peak in the vicinity of , but the actual resonance frequency has some dependency on Rs.

Let us now consider the “tuned＂ stage of Figure 3.2(a), where an LC tank operatesas

the load. At resonance, and the voltage gain equal (Note that the gain of the circuit is very small at frequencies near zero). Also from Figure 3.2(b), the frequency dependent phase shift of the tank never reaches .Thus, the circuit does not oscillate.

Figure 3.3 Output signal levels in a tuned stage

Before modifying the circuit for oscillatory behavior, let us observe another interesting property of the gain stage of Figure 3.2 (a) that distinguishes it from a common-source topology using a resistive load. Suppose, as shown in Figure 4.3, the stage is biased at a drain current. If the series resistance of Lp is small, the dc level of Vout is close to Vdd.

We expect Vout to be an inverted sinusoid with an average value near Vdd because the inductor cannot sustain a large dc drop. In other average value of Vout deviates significantly fromVdd, then the inductor series resistance must carry an average current greater than I1. Thus, the peak output level in fact exceeds the supply voltage, an import and often useful attribute of the LC load. For example, with proper design, the output peak-to-peak swing can be large than Vdd.

3.2 LC cross coupled oscillator theory

Calculating the impedance seen at the collector of Q1 and Q2 as shown in Figure 3.4(a),

we note that positive feedback yields (shown in Figure 3.4 (b) and Equations 3.6-3.10). Thus, if |Rin| is larger than or equal to the equivalent parallel resistance of the tank, the circuit oscillates. This topology is called a negative-Gm oscillator.

Figure 3.4 (a) Circuit to calculate the input impedance of cross coupled

Figure 3.4 (a) Circuit to calculate the input impedance of cross coupled