In our study of RF transceivers in Chapter 4, we noted the extensive use of oscillators in both the transmit and receive paths. Interestingly, in most systems, one input of every mixer is driven by a periodic signal, hence the need for oscillators. This chapter deals with the analysis and design of oscillators. The outline is shown below.

An oscillator used in an RF transceiver must satisfy two sets of requirements: (1) system specifications, e.g., the frequency of operation and the “purity” of the output, and (2) “interface” specifications, e.g., drive capability or output swing. In this section, we study the oscillator performance parameters and their role in the overall system.

An RF oscillator must be designed such that its frequency can be varied (tuned) across a certain range. This range includes two components: (1) the system specification; for example, a 900-MHz GSM direct-conversion receiver may tune the LO from 935 MHz to 960 MHz; (2) additional margin to cover process and temperature variations and errors due to modeling inaccuracies. The latter component typically amounts to several percent.

The actual frequency range of an oscillator may also depend on whether quadrature outputs are required and/or injection pulling is of concern (Chapter 4). A direct-conversion transceiver employs quadrature phases of the carrier, necessitating that either the oscillator directly generate quadrature outputs or it run at twice the required frequency so that a ÷2 stage can produce such outputs. For example, in the hybrid topology of Fig. 8.1, the LO must still provide quadrature phases in the 5-GHz range—but it is prone to injection pulling by the PA output. We address the problem of quadrature generation in Section 8.11.

How high a frequency can one expect of a CMOS oscillator? While oscillation frequencies as high as 300 GHz have been demonstrated [1], in practice, a number of serious trade-offs emerge that become much more pronounced at higher operation frequencies. We analyze these trade-offs later in this chapter.

As exemplified by the arrangement shown in Fig. 8.1, the oscillators in an RF system drive mixers and frequency dividers. As such, they must produce sufficiently large output swings to ensure nearly complete switching of the transistors in the subsequent stages. Furthermore, as studied in Section 8.7, excessively low output swings exacerbate the effect of the internal noise of the oscillator. With a 1-V supply, a typical single-ended swing may be around 0.6 to 0.8 V_{pp}. A buffer may follow the oscillator to amplify the swings and/or drive the subsequent stage.

Oscillators may need to drive a large load capacitance. Figure 8.2 depicts a typical arrangement for the receive path. In addition to the downconversion mixers, the oscillator must also drive a frequency divider, denoted by a ÷*N* block. This is because a loop called the “frequency synthesizer” must precisely control the frequency of the oscillator, requiring a divider (Chapter 10). In other words, the LO must drive the input capacitance of at least one mixer and one divider.

Interestingly, typical mixers and dividers exhibit a trade-off between the minimum LO swing with which they can operate properly and the capacitance that they present at their LO port. This can be seen in the representative stage shown in Fig. 8.3(a), wherein it is desirable to switch *M*_{1} and *M*_{2} as abruptly as possible (Chapter 6). To this end, we can select large LO swings so that *V*_{GS1} − *V*_{GS2} rapidly reaches a large value, turning off one transistor [Fig. 8.3(b)]. Alternatively, we can employ smaller LO swings but *wider* transistors so that they steer their current with a smaller differential input.

The issue of capacitive loading becomes more serious in transmitters. As explained in Example 6.37, the PA input capacitance “propagates” to the LO port of the upconversion mixers.

To alleviate the loading presented by mixers and dividers and perhaps amplify the swings, we can follow the LO with a buffer, e.g., a differential pair. Note that in Fig. 8.2, *two* buffers are necessary for the quadrature phases. The buffers consume additional power and may require inductive loads—owing to speed limitations or the need for swings above the supply voltage (Chapter 6). The additional inductors complicate the layout and the routing of the high-frequency signals.

The spectrum of an oscillator in practice deviates from an impulse and is “broadened” by the noise of its constituent devices. Called “phase noise,” this phenomenon has a profound effect on RF receivers and transmitters (Section 8.7). Unfortunately, phase noise bears direct trade-offs with the tuning range and power dissipation of oscillators, making the design more challenging. Since the phase noise of LC oscillators is inversely proportional to the *Q* of their tank(s), we will pay particular attention to factors that degrade the *Q*.

What is the desired output waveform of an RF oscillator? Recall from the analysis of mixers in Chapter 6 that *abrupt* LO transitions reduce the noise and increase the conversion gain. Moreover, effects such as direct feedthrough are suppressed if the LO signal has a 50% duty cycle. Sharp transitions also improve the performance of frequency dividers (Chapter 10). Thus, the ideal LO waveform in most cases is a square wave.

In practice, it is difficult to generate square LO waveforms. This is primarily because the LO circuitry itself and the buffer(s) following it typically incorporate (narrowband) resonant loads, thereby attenuating the harmonics. For this reason, as illustrated in Fig. 8.3, the LO amplitude is chosen large and/or the switching transistors wide so as to approximate abrupt current switching.

A number of considerations call for *differential* LO waveforms. First, as observed in Chapter 6, *balanced* mixers outperform unbalanced topologies in terms of gain, noise, and dc offsets. Second, the leakage of the LO to the input is generally smaller with differential waveforms.

The frequency of an oscillator may vary with the supply voltage, an undesirable effect because it translates supply noise to frequency (and phase) noise. For example, external or internal voltage regulators may suffer from substantial flicker noise, which cannot be easily removed by bypass capacitors due to its low-frequency contents. This noise therefore modulates the oscillation frequency (Fig. 8.5).

The power drained by the LO and its buffer(s) proves critical in some applications as it trades with the phase noise and tuning range. Thus, many techniques have been introduced that lower the phase noise for a given power dissipation.

An oscillator generates a periodic output. As such, the circuit must involve a self-sustaining mechanism that allows its own noise to grow and eventually become a periodic signal.

An oscillator may be viewed as a “badly-designed” negative-feedback amplifier—so badly-designed that it has a zero or negative phase margin. While the art of oscillator design entails much more than an unstable amplifier, this view provides a good starting point for our study. Consider the simple linear negative-feedback system depicted in Fig. 8.6, where

What happens if at a sinusoidal frequency, *ω*_{1}, *H*(*s* = *jω*_{1}) becomes equal to −1? The gain from the input to the output goes to infinity, allowing the circuit to amplify a noise component at *ω*_{1} indefinitely. That is, the circuit can sustain an output at *ω*_{1}. From another point of view, the closed-loop system exhibits two imaginary poles given by ±*jω*_{1}.

The above example leads to a general and powerful analytical point: in the small-signal model of an oscillator, the impedance seen between *any* two nodes (one of which is not ground) in the signal path goes to infinity at the oscillation frequency, *ω*_{1}, because a noise current at *ω*_{1} injected between these two nodes produces an infinitely large swing. This observation can be used to determine the oscillation condition and frequency.

Since *H*(*s*) is a complex function, the condition *H*(*jω*_{1}) = −1 can equivalently be expressed as

which are called “Barkhausen’s criteria” for oscillation. Let us examine these two conditions to develop more insight. We recognize that a signal at *ω*_{1} experiences a gain of unity and a phase shift of 180° as it travels through *H*(*s*) [Fig. 8.8(a)]. Bearing in mind that the system is originally designed to have *negative* feedback (as denoted by the input subtractor), we conclude that the signal at *ω*_{1} experiences a *total* phase shift of 360° [Fig. 8.8(b)] as it travels around the loop. This is, of course, to be expected: for the circuit to reach steady state, the signal returning to *A* must exactly coincide with the signal that started at *A*. We call ∠*H*(*jω*_{1}) a “frequency-dependent” phase shift to distinguish it from the 180° phase due to negative feedback.

The above point can also be viewed as follows. Even though the system was originally configured to have negative feedback, *H*(*s*) is so “sluggish” that it contributes an additional phase shift of 180° at *ω*_{1}, thereby creating *positive* feedback at this frequency.

What is the significance of |*H*(*jω*_{1})| = 1? For a noise component at *ω*_{1} to “build up” as it circulates around the loop with positive feedback, the loop gain must be at least unity. Figure 8.9 illustrates the “startup” of the oscillator if |*H*(*jω*_{1})| = 1 and ∠*H*(*jω*_{1}) = 180°. An input at *ω*_{1} propagates through *H*(*s*), emerging unattenuated but inverted. The result is *subtracted* from the input, yielding a waveform with twice the amplitude. This growth continues with time. We call |*H*(*jω*_{1})| = 1 the “startup” condition.

What happens if |*H*(*jω*_{1})| *>* 1 and ∠*H*(*jω*_{1}) = 180°? The growth shown in Fig. 8.9 still occurs but at a faster rate because the returning waveform is amplified by the loop. Note that the closed-loop poles now lie in the right half plane. The amplitude growth eventually ceases due to circuit nonlinearities. We elaborate on these points later in this chapter.

Our study thus far allows us to predict the frequency of oscillation: we seek the frequency at which the total phase shift around the loop is 360° and determine whether the loop gain reaches unity at this frequency. (An exception is described in the example below.) This calculation, however, does not predict the oscillation *amplitude*. In a perfectly linear loop, the oscillation amplitude is simply given by the initial conditions residing on the storage elements if the loop gain is equal to unity at the oscillation frequency. The following example illustrates this point.

Other oscillators behave differently from the two-integrator loop: they may begin to oscillate at a frequency at which the loop gain is *higher* than unity, thereby experiencing an exponential growth in their output amplitude. In an actual circuit, the growth eventually stops due to the saturating behavior of the amplifier(s) in the loop. For example, consider the cascade of three CMOS inverters depicted in Fig. 8.11 (called a “ring oscillator”). If the circuit is released with *X*, *Y*, and *Z* at the trip point of the inverters, then each stage operates as an amplifier, leading to an oscillation frequency at which each inverter contributes a frequency-dependent phase shift of 60°. (The three inversions make this a negative-feedback loop at low frequencies.) With the high loop gain, the oscillation amplitude grows exponentially until the transistors enter the triode region at the peaks, thus lowering the gain. In the steady state, the output of each inverter swings from nearly zero to nearly *V _{DD}*.

In most oscillator topologies of interest to us, the voltage swings are defined by the saturating behavior of differential pairs. The following example elaborates on this point.

In the previous section, we considered oscillators as negative-feedback systems that experience sufficient positive feedback at some frequency. An alternative perspective views oscillators as two one-port components, namely, a lossy resonator and an active circuit that cancels the loss. This perspective provides additional insight and is described in this section.

Suppose, as shown in Fig. 8.13(a), a current impulse, *I*_{0}*δ*(*t*), is applied to a lossless tank. The impulse is entirely absorbed by *C*_{1} (why?), generating a voltage of *I*_{0}/*C*_{1}. The charge on *C*_{1} then begins to flow through *L*_{1}, and the output voltage falls. When *V _{out}* reaches zero,

Now, let us assume a lossy tank. Depicted in Fig. 8.13(b), such a circuit behaves similarly except that *R _{p}* drains and “burns” some of the capacitor energy in every cycle, causing an exponential decay in the amplitude. We therefore surmise that, if an active circuit replenishes the energy lost in each period, then the oscillation can be sustained. In fact, we predict that an active circuit exhibiting an input resistance of −

How can a circuit present a negative (small-signal) input resistance? Figure 8.14(a) shows an example, where two capacitors are tied from the gate and drain of a transistor to its source. The impedance *Z _{in}* can be obtained by noting that

It follows that

For a sinusoidal input, *s* = *jω*,

Thus, the input impedance can be viewed as a series combination of *C*_{1}, *C*_{2}, and a *negative* resistance equal to −*g _{m}*/(

Having found a negative resistance, we can now attach it to a lossy tank so as to construct an oscillator. Since the capacitive component in Eq. (8.24) can become part of the tank, we simply connect an inductor to the negative-resistance port (Fig. 8.15), seeking the condition for oscillation. In this case, it is simpler to model the loss of the inductor by a series resistance, *R _{S}*. The circuit oscillates if

Under this condition, the circuit reduces to *L*_{1} and the series combination of *C*_{1} and *C*_{2}, exhibiting an oscillation frequency of

As expected, for oscillation to occur, the transistor in Fig. 8.14(a) must provide sufficient “strength” (transconductance). In fact, (8.30) implies that the minimum allowable *g _{m}* is obtained if

In this section, we develop an LC oscillator topology that, owing to its robust operation, has become the dominant choice in RF applications. We begin the development with a feedback system, but will discover that the result also lends itself to the one-port view described in Section 8.2.2.

We wish to build a negative-feedback oscillatory system using “LC-tuned” amplifier stages. Figure 8.16(a) shows such a stage, where *C*_{1} denotes the total capacitance seen at the output node and *R _{p}* the equivalent parallel resistance of the tank at the resonance frequency. We neglect

Let us examine the frequency response of the stage. At very low frequencies, *L*_{1} dominates the load and

That is, |*V _{out}*/

The phase shift from the input to the output is thus equal to −180°. At very high frequencies, *C*_{1} dominates, yielding

Thus, |*V _{out}*/

Can the circuit of Fig. 8.16(a) oscillate if its input and output are shorted? As evidenced by the open-loop magnitude and phase plots shown in Fig. 8.16(b), no frequency satisfies Barkhausen’s criteria; the total phase shift fails to reach 360° at any frequency.

Upon closer examination, we recognize that the circuit provides a phase shift of 180° with possibly adequate gain (*g _{m}R_{p}*) at

The above circuit can be redrawn as shown in Fig. 8.18(a) and is called a “cross-coupled” oscillator due to the connection of *M*_{1} and *M*_{2}. Forming the core of most RF oscillators used in practice, this topology entails many interesting properties and will be studied from different perspectives in this chapter.

Let us compute the oscillation frequency of the circuit. The capacitance at *X* includes *C*_{GS2}, *C*_{DB1}, and the effect of *C*_{GD1} and *C*_{GD2}. We note that (a) *C*_{GD1} and *C*_{GD2} are in parallel, and (b) the total voltage change across *C*_{GD1} + *C*_{GD2} is equal to twice the voltage change at *X* (or *Y*) because *V _{X}* and

Here, *C*_{1} denotes the parasitic capacitance of *L*_{1} plus the input capacitance of the next stage.

The oscillator of Fig. 8.18(a) suffers from poorly-defined bias currents. Since the average *V _{GS}* of each transistor is equal to

The above-supply swings in the cross-coupled oscillator of Fig. 8.18(b) raise concern with respect to transistor reliability. The instantaneous voltage difference between any two terminals of *M*_{1} or *M*_{2} must remain below the maximum value allowed by the technology. Figure 8.19 shows a “snapshot” of the circuit when *M*_{1} is off and *M*_{2} is on. Each transistor may experience stress under the following conditions: (1) The drain reaches *V _{DD}* +

The reader may wonder how the inductance value and the device dimensions are selected in the cross-coupled oscillator. We defer the design procedure to after we have studied voltage-controlled oscillators and phase noise (Section 8.8).

While conceived as a feedback system, the cross-coupled oscillator also lends itself to the one-port view described in Section 8.2.2. Let us first redraw the circuit as shown in Fig. 8.21(a) and note that, for small differential waveforms at *X* and *Y*, *V _{N}* does not change even if it is not connected to

It follows that

which, for *g*_{m1} = *g*_{m2} = *g _{m}*, reduces to

For oscillation to occur, the negative resistance must cancel the loss of the tank:

and hence

As expected, this condition is identical to that expressed by Eq. (8.34).^{3}

The foregoing studies may suggest that the *g _{m}* of the cross-coupled transistors in Fig. 8.18(b) can be chosen slightly greater than

As observed in Section 8.2.2, the circuit of Fig. 8.14(a) can be attached to an inductor so as to form an oscillator. Note that the derivation of the impedance *Z _{in}* does not assume any terminal is grounded. Thus, three different oscillator topologies can be obtained by grounding each of the transistor terminals. Figures 8.22(a), (b), and (c) depict the resulting circuits if the source, the gate, or the drain is (ac) grounded, respectively. In each case, a current source defines the bias current of the transistor. [The gate of

It is important to bear in mind that the operation frequency and startup condition of all three oscillators in Fig. 8.22 are given by Eqs. (8.26) and (8.30), respectively. Specifically, the transistor must provide sufficient transconductance to satisfy

if *C*_{1} = *C*_{2}. This condition is more stringent than Eq. (8.34) for the cross-coupled oscillator, suggesting that the circuits of Fig. 8.22 may fail to oscillate if the inductor *Q* is not very high. This is the principal disadvantage of these oscillators and the reason for their lack of popularity.

Another drawback of the circuits shown in Fig. 8.22 is that they produce only single-ended outputs. It is possible to couple two copies of one oscillator so that they operate differentially. Shown in Fig. 8.23 is an example, where two instances of the oscillator in Fig. 8.22(c) are coupled at node *P*. Resistor *R*_{1} establishes a dc level equal to *V _{DD}* at

Even with differential outputs, the circuit of Fig. 8.23 may be inferior to the cross-coupled oscillator of Fig. 8.18(b)—not only for the more stringent startup condition but also because the noise of *I*_{1} and *I*_{2} directly corrupts the oscillation. The circuit nonetheless has been used in some designs.

Most oscillators must be tuned over a certain frequency range. We therefore wish to construct oscillators whose frequency can be varied electronically. “Voltage-controlled oscillators” (VCOs) are an example of such circuits.

Figure 8.24 conceptually shows the desired behavior of a VCO. The output frequency varies from *ω*_{1} to *ω*_{2} (the required tuning range) as the control voltage, *V _{cont}*, goes from

where *ω*_{0} denotes the intercept point on the vertical axis. As explained in Chapter 9, it is desirable that this characteristic be relatively linear, i.e., *K _{VCO}* not change significantly across the tuning range.

In order to vary the frequency of an LC oscillator, the resonance frequency of its tank(s) must be varied. Since it is difficult to vary the inductance electronically, we only vary the capacitance by means of a varactor. As explained in Chapter 7, MOS varactors are more commonly used than *pn* junctions, especially in low-voltage design. We thus construct our first VCO as shown in Fig. 8.25(a), where varactors *M*_{V1} and *M*_{V2} appear in *parallel* with the tanks (if *V _{cont}* is provided by an ideal voltage source). Note that the gates of the varactors are tied to the oscillator nodes and the source/drain/

Since the gates of *M*_{V1} and *M*_{V2} reside at an average level equal to *V _{DD}*, their gate-source voltage remains positive and their capacitance

where *C _{var}* denotes the average value of each varactor’s capacitance.

The reader may wonder why capacitors *C*_{1} have been included in the oscillator of Fig. 8.25(a). It appears that, without *C*_{1}, the varactors can vary the frequency to a greater extent, thereby providing a wider tuning range. This is indeed true, and we rarely need to add a constant capacitance to the tank deliberately. In other words, *C*_{1} simply models the inevitable capacitances appearing at *X* and *Y*: (1) *C _{GS}*,

The above VCO topology merits two remarks. First, the varactors are stressed for part of the period if *V _{cont}* is near ground and

As explained in Chapter 7, symmetric spiral inductors excited by differential waveforms exhibit a higher *Q* than their single-ended counterparts. For this reason, *L*_{1} and *L*_{2} in Fig. 8.25 are typically realized as a single symmetric structure. Figure 8.26 illustrates the idea and its circuit representation. The point of symmetry of the inductor (its “center tap”) is tied to *V _{DD}*. In some of our analyses, we omit the center tap connection for the sake of simplicity.

The VCO of Fig. 8.25(a) provides an output CM level near *V _{DD}*, an advantage or disadvantage depending on the next stage (Section 8.9).

While a robust, versatile topology, the cross-coupled VCO of Fig. 8.25(a) suffers from a narrow tuning range. As mentioned above, the three components comprising *C*_{1} tend to limit the effect of the varactor capacitance variation. Since in (8.50), *C _{var}* tends to be a small fraction of the total capacitance, we make a crude approximation,

If the varactor capacitance varies from *C*_{var1} to *C*_{var2}, then the tuning range is given by

For example, if *C*_{var2} − *C*_{var1} = 20%*C*_{1}, then the tuning range is about ±5% around the center frequency.

What limits the capacitance range of the varactor, *C*_{var2} − *C*_{var1}? We note from Chapter 7 that *C*_{var2} − *C*_{var1} trades with the *Q* of the varactor: a longer channel reduces the relative contribution of the gate-drain and gate-source overlap capacitances, widening the range but lowering the *Q*. Thus, the tuning range trades with the overall tank *Q* (and hence with the phase noise).

Another limitation on *C*_{var2} − *C*_{var1} arises from the available range for the control voltage of the oscillator, *V _{cont}* in Fig. 8.25(a). This voltage is generated by a “charge pump” (Chapter 9), which, as any other analog circuit, suffers from a limited output voltage range. For example, a charge pump running from a 1-V supply may not be able to generate an output below 0.1 V or above 0.9 V. The characteristic of Fig. 8.25(b) therefore reduces to that depicted in Fig. 8.27.

The foregoing tuning limitations prove serious in LC VCO design. We introduce in Section 8.6 a number of oscillator topologies that provide a wider tuning range—but at the cost of other aspects of the performance.

As observed in the previous section, the varactor capacitance is but a small fraction of the total tank capacitance. We therefore surmise that the resistive loss of the varactor lowers the overall *Q* of the tank only to some extent. Let us begin with a fundamental observation.

To quantify the effect of varactor loss, consider the tank shown in Fig. 8.29(a), where *R*_{p1} models the loss of the inductor and *R _{var}* the equivalent series resistance of the varactor. We wish to compute the

To utilize our previous results, we combine *C*_{1} and *C _{var}*. The

Recognizing that *Q _{var}* = (

In other words, the *Q* of the varactor is “boosted” by a factor of 1 + *C*_{1}/*C _{var}*. The overall tank

For frequencies as high as several tens of gigahertz, the first term in Eq. (8.64) is dominant (unless a long channel is chosen for the varactors).

Equation (8.64) can be generalized if the tank consists of an ideal capacitor, *C*_{1}, and lossy capacitors, *C*_{2}-*C _{n}*, that exhibit a series resistance of

where *C _{tot}* =

The tuning range obtained from the C–V characteristic depicted in Fig. 8.27 may prove prohibitively narrow, particularly because the capacitance range corresponding to *negative V _{GS}* (for

Figure 8.30(a) shows one such topology. Unlike the tail-biased configuration studied in Section 8.3, this circuit defines the bias currents of *M*_{1} and *M*_{2} by a *top* current source, *I _{DD}*. We analyze this circuit by first computing the output common-mode level. In the absence of oscillation, the circuit reduces to that shown in Fig. 8.30(b), where

We select the transistor dimensions such that the CM level is approximately equal to *V _{DD}*/2. Consequently, as

The startup condition, oscillation frequency, and output swing of the oscillator shown in Fig. 8.30(a) are similar to those derived for the tail-biased circuit of Fig. 8.18(b). Also, *L*_{1} and *L*_{2} are realized as a single symmetric inductor so as to achieve a higher *Q*; the center tap of the inductor is tied to *I _{DD}*.

While providing a wider range than its tail-biased counterpart, the topology of Fig. 8.30(a) suffers from a higher phase noise. As studied in Section 8.7, this penalty arises primarily from the modulation of the output CM level (and hence the varactors) by the noise current of *I _{DD}*, as evidenced by Eq. (8.66). This effect does not occur in the tail-biased oscillator because the output CM level is “pinned” at

In order to avoid varactor modulation due to the noise of the bias current source, we return to the tail-biased topology but employ *ac coupling* between the varactors and the core so as to allow positive and negative voltages across the varactors. Illustrated in Fig. 8.32(a), the idea is to define the dc voltage at the gate of the varactors by *V _{b}* (≈

The principal drawback of the above circuit stems from the parasitics of the coupling capacitors. In Fig. 8.32(a), *C*_{S1} and *C*_{S2} must be *much greater* than the maximum capacitance of the varactors, *C _{max}*, so that the capacitance range presented by the varactors to the tanks does not shrink substantially. If

For example, if *C _{S}* = 10

The choice of *C _{S}* = 10

Let us now study the effect of the parasitics of *C*_{S1} and *C*_{S2} in Fig. 8.32(a). From Eq. (8.53), we note that a larger *C*_{1} further limits the tuning range. In other words, the numerator of (8.53) *decreases* due to the series effect of *C _{S}*, and the denominator of (8.53)

In the above study, we have assumed that *C _{b}* appears at nodes

A capacitor structure that exhibits lower parasitics than the metal sandwich geometry of Fig. 8.33(a) is shown in Fig. 8.34(a). Called a “fringe” or “lateral-field” capacitor, this topology incorporates closely-spaced narrow metal lines to maximize the fringe capacitance between them. The capacitance per unit volume is larger than that of the metal sandwich, leading to a smaller parasitic.

Three other issues in Fig. 8.32(a) merit consideration. First, since *R*_{1} and *R*_{2} appear approximately in parallel with the tanks, their value must be chosen much greater than *R _{p}*. (Even a tenfold ratio proves inadequate as it lowers the

Another VCO topology that naturally provides an output CM level approximately equal to *V _{DD}*/2 is shown in Fig. 8.35. The circuit can be viewed as two back-to-back CMOS inverters, except that the sources of the NMOS devices are tied to a tail current, or as a cross-coupled NMOS pair and a cross-coupled PMOS pair sharing the same bias current. Proper choice of device dimensions and

In this circuit, the bias current is “reused” by the PMOS devices, providing a higher transconductance. But a more important advantage of the above topology over those in Figs. 8.25(a), 8.30(a), and 8.32(a) is that it produces *twice* the voltage swing for a given bias current and inductor design. To understand this point, we assume *L _{XY}* in the complementary topology is equal to

The circuit of Fig. 8.35 nonetheless suffers from two drawbacks. First, for |*V*_{GS3}| + *V*_{GS1} + *V _{ISS}* to be equal to

In addition to the narrow varactor capacitance range, another factor that limits the *useful* tuning range is the variation of the oscillation amplitude. As the capacitance attached to the tank increases, the amplitude tends to decrease. To formulate this effect, suppose the tank inductor exhibits only a *series* resistance, *R _{S}*, (due to metal resistance and skin effect). Recall from Chapter 2 that, for a narrow frequency range and a

and hence

Thus, *R _{p}* falls in proportion to

Our study of varactor tuning in Example 8.18 points to a relatively narrow range. The use of large varactors leads to a high *K _{VCO}*, making the circuit sensitive to noise on the control voltage. In applications where a substantially wider tuning range is necessary, “discrete tuning” may be added to the VCO so as to achieve a capacitance range well beyond

The overall tuning range can be calculated as follows. With ideal switches and unit capacitors, the lowest frequency is obtained if all of the capacitors are switched in and the varactor is at its maximum value, *C _{max}*:

The highest frequency occurs if the unit capacitors are switched out and the varactor is at its minimum value, *C _{min}*:

Of course, as expressed by Eq. (8.77), the oscillation amplitude may vary considerably across this range, requiring “overdesign” at *ω _{max}* (or calibration) so as to obtain reasonable swings at

The discrete tuning technique shown in Fig. 8.38(a) entails several difficult issues. First, the on-resistance, *R _{on}*, of the switches that control the unit capacitors degrades the

Can we simply increase the width of the switch transistors in Fig. 8.38(a) so as to minimize the effect of *R _{on}*? Unfortunately, wider switches introduce a larger capacitance from the bottom plate of the unit capacitors to ground, thereby presenting a substantial capacitance to the tanks when the switches are

This trade-off between the *Q* and the tuning range limits the use of discrete tuning.

The problem of switch on-resistance can be alleviated by exploiting the differential operation of the oscillator. Illustrated in Fig. 8.42(a), the idea is to place the main switch, *S*_{1}, between nodes *A* and *B* so that, with differential swings at these nodes, only *half* of *R*_{on1} appears in series with each unit capacitor [Fig. 8.42(b)]. This allows a twofold reduction in the switch width for a given resistance. Switches *S*_{2} and *S*_{3} are minimum-size devices, merely defining the CM level of *A* and *B*.

The second issue in discrete tuning relates to potential “blind” zones. Suppose, as shown in Fig. 8.43(a), unit capacitor number *j* is switched out, creating a frequency change equal to *ω*_{4} − *ω*_{2} ≈ *ω*_{3} − *ω*_{1}, but the fine tuning range provided by the varactor, *ω*_{4} − *ω*_{3}, is *less* than *ω*_{4} − *ω*_{2}. Then, the oscillator fails to cover the range between *ω*_{2} and *ω*_{3} for any combination of fine and coarse controls.

To avoid blind zones, each two consecutive tuning characteristics must have some overlap. Depicted in Fig. 8.43(b), this precaution translates to smaller unit capacitors but a larger number of them and hence a complex layout. As explained in Chapter 13, the unit capacitors can be chosen unequal to mitigate this issue. Note that the overlap is also necessary to avoid excessive *variation* of *K _{VCO}* near the ends of each tuning curve. For example, around

Some recent designs have employed oscillators with only discrete tuning. Called “digitally-controlled oscillators” (DCOs), such circuits must employ very fine frequency stops. Examples are described in [2].

In Chapter 13, we design and simulate a VCO with continuous and discrete tuning for 11a/g applications.

The design of VCOs must deal with trade-offs among tuning range, phase noise, and power dissipation. Our study has thus far focused on the task of tuning. We now turn our attention to phase noise.

An ideal oscillator produces a perfectly-periodic output of the form *x*(*t*) = *A* cos *ω _{c}t*. The zero crossings occur at exact integer multiples of

The waveforms in Fig. 8.44 can also be viewed from another, slightly different, perspective. We can say that the *period* remains constant if *x*(*t*) = *A* cos *ω _{c}t* but varies randomly if

Our focus on noise in the zero crossings rather than noise on the amplitude arises from the assumption that the latter is removed by hard switching in stages following the oscillator. For example, the switching transistors in an active mixer spend little time near equilibrium, “masking” most of the LO amplitude noise for the rest of the time.

The spectrum of Fig. 8.45(b) can be related to the time-domain expression. Since *φ _{n}*(

That is, the spectrum of *x*(*t*) consists of an impulse at *ω _{c}* and the spectrum of

In phase noise calculations, many factors of 2 or 4 appear at different stages and must be carefully taken into account. For example, as illustrated in Fig. 8.47, (1) since *φ _{n}*(

How is the phase noise quantified? Since the phase noise falls at frequencies farther from *ω _{c}*, it must be specified at a certain “frequency offset,” i.e., a certain difference with respect to

In practice, the phase noise reaches a constant floor at large frequency offsets (beyond a few megahertz) (Fig. 8.49). We call the regions near and far from the carrier the “close-in” and the “far-out” phase noise, respectively, although the border between the two is vague.

To understand the effect of phase noise in RF systems, let us consider the receiver front end shown in Fig. 8.50(a) and study the downconverted spectrum. Referring to the ideal case depicted in Fig. 8.50(b), we observe that the desired channel is convolved with the impulse at *ω _{LO}*, yielding an IF signal at

Reciprocal mixing becomes critical in receivers that may sense large interferers. The LO phase noise must then be so small that, when integrated across the desired channel, it produces negligible corruption.

Phase noise also manifests itself in transmitters. Shown in Fig. 8.52 is a scenario where two users are located in close proximity, with user #1 transmitting a high-power signal at *f*_{1} and user #2 receiving this signal and a weak signal at *f*_{2}. If *f*_{1} and *f*_{2} are only a few channels apart, the phase noise skirt masking the signal received by user #2 greatly corrupts it even *before* downconversion.

The LO phase noise also corrupts phase-modulated signals in the process of upconversion or downconversion. Since the phase noise is indistinguishable from phase (or frequency) modulation, the mixing of the signal with a noisy LO in the TX or RX path corrupts the information carried by the signal. For example, a QPSK signal containing phase noise can be expressed as

revealing that the amplitude is unaffected by phase noise. Thus, the constellation points experience only random rotation around the origin (Fig. 8.53). If large enough, phase noise and other nonidealities move a constellation point to another quadrant, creating an error.

Oscillator phase noise has been under study for decades [3]–[17], leading to a multitude of analysis techniques in the frequency and time domains. The calculation of phase noise by hand still remains tedious, but simulation tools such as Cadence’s SpectreRF have greatly simplified the task. Nonetheless, a solid understanding of the mechanisms that give rise to phase noise proves essential to oscillator design. In this section, we analyze these mechanisms. In particular, we must answer two important questions: (1) how much and at what point in an oscillation cycle does each device “inject” noise? (2) how does the injected noise produce phase noise in the output voltage waveform?

In Chapters 2 and 7, we derived various expressions for the *Q* of an LC tank. We know intuitively that a high *Q* signifies a sharper resonance, i.e., a higher selectivity. Another definition of the *Q* that is especially well-suited to oscillators is illustrated in Fig. 8.55. Here, the circuit is viewed as a feedback system and the *phase* of the *open-loop* transfer function, *φ*(*ω*), is examined at the resonance frequency, *ω*_{0}. The “open-loop” *Q* is defined as

This definition offers an interesting insight if we recall that for steady oscillation, the total phase shift around the loop must be 360° (or zero). Suppose the noise injected by the devices attempts to deviate the frequency from *ω*_{0}. From Fig. 8.55, such a deviation translates to a change in the total phase shift around the loop, violating Barkhausen’s criterion and forcing the oscillator to return to *ω*_{0}. The larger the slope of *φ*(*jω*), the greater is this “restoration” force; i.e., oscillators with a high open-loop *Q* tend to spend less time at frequencies other than *ω*_{0}. In Problem 8.10, we prove that this definition of *Q* coincides with our original definition, *Q* = *R _{p}*/(

While the open-loop *Q* indicates how much an oscillator “rejects” the noise, the phase noise depends on three other factors as well: (1) the *amount* of noise that different devices inject, (2) the point in time during a cycle at which the devices inject noise (some parts of the waveform are more sensitive than others), and (3) the output voltage swing (carrier power). We elaborate on these as we analyze phase noise.

As our first step toward formulating the phase noise, we wish to understand what happens if noise is injected into an oscillatory circuit. Employing the feedback model, we represent the noise as an additive term [Fig. 8.57(a)] and write

In the vicinity of the oscillation frequency, i.e., at *ω* = *ω*_{0} + Δ*ω*, we can approximate *H*(*jω*) with the first two terms in its Taylor series:

If *H*(*jω*_{0}) = −1 and Δ*ωdH*/*dω* 1, then Eq. (8.100) reduces to

In other words, as shown in Fig. 8.57(b), the noise spectrum is “shaped” by

To determine the shape of |*dH*/*dω*|^{2}, we write *H*(*jω*) in polar form, *H*(*jω*) = |*H*| exp(*jφ*) and differentiate with respect to *ω*,

It follows that

This equation leads to a general definition of *Q* [4], but we limit our study here to simple LC oscillators. Note that (a) in an LC oscillator, the term |*d*|*H*|/*dω*|^{2} is much less than |*dφ*/*dω*|^{2} in the vicinity of the resonance frequency, and (b) |*H*| is close to unity for steady oscillations. The right-hand side of Eq. (8.105) therefore reduces to |*dφ*/*dω*|^{2}, yielding

From (8.95),

Known as “Leeson’s Equation” [3], this result reaffirms our intuition that the open-loop *Q* signifies how much the oscillator rejects the noise.

In Problem 8.11, we prove that, if the feedback path has a transfer function *G*(*s*) (Fig. 8.59), then

where the open-loop *Q* is given by

The foregoing development suggests that the total noise at the output of an oscillator can be obtained according to a number of transfer functions similar to Eq. (8.107) from each noise source to the output. Such an approach begins with a small-signal (linear) model and can account for some of the nonidealities [4]. However, the small-signal model may ignore some important effects, e.g., the noise of the tail current source, or face other difficulties. The following example illustrates this point.

The result expressed by (8.107) and exemplified by (8.111) yields the total noise that is *added* to the oscillation waveform at the output. We must now determine how and to what extent additive noise corrupts the *phase*.

Let us begin with the simple case depicted in Fig. 8.62(a). The carrier appears at *ω*_{0} and the additive noise in a 1-Hz bandwidth centered at *ω*_{0} + Δ*ω* is modeled by an impulse. In the time domain, the overall waveform is *x*(*t*) = *A* cos *ω*_{0}*t* + *a* cos(*ω*_{0} + Δ*ω*)*t* where *a* *A*. Intuitively, we expect the additive component to produce both amplitude and phase modulation. To understand this point, we represent the carrier by a phasor of magnitude *A* that rotates at a rate of *ω*_{0} [Fig. 8.62(b)]. The component at *ω*_{0} + Δ*ω* adds *vectorially* to the carrier, i.e., it appears as a small phasor at the tip of the carrier phasor and rotates at a rate of *ω*_{0} + Δ*ω*. At any point in time, the small phasor can be expressed as the sum of two other phasors, one aligned with *A* and the other perpendicular to it. The former modulates the amplitude and the latter, the phase. Figure 8.62(c) illustrates the behavior in the time domain.

In order to compute the phase modulation resulting from a small sinusoid at *ω*_{0} + Δ*ω*, we make two important observations. First, as described in Chapter 3, the spectrum of Fig. 8.62(a) can be written as the sum of an AM signal and an FM signal. Second, the phase of the overall signal is obtained by applying the composite signal to a hard limiter, i.e., a circuit that clips the amplitude and hence removes AM. From Chapter 3, the output of the limiter can be written as

We recognize the phase component, (2*a*/*A*) sin Δ*ωt*, as simply the original additive component at *ω*_{0} + Δ*ω*, but translated down by *ω*_{0}, shifted by 90°, and normalized to *A*/2. We therefore expect that narrowband random additive noise in the vicinity of *ω*_{0} results in a phase whose spectrum has the *same* shape as that of the additive noise but translated by *ω*_{0} and normalized to *A*/2.

This conjecture can be proved analytically. We write *x*(*t*) = *A* cos *ω*_{0}*t* + *n*(*t*), where *n*(*t*) denotes the narrowband additive noise (voltage or current). It can be proved that narrowband noise in the vicinity of *ω*_{0} can be expressed in terms of its *quadrature* components [9]:

where *n _{I}*(

Expressing Eq. (8.115) in polar form, we have

Since *n _{I}*(

as postulated previously. It follows that

Note that *A* is the peak (not the rms) amplitude of the carrier. In Problem 8.12, we prove that half of the noise power is carried by the AM sidebands and the other half by the PM sidebands.

We are ultimately interested in the spectrum of the RF waveform, *x*(*t*), but excluding its AM noise. We have

Thus, the power spectral density of *x*(*t*) consists of two impulses at ±*ω*_{0}, each with a power of *A*^{2}/4, and *S _{nQ}*/4 centered around ±

The foregoing development can be summarized as follows (Fig. 8.64). Additive noise around ±*ω*_{0} having a two-sided spectral density with a peak of *η* results in a phase noise spectrum around *ω*_{0} having a normalized one-sided spectral density with a peak of 2*η*/*A*^{2}, where *A* is the *peak* amplitude of the carrier.

The derivations leading to Eq. (8.111) have assumed that the noise of each transistor can be represented by a *constant* spectral density; however, as the transistors experience large-signal excursions, their transconductance and hence noise power varies. Since oscillators perform this noise modulation periodically, we say such noise sources are “cyclostationary,” i.e., their spectrum varies periodically. We begin our analysis with an observation made in Chapter 6 regarding cyclostationary white noise: white noise multiplied by a periodic envelope in the time domain remains white. For example, if white noise is switched on and off with 50% duty cycle, the result is still white but has half the spectral density.

In order to study the effect of cyclostationary noise, we return to the original cross-coupled oscillator and, from Fig. 8.65(a), recognize that (1) when *V _{X}* reaches a maximum and

The noise envelope waveform can be determined by simulations, but let us approximate the envelope by a sinusoid, 0.5 cos 2*ω*_{0}*t* + 0.5. The reader can show that white noise multiplied by such an envelope results in white noise with three-eighths the spectral density. Thus, we simply multiply the noise contribution of *M*_{1} and *M*_{2}, *kTγ g _{m}*, by 3/8.

How about the noise of the tanks? We observe that the noise of *R _{p}* in Fig. 8.58 is stationary. In other words, the two-sided tank noise contribution is equal to 2

In addition to cyclostationary noise, the time variation of the resistance presented by the cross-coupled pair also complicates the analysis. However, since we have taken a “macroscopic” view of cyclostationary noise and modeled it by an equivalent white noise, we may consider a *time average* of the resistance as well.

We have noted that the resistance seen between the drains of *M*_{1} and *M*_{2} in Fig. 8.65(a) periodically varies from −2/*g _{m}* to nearly infinity. The corresponding conductance,

What can we say about −*G _{avg}*? If −

We now consolidate our formulations of (a) conversion of additive noise to phase noise, (b) cyclostationary noise, and (c) time-varying resistance. Our analysis proceeds as follows:

1. We compute the average spectral density of the noise current injected by the cross-coupled pair. If a sinusoidal envelope is assumed, the two-sided spectral density amounts to *kTγ g _{m}* × (3/8), where

2. To this we add the noise current of *R _{p}*, obtaining (3/8)

3. We multiply the above spectral density by the squared magnitude of the net impedance seen between the output nodes. Since *G _{avg}* = 1/

which, for *ω* = *ω*_{0} + Δ*ω* and Δ*ω* *ω*_{0}, reduces to

The factor of 3/8 depends on the noise envelope waveform and must be obtained by careful simulations.

4. From Fig. 8.64, we divide this result by *A*^{2}/2 to obtain the one-sided phase noise spectrum around *ω*_{0}. Note that in Fig. 8.65(a), *A* = (4/*π*)(*I _{SS}R_{p}*/2) = (2/

As the tail current and hence the output swings increase, rises much more sharply than *g _{m}*, thereby lowering the phase noise (so long as the transistors do not enter the deep triode region).

A closer examination of the cross-coupled oscillator reveals that the phase noise is in fact independent of the transconductance of the transistors [10, 11, 17]. This can be qualitatively justified as follows. Suppose the widths of the two transistors are increased while the output voltage swing and frequency are kept constant. The transistors can now steer their tail current with a smaller voltage swing, thus experiencing sharper current switching (Fig. 8.68). That is, *M*_{1} and *M*_{2} spend less time injecting noise into the tank. However, the higher transconductance translates to a higher amount of injected noise, as evident from the noise envelope. It turns out that the decrease in the width and the increase in the height of the noise envelope pulses cancel each other, and *g _{m}* can be simply replaced with 2/

What happens if one of the transistors enters the deep triode region? As depicted in Fig. 8.69(a), the corresponding tank is temporarily connected to the tail capacitance through the on-resistance of the transistor, degrading the *Q*. Transforming the series combination of *R _{on}* and

The key result here is that, as the tail current is increased, the (relative) phase noise continues to decline up to the point where the transistors enter the triode region. Beyond this point, a higher tail current raises the output swing more gradually, but the overall tank *Q* begins to fall, yielding no significant improvement in the phase noise. Of course, this trend depends on the value of *C _{T}* and may be pronounced only in some designs. This capacitance may be large due to the parasitics of

The approach described in this section follows that in [6]. Consider an ideal LC tank that, due to an initial condition, produces a sinusoidal output [Fig. 8.70(a)]. During the oscillation, *L*_{1} and *C*_{1} exchange the initial energy, with the former carrying the entire energy at the zero crossings and the latter, at the peaks. Let us assume that the circuit begins with an initial voltage of *V*_{0} across the capacitor. Now, suppose an impulse of current is injected into the oscillating tank at the peak of the output voltage [Fig. 8.70(b)], producing a voltage step across *C*_{1}. If^{8}

then the additional energy gives rise to a larger oscillation amplitude:

The key point here is that the injection at the peak does not disturb the *phase* of the oscillation (as shown in the example below).

Next, let us assume the impulse of current is injected at a zero crossing point. A voltage step is again created but leading to a *phase* jump [Fig. 8.70(c)]. Since the voltage jumps from 0 to *I*_{1}/*C*_{1}, the phase is disturbed by an amount equal to sin^{−1}[*I*_{1}/(*C*_{1}*V*_{0})]. We therefore conclude that noise creates only amplitude modulation if injected at the peaks and only phase modulation if injected at the zero crossings.

The foregoing observations suggest the need for a method of quantifying how and when each source of noise in an oscillator “hits” the output waveform. While the transistors turn on and off, a noise source may only appear near the peaks of the output voltage, contributing negligible phase noise, whereas another may hit the zero crossings, producing substantial phase noise. To this end, we define a linear, *time-variant* system from each noise source to the output phase. The linearity property is justified because the noise levels are very small, and the time variance is necessary to capture the effect of the time at which the noise appears at the output.

For a linear, time-variant system, the convolution property holds, but the impulse response varies with time. Thus, the output phase in response to a noise *n*(*t*) is given by

where *h*(*t*, *τ*) is the time-variant impulse response from *n*(*t*) to *φ*(*t*). In an oscillator, *h*(*t*, *τ*) varies *periodically*: as illustrated in Fig. 8.72, a noise impulse injected at *t* = *t*_{1} or integer multiples of the period thereafter produces the same phase change. Now, the task of output phase noise calculation consists of computing *h*(*t*, *τ*) for each noise source and convolving it with the noise waveform. The impulse response, *h*(*t*, *τ*), is called the “impulse sensitivity function” (ISF) in [6].

Let us now return to Eq. (8.127) and determine how the convolution is carried out. It is instructive to begin with a linear, time-invariant system. A given input, *x*(*t*), can be approximated by a series of time-domain impulses, each carrying the energy of *x*(*t*) in a short time span [Fig. 8.74(a)]:

Each impulse produces the time-invariant impulse response of the system at *t _{n}*. Thus,

Now, consider the time-variant system shown in Fig. 8.74(b). In this case, the time-shifted versions of *h*(*t*) may be different, and we denote them by *h*_{1}(*t*), *h*_{2}(*t*), ..., *h _{n}*(

How do we express these impulse responses as a continuous-time function? We simply write them as *h*(*t*, *τ*), where *τ* is the specific time shift. For example, *h*_{1}(*t*) = *h*(*t*, 1 ns), *h*_{2}(*t*) = *h*(*t*, 2 ns), etc. Thus,

The reader may find the foregoing example confusing: if the lossless tank with its nonzero initial condition is viewed as an oscillator with infinite Q, why is the phase noise not *zero*? This confusion is resolved if we recognize that, as *Q* → ∞, the width and bias current of the transistor needed to sustain oscillation become infinitesimally small. The transistor thus injects nearly zero noise; i.e., if *i _{n}*(

Due to its periodic nature, the impulse response of oscillators can be expressed as a Fourier series:

where *a*_{0} is the average (or “dc”) value of *h*(*t*, *τ*). In the LC tank studied above, *a _{j}* = 0 for all

From Example 8.35, the integration is equivalent to a transfer function of (*jω*)^{−1} and hence

That is, *low-frequency* components in *i _{n}*(

Note that in this case, *a*_{0} represents the dc term of the impulse response from the *gate voltage* of the transistors to the output phase. Since *a*_{0} relates to the symmetry of *h*(*t*, *τ*), low upconversion of 1/*f* noise requires a circuit design that exhibits an odd-symmetric *h*(*t*, *τ*) [6]. However, the 1/*f* noise of different transistors in the circuit may see different impulse responses, and it may therefore be impossible to minimize the upconversion of *all* 1/*f* noise sources. For example, in the circuit of Fig. 8.35, it is possible to make *h*(*t*, *τ*) from the gates of *M*_{1}-*M*_{4} to the output symmetric, but not from the tail current source to the output. As *I _{SS}* slowly fluctuates, so does the output CM level and hence the oscillation frequency. In general, the phase noise spectrum assumes the shape shown in Fig. 8.77.

Let us now turn our attention to the remaining terms in Eq. (8.144). As mentioned in Example 8.35, *a*_{1} cos(*ω*_{0}*t* + *φ*_{1}) translates noise frequencies around *ω*_{0} to the vicinity of zero and into phase noise. By the same token, *a _{m}* cos(

We must also incorporate the effect of cyclostationary noise. As explained in Section 8.7.3, such noise can be viewed as stationary noise, *n*(*t*), multiplied by a periodic envelope, *e*(*t*). Equation (8.138) can thus be written as

implying that *e*(*t*)*h*(*t*, *τ*) can be viewed as an “effective” impulse response [6]. In other words, the effect of *n*(*t*) on phase noise ultimately depends on the *product* of the cyclostationary noise envelope and *h*(*t*, *τ*).

This approach to phase noise analysis generally requires that both the noise envelope and the impulse response be determined from multiple simulations for each device. Design optimization may therefore prove a lengthy task.

Each of the two analysis approaches described thus far imparts its own insights and finds its own utility in circuit design. However, there are other phase noise mechanisms that can be better understood by other analysis techniques. The next section is an example.

Oscillators typically employ a bias current source so as to minimize sensitivity to the supply voltage and noise therein. We wish to study the phase noise contributed by this current source. Figure 8.79 summarizes the tail-related noise mechanisms studied here.

Consider the topology shown in Fig. 8.80(a), where *I _{n}* models the noise of

We now examine the effect of different noise frequencies upon the performance of the oscillator in Fig. 8.80(a). The flicker noise in *I _{n}* slowly varies the bias current and hence the output voltage swing (4

How about the noise around *ω*_{0}? This noise component is mixed with the harmonics of the square wave, *ω*_{0}, 3*ω*_{0}, 5*ω*_{0},..., landing at 0, 2*ω*_{0}, 4*ω*_{0},.... Thus, this component is negligible.

The noise around 2*ω*_{0}, on the other hand, markedly impacts the performance [12, 7]. As illustrated in Fig. 8.81(a), a noise component slightly below 2*ω*_{0} is mixed with the first and third harmonics of the square wave, thereby falling at slightly below and above *ω*_{0} but with *different* amplitudes and polarities. To determine whether these components produce AM or FM, we express the oscillator output as cos *ω*_{0}*t* and its third harmonic as (−1/3) cos(3*ω*_{0}*t*). For a tail current noise component, *I*_{0} cos(2*ω*_{0} − Δ*ω*)*t*, the differential output current of *M*_{1} and *M*_{2} emerges as

As explained in Chapter 3, two equal cosine sidebands having opposite signs surrounding a cosine carrier represent FM. In the above equation, however, the two sidebands have unequal magnitudes, creating some AM as well. Writing (*I*_{0}/2) cos(*ω*_{0} − Δ*ω*)*t* = (*I*_{0}/6) cos(*ω*_{0} − Δ*ω*)*t* + (*I*_{0}/3) cos(*ω*_{0} − Δ*ω*)*t* and extracting from the second term its PM components [Fig. 8.81(b)] as (*I*_{0}/6) cos(*ω*_{0} − Δ*ω*)*t* − (*I*_{0}/6) cos(*ω*_{0} + Δ*ω*)*t*, we obtain the overall PM sidebands:

The proportionality factor is related to the conversion gain of the mixing action, as illustrated by the following example.

To obtain the phase noise in the output *voltage*, (1) the current sidebands computed in the above example must be multiplied by the impedance of the tank at a frequency offset of ±Δ*ω*, and (2) the result must be normalized to the oscillation amplitude. Note that the current components see the *lossless* impedance of the tank once they are injected into the output nodes because the average negative conductance presented by *M*_{1} and *M*_{2} cancels the loss. This impedance is given by −*j*/(2*C*_{1}Δ*ω*) if Δ*ω* *ω*_{0}. Thus, the relative phase noise can be expressed as

The thermal noise near higher even harmonics of *ω*_{0} plays a similar role, producing FM sidebands around *ω*_{0}. It can be shown that the summation of all of the sideband powers results in the following phase noise expression due to the tail current source [8, 10]:

Let us now consider the noise of the top current source in Fig. 8.30(a). We wish to formulate the frequency modulation resulting from this noise. Suppose *I _{DD}* contains a noise current

This change is indistinguishable from an equal but opposite change in the control voltage, *V _{cont}*. As explained in Section 8.10, the output waveform can be expressed as

where the second term in the square brackets is the resulting phase noise, *φ _{n}*(

We recognize that *low-frequency* components in *i _{n}*(

Let us summarize our findings thus far. In the tail-biased oscillator (and in the top-biased oscillator), the noise near zero frequency introduces amplitude modulation, whereas that near even harmonics of *ω*_{0} leads to phase noise. [In the top-biased oscillator, low-frequency noise in the current source also modulates the output CM level, producing phase noise (Example 8.16).]

The amplitude modulation resulting from the bias current noise does translate to phase noise in the presence of nonlinear capacitances in the tanks [13, 14]. To understand this point, we return to our AM/PM modulation study in Chapter 2 and make the following observations. Since the varactor capacitance varies periodically with time, it can be expressed as a Fourier series:

where *C _{avg}* denotes the “dc” value. If noise in the circuit modulates

Consider the tank shown in Fig. 8.83(a), and first assume that the voltage dependence of *C*_{1} is odd-symmetric around the vertial axis, e.g., *C*_{1} = *C*_{0}(1 + *αV*). In this case, *C _{avg}* is independent of the signal amplitude because the capacitance spends equal amounts of time above and below

The above results change if *C*_{1} exhibits *even-order* voltage dependence, e.g., *C*_{1} = *C*_{0}(1 + *α*_{1}*V* + *α*_{2}*V*^{2}). Now, the capacitance changes more sharply for negative or positive voltages, yielding an average that depends on the current amplitude [Fig. 8.83(c)]. We therefore observe that, in an oscillator employing such a tank, slow modulation of the amplitude varies the average tank resonance frequency and hence the frequency of oscillation. The phase noise resulting from the low-frequency bias current noise is computed in [13]. We study a simplified case in Problem 8.13.

It is important to note that the tail current in Fig. 8.35 introduces phase noise via three distinct mechanisms: (1) its flicker noise modulates the output CM level and hence the varactors; (2) its flicker noise produces AM at the output and hence phase noise through AM/PM conversion; (3) its thermal noise at 2*ω*_{0} gives rise to phase noise.

Our studies in this chapter point to direct trade-offs among the phase noise, power dissipation, and tuning range of VCOs. For example, as explained in Chapter 7, varactors themselves suffer from a trade-off between their capacitance range and their *Q*. We also recall from Leeson’s equation, Eq. (8.107), that phase noise rises with the oscillation frequency if the *Q* does not increase proportionally.

A figure of merit (FOM) that encapsulates some of these trade-offs is defined as

where the phase noise is multiplied by the square of the offset frequency at which it is measured so as to perform normalization. Attention must be paid to the unit of phase noise (noise power normalized to carrier power) in this expression. Note that the product of power dissipation and phase noise has a unit of W/Hz. Another FOM that additionally represents the trade-offs with the tuning range is

State-of-the-art CMOS VCOs in the range of several gigahertz achieve an FOM_{2} around 190 dB. In general, the phase noise in the above expressions refers to the worst-case value, typically at the the highest oscillation frequency. Also, note that these FOMs do not account for the load driven by the VCO.

Following our study of tuning techniques and phase noise issues, we now describe a procedure for the design of LC VCOs. We focus on the topology shown in Fig. 8.25(a) and assume the following parameters are given: the center frequency, *ω*_{0}, the output voltage swing, the power dissipation, and the load capacitance, *C _{L}*. Even though some of these parameters may not be known at the outset, it is helpful to select some reasonable values and iterate if necessary. Of course, the output swing must be chosen so as not to stress the transistors.

The procedure consists of six steps:

1. Based on the power budget and hence the maximum allowable *I _{SS}*, select the tank parallel resistance,

2. Select the *smallest* inductor value that yields a parallel resistance of *R _{p}* at

3. Determine the dimensions of *M*_{1} and *M*_{2} such that they experience nearly complete switching with the given voltage swings. To minimize their capacitance contributions, choose minimum channel length for the transistors.

4. Noting that the transistor, inductor, and load capacitances amount to a total of *C _{GS}* + 4

5. Using proper varactor models, determine the minimum capacitance of such a varactor, *C _{var,min}*, and compute the upper end of the tuning range,

6. If *ω _{max}* is quite higher than necessary, increase

The above procedure yields a design that achieves the *maximum* tuning range subject to known values of *ω*_{0}, the output swing, the power dissipation, and *C _{L}*. If the varactor

We carry out the detailed design of a 12-GHz VCO for 11a/g applications in Chapter 13.

A great deal of effort has been expended on relaxing the trade-offs among phase noise, power dissipation, and tuning range of VCOs. In this section, we study several examples.

In order to reduce the phase noise due to flicker noise, the generic cross-coupled oscillator can incorporate PMOS transistors rather than NMOS devices. Figure 8.85 depicts the PMOS counterparts of the circuits shown in Figs. 8.25(a) and 8.30(a). Since PMOS devices exhibit substantially less flicker noise, the close-in phase noise of these oscillators is typically 5 to 10 dB lower. The principal drawback of these topologies is their limited speed, an issue that arises only as frequencies exceeding tens of gigahertz are sought.

As explained in Section 8.7.5, the noise current at 2*ω*_{0} in the tail current source translates to phase noise around *ω*_{0}. This and higher noise harmonics can be removed by a capacitor as shown in Fig. 8.86(a). However, if *M*_{1} and *M*_{2} enter the deep triode region during oscillation, then two effects raise the phase noise: (1) the on-resistance of each transistor now degrades the *Q* of the tank [Fig. 8.69(a)] [16], and (2) the impulse response (ISF) from the noise of each transistor to the output phase becomes substantially larger [17]. This issue can be resolved by means of two techniques. If operation in the triode region must be avoided but large output swings are desired, capacitive coupling can be inserted in the loop [Fig. 8.86(b)]. Here, *V _{b}* is chosen so that the peak voltage at the gate of each transistor does not exceed its minimum drain voltage by more than one threshold. A Class-C oscillator similar to this topology is described in [17].

The bias voltage, *V _{b}*, in Fig. 8.86(b) must remain high enough to provide sufficient

The second approach is to allow *M*_{1} and *M*_{2} in Fig. 8.86(a) to enter the triode region but remove the effect of the tail capacitance at 2*ω*_{0}. Illustrated in Fig. 8.88 [16], the idea is to insert inductor *L _{T}* in series with the tail node and choose its value such that it resonates with the parasitic capacitance,

Each oscillator in an RF system typically drives a mixer and a frequency divider, experiencing their input capacitances. Moreover, the LO output common-mode level must be compatible with the input CM level of these circuits. We study this compatibility issue here.

The output CM level of the oscillators studied in this chapter is around *V _{DD}*,

We consider two approaches to providing CM compatibility. Shown in Fig. 8.91(a), the first employs capacitive coupling and selects *V _{b}* such that

The second approach to CM compatibility interposes a buffer between the LO and the mixer. In fact, if the mixer input capacitance excessively loads the LO, or if long, lossy interconnects appear in the layout between the LO and the mixer, then such a buffer proves indispensable. Depicted in Fig. 8.91(b) is an example, where an inductively-loaded differential pair serves as a buffer, providing an output CM level equal to *V _{DD}* −

Similar considerations apply to the interface between an oscillator and a frequency divider. For example, in Fig. 8.92(a), the divider input CM level must be well below *V _{DD}* to ensure the current-steering transistors

Our definition of voltage-controlled oscillators in Section 8.5 relates the output frequency to the control voltage by a linear, static equation, *ω _{out}* =

Let us now consider an unmodulated sinusoid, *V*_{1}(*t*) = *V*_{0} sin *ω*_{1}*t*. Called the “total phase,” the argument of the sine, *ω*_{1}*t*, varies linearly with time in this case, exhibiting a slope of *ω*_{1} [Fig. 8.93(a)]. We say the phase “accumulates” at a rate of *ω*_{1}. In other words, if *ω*_{1} is increased to *ω*_{2}, then the phase accumulates faster, crossing multiples of *π* at a higher rate. It is therefore plausible to define the instantaneous frequency as the time derivative of the phase:

Conversely,

The initial phase, *φ*_{0}, is usually unimportant and assumed zero hereafter. Since a VCO exhibits an output frequency given by *ω*_{0} + *K _{VCO}V_{cont}*, we can express its output waveform as

Comparing this result with that in Chapter 3, we recognize that a VCO is simply a frequency modulator. For example, the narrowband FM approximation holds here as well. Note the difference between Eqs. (8.167) and (8.172).

In the analysis of phase-locked frequency synthesizers (Chapter 10), we are concerned with only the second term in the argument of Eq. (8.172). Called the “excess phase,” this term represents an *integrator* behavior for the VCO. In other words, if the quantity of interest at the output of the VCO is the excess phase, *φ _{ex}*, then

and hence

The important observation here is that the output *frequency* of a VCO (almost) instantaneously changes in response to a change in *V _{cont}*, whereas the output

In our study of transceiver architectures in Chapter 4, we observed the need for quadrature LO phases in downconversion and upconversion operations. We also noted that flipflop-based divide-by-two circuits generate quadrature phases, but they restrict the maximum LO frequency. In applications where dividers do not offer sufficient speed, we may employ polyphase filters or quadrature oscillators instead. In this section, we study the latter.

Two identical oscillators can be “coupled” such that they operate in-quadrature. We therefore begin our study with the concept of coupling (or injecting) a signal to an oscillator. Figure 8.96 depicts an example, where the input voltage is converted to current and injected into the oscillator. The differential pair is a natural means of coupling because the cross-coupled pair can also be viewed as a circuit that steers and injects current into the tanks. If the two pairs completely steer their respective tail currents, then the “coupling factor” is equal to *I*_{1}/*I _{SS}*.

Let us now consider two identical oscillators that are unilaterally coupled. Shown in Fig. 8.97 are two possibilities, with “in-phase” and “anti-phase” coupling. The coupling factors have the same sign in the former and opposite signs in the latter. We analyze these topologies using both the feedback model and the one-port model of the oscillators. Note that the tuning techniques described earlier in this chapter apply to these topologies as well.

The circuits of Fig. 8.97 can be mapped to two coupled feedback oscillators as shown in Fig. 8.98, where |*α*_{1}| = |*α*_{2}| and the sign of *α*_{1}*α*_{2} determines in-phase or anti-phase coupling [18]. The output of the top adder is equal to *α*_{1}*Y* − *X*, yielding

Similarly, the bottom oscillator produces

Multiplying both sides of (8.178) by *α*_{2}*X* and both sides of (8.179) by *α*_{1}*Y* and subtracting the results, we have

As explained below, 1 + *H*(*s*) ≠ 0 at the oscillation frequency, and hence

If *α*_{1} = *α*_{2} (in-phase coupling), then *X* = ±*Y*, i.e., the two oscillators operate with a zero or 180° phase difference.

The in-phase operation is not particularly useful as the two oscillators can be simply merged into one (as in Fig. 8.84). On the other hand, if *α*_{1} = − *α*_{2} (anti-phase coupling), then Eq. (8.181) yields

i.e., the outputs bear a phase difference of + 90° or −90°. In this case, Eq. (8.182) is revised to

It is possible to gain additional insight through the use of the one-port model for each oscillator. A single oscillator experiencing unilateral coupling can be represented as shown in Fig. 8.100(a), where *G _{m}* denotes the transconductance of the coupling differential pair (

The parallel combination of *Z _{T}* and −

Multiplying both sides of (8.186) by *V _{X}* and both sides of (8.187) by

Since the parallel combination of *Z _{T}* and −

implying that, if *G*_{m1} = − *G*_{m2}, then the two oscillators operate in quadrature. Since each oscillator receives energy from the other, the startup condition need not be as stringent as *Z _{T}*(

Unilaterally-coupled oscillators exhibit interesting attributes. Let us consider the case of in-phase coupling as a starting point. As shown in Fig. 8.101, we construct a phasor diagram of the circuit’s voltages and currents. This is accomplished by noting that (1) *V _{A}* and

Can the circuit operate with *I*_{D3} *opposing I*_{D1}, i.e., with *V _{C}* = −

We now repeat this study for anti-phase coupling. Since the two differential oscillators operate in quadrature, the voltage and current phasors appear as in Fig. 8.102, with the drain current phasor of each transistor still aligned with its gate voltage phasor. In this case, the total current flowing through each tank consists of two *orthogonal* phasors; e.g., *Z _{A}* carries

How can the vector sum *I*_{D1} + *I*_{D3} yield *V _{A}*? Depicted in Fig. 8.103(a), the resultant,

i.e., the necessary rotation is determined by the coupling factor. If *Z _{A}* = (

Since *ω*_{osc1} − *ω*_{0} = Δ*ω* *ω*_{0}, the argument of tan^{−1} on the left-hand side can be simplified by writing *ω*_{osc1} ≈ *ω*_{0} in the numerator and in the denominator:

We also recognize that tan^{−1} *a* ≈ *π*/2 − 1/*a* if *a* 1 and apply this approximation to the left-hand side:

Since 2*R _{p}C*

Equation (8.95) yields the same result.

The above example implies that quadrature oscillators may operate at either one of the two frequencies above and below *ω*_{0}. In fact, Eq. (8.185) also predicts the same results: since *H*(*jω*_{0}) is a *complex* number, oscillation must depart from resonance, and since both 1 + *jα*_{1} and 1 − *jα*_{1} are acceptable, two solutions above and below resonance exist. Observed in practice as well [19], this property proves a serious drawback. In transient circuit *simulations*, the oscillator typically operates at *ω*_{osc2}, exhibiting little tendency to start in the higher mode even with different initial conditions. Nonetheless, it is possible to devise a simulation that reveals the possibility of oscillation at either frequency. This is explained in Appendix A.

It is important to make two observations at this point. (1) The foregoing derivations do not impose a *lower* bound on the coupling factor, i.e., quadrature operation appears to occur with arbitrarily small coupling factors. Unfortunately, in the presence of *mismatches* between the natural frequencies of the two oscillators, a small coupling factor may not guarantee “locking.” As a result, each oscillator tends to operate at its own *ω*_{0} while it is also “pulled” by the other. The overall circuit exhibits spurious components due to this mutual injection pulling behavior (Fig. 8.106) [20]. To avoid this phenomenon, the coupling factor must be at least equal to [20]

We typically choose an *α* in the range of 0.2 to 0.25. (2) As the coupling factor increases, two issues become more serious: (a) *ω*_{osc1} and *ω*_{osc2} diverge further, making it difficult to target the desired frequency range if both can occur; and (b) the phase noise of the circuit rises, with the flicker noise of the coupling transistors contributing significantly at low frequency offsets [21, 27]. This can be seen by noting that Δ*ω* in Eq. (8.194) is a function of the coupling factor, *I*_{D3}/*I*_{D1}, which itself varies slowly with the flicker noise of the coupling transistors and their tail currents [21] (Problem 8.16). It follows that the choice of the coupling factor entails a trade-off between proper, spurious-free operation and phase noise. For a given power dissipation, the phase noise of quadrature oscillators is typically 3 to 5 dB higher [23] than a single oscillator.

The mismatches between the two oscillator cores and the coupling pairs result in phase and amplitude mismatch between the quadrature outputs. These effects are studied in [22, 24].

A number of quadrature oscillator topologies have been proposed that alleviate the tradeoffs mentioned above. The principal drawback of the generic configuration studied thus far is that the coupling pairs introduce significant phase noise. We therefore postulate that, if the quadrature relationship between the two core oscillators is established by a different means, then phase noise can be reduced.

Consider the two oscillators shown in Fig. 8.107(a) and suppose they are somehow forced to operate in quadrature at a frequency of *ω _{osc}*. The tail nodes,

The topology of Fig. 8.107(b) merits several remarks. First, since the coupling pairs used in the generic circuit of Fig. 8.97 are absent, the two core oscillators operate at their tanks’ resonance frequency, *ω _{osc}*, rather than depart so as to produce additional phase shift. This important attribute means that this approach avoids the frequency ambiguity suggested by Eqs. (8.194) and (8.195). Moreover, it improves the phase noise. Second, in a manner similar to that illustrated in Fig. 8.88, the resonance of

In the presence of mismatches between the core oscillators of Fig. 8.107(b), both the voltage swings at *A* and *B* and the mutual coupling between *L*_{1} and *L*_{2} must exceed a certain minimum to guarantee lock. Note that *I _{A}* is commutated by

The above example reveals that techniques that reduce the deviation of the oscillation frequency from the resonance frequency may also lower the flicker noise contribution of the coupling transistors. Specifically, it is possible to introduce additional phase shift in the coupling network by means of passive devices so as to drive Δ*ω* in Eqs. (8.194) and (8.195) toward zero. Shown in Fig. 8.109(a) is an example where the degeneration network yields an overall transconductance of

As depicted in Fig. 8.109(b), the phase reaches several tens of degrees between the zero and pole frequencies. However, the degeneration does reduce the coupling factor, requiring larger transistors and bias currents.

In order to avoid the flicker noise of the coupling devices, one can perform the coupling through the *bulk* of the main transistors. Illustrated in Fig. 8.110 [26], the idea is to apply the differential output of one oscillator to the *n*-well of the cross-coupled transistors in the other.^{11} The large resistors *R*_{1} and *R*_{2} set the bias voltage of the *n*-wells to *V _{P}*. Also,

In order to examine the tendency of the quadrature oscillator to operate at the frequencies above and below *ω*_{0} [Figs. 8.103(b) and 8.104(c)], we simulate the circuit as follows. First, we reconfigure the circuit so that it operates with *in-phase* coupling and hence at *ω*_{0}. This simulation provides the exact value of *ω*_{0} in the presence of all capacitances.

Next, we apply anti-phase coupling and simulate the circuit, obtaining the exact value of *ω*_{osc2} (or *ω*_{osc1} if the oscillator prefers the higher mode). Since *ω*_{0} − *ω*_{osc2} ≈ *ω*_{osc1} − *ω*_{0}, we now have a relatively accurate value for *ω*_{osc1}.

Last, we *inject* a sinusoidal current of frequency *ω*_{osc1} into the oscillator, *I _{inj}* =

We should also mention that a realistic inductor model is essential to proper simulation of quadrature oscillators. Without the parallel or series resistances that model various loss mechanisms (Chapter 7), the circuit may behave strangely in simulations.

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8.1. Determine the input admittance of the circuit shown in Fig. 8.4 and find its real part.

8.2. Suppose *H*(*s*) in Fig. 8.6 satisfies the following conditions at a frequency *ω*_{1}: |*H*(*jω*_{1}| = 1 but ∠*H*(*jω*_{1}) = 170°. Explain what happens.

8.3. Repeat the above problem if |*H*(*jω*_{1}| *<* 1 but ∠*H*(*jω*_{1}) = 180°.

8.4. Analyze the oscillator of Fig. 8.15(b) if *C _{GD}* is not neglected.

8.5. Can any feedback oscillator that employs a lossy resonator be viewed as the one-port system of Fig. 8.13(c)?

8.6. Suppose the inductors in the oscillator of Fig. 8.17(a) exhibit a mismatch of Δ*L*. Determine the oscillation frequency by calculating the frequency at which the total phase shift around the loop reaches 360°.

8.7. Prove that the series combination of the two tanks in Fig. 8.21(a) can be replaced with one tank as shown in Fig. 8.21(b).

8.8. Compute the tuning range in Example 8.18 if *C _{b}* is placed at nodes

8.9. Why do the PMOS devices in Fig. 8.36 carry a current of *I _{SS}*?

8.10. For a CS stage loaded by a second-order parallel RLC tank, prove that *R _{p}*/(

8.11. Prove that the noise shaping in the system shown in Fig. 8.59 is given by Eq. (8.108).

8.12. Assuming *x*(*t*) = *A* cos *ω*_{0}*t* + *n*(*t*) = *A* cos *ω*_{0}*t* + *n _{I}*(

8.13. Suppose the VCO of Fig. 8.25(a) employs varactors whose capacitance is given by *C _{var}* =

(a) Determine the AM noise resulting from *I _{n}*.

(b) Determine how the average value of the varactor capacitance varies with *I _{n}*.

(c) Compute the phase modulation as a result of the tank resonance frequency modulation.

8.14. Prove that the peak drain voltage swing in Fig. 8.86(b) is no more than roughly *V _{DD}*− 2(

8.15. Assuming *Z _{T}* = (

8.16. For small-signal operation, Eq. (8.194) can be written as

Now suppose the tail current of the coupling transistors, *I*_{T1} contains a flicker noise component, *I _{n}*

8.17. In the VCO circuit shown in Fig. 8.112, the voltage dependence of each varactor can be expressed as *C _{var}* =

(a) Compute the “gain” from *I _{SS}* to the output frequency,

(b) Assume *I _{SS}* has a noise component that can be expressed as

8.18. The circuit shown in Fig. 8.113 is a simplified model of a “dual-mode” oscillator [28]. The voltage-dependent current source models a transistor. The circuit oscillates if *Z _{in}* goes to infinity for

(a) Determine the input impedance *Z _{in}*.

(b) Set the denominator of *Z _{in}* to zero for

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