Thesis Organization

In the chapter 2 of the thesis, some basic concepts of RF design are introduced.

These basic concepts which include the introduction of receiver architecture, noise and linearity provide the guidance for RF circuit design.

In the chapter 3 of the thesis, the design consideration of some circuit blocks which include LNA and mixer is introduced. Based on these circuits, a mixer and a front-end circuit are designed and verified in later chapter.

In the chapter 4 of the thesis, -5dBm IIP3 UWB LNA using complex derivative cancellation technique are designed. The multiple gated transistors configuration is introduced and a compact equivalent circuit using a complex transconductance is proposed for broadband linearity improvement design in this configuration. Following the above analysis, a low-power and high-linearity UWB LNA is designed. Finally, measurement result of the LNA chip fabricated by TSMC 0.18um CMOS technology is discussed.

In the chapter 5 of the thesis, a low-power, high linearity UWB front-end circuit is designed. The first stage is the LNA which is the same as that in chapter 4. The second stage is the active balun which provides the signal into differential form. The last stage is the double balance Gilbert cell mixer which has an acceptable linearity for the receiver. Overall front-end circuit is implemented.

In the chapter 6, the work is summarized and concluded.

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Chapter 2

Basic Concepts in RF Circuit

2.1 Receiver Architecture

2.1.1 Heterodyne Receiver

A simple heterodyne architecture is shown in Fig. 2.1. This architecture is the most reliable reception technique today. But if the cost, complexity, integration and power dissipation are the primary criteria, the heterodyne receiver will become unsuitable due to its complexity and the need for a large number of external components.

In heterodyne architectures, the signal band is translated into much lower frequencies by a down-conversion mixer, and the filters are used to select the band and channel of interest. In general, a low noise amplifier is placed in front of the down-conversion mixer to suppress the noise of the down-conversion mixer.

Frequency planning is important in the heterodyne receiver. For high-side injection, an undesired signal (image) at the frequency of ωIM=ωLO+(ωLO-ωRF) is translated into the same intermediate frequency (IF) as the desired signal. Similarly, for low-side injection, the image frequency is at ωIM=ωLO-(ωLO-ωRF). Therefore the image would cause extra noise at the intermediate frequency. As shown in Fig. 2.2, some techniques are necessary to suppress the image, such as image reject filter. But here comes the question: how to choose the intermediate frequency? If 2ωIF is sufficiently large, the rejection of a image filter will have a relatively small loss in the signal band and a large attenuation in the image band. But a lower 2ωIF will relax the quality factor of the channel select filter to get great suppression of nearby interferers.

Therefore a trade-off between image rejection and channel selection must be taken.

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2.1.2 Homodyne Receiver

The homodyne receiver is also called “direct-conversion” or “zero-IF”

architecture, since the RF signal is directly down-converted to the baseband in the first down conversion. In the homodyne receiver, the LO frequency is equal to the input carrier frequency, and channel selection requires only a low-pass filter with relatively sharp cutoff characteristics. The simple homodyne architecture is shown in Fig. 2.3. However, quadrature outputs are needed for frequency and phase-modulated signals, since the two sides of FM or QPSK spectra carry different information.

In recent years, this architecture becomes the topic of active research gradually and it may be due to the following reasons:

Fig. 2.1 Simple heterodyne architecture

Fig. 2.2 Rejection of image versus suppression of interferers (a) large ω (b) small _IF ω IF

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(1) The problem of image is removed due to ωIF = 0. Therefore no image filter is required, and the LNA need not drive a 50-Ω load.

(2) It is attractive for monolithic integration because this architecture needs less external components.

For the above reasons, this architecture is suitable for low-power and single-chip design. But some other issues that do not exist or are not as serious in a heterodyne receiver must be entailed, such as channel selection, DC offset, I/Q mismatch, even-order distortion, and flicker noise.

2.2 Noise Basic

Noise can be generally defined as any random interference unrelating to the signal of interest, and it is characterized by a PDF and a PSD. In analog circuits, the signal-to-noise ratio (SNR), defined as the ratio of the signal power to the total noise power, is an important parameter. But in RF design, most of the front-end receiver blocks are characterized in terms of their noise figure, which is a measure of SNR degradation resulting from the added noise from the circuit/system but rather than the input-referred noise. Noise factor can be expressed as following:

Total output noise power Noise Factor

Output noise power due to input source

= (2-1)

Fig. 2.3 Simple homodyne receiver architecture

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The noise figure (NF) is simply the noise factor expressed in decibels. If there is no noise in the system, then noise figure is 0 dB regardless of the gain. In reality, the finite noise of a system degrades the SNR, yielding noise figure > 0 dB. For those whose noise factor is close to unity, noise temperature (TN) is an alternative way to express the effect of noise contribution. Noise temperature is the description of the noise performance in higher-resolution, and is defined as the required temperature increment for the source resistance. Noise temperature is calculated by all of the output noise at the reference temperature Tref (which is 290 K). It relates to the noise factor as following:

N

N ref

ref

Noise Factor 1 T T T (Noise Factor-1)

= +T ⇒ = ⋅ (2-2)

2.2.1 Noise Source

Thermal noise:

Thermally agitated charge carriers in a conductor constitute a randomly varying current that gives rise to a random voltage resulting from the Brownian motion.

Thermal noise is often called Johnson noise or Nyquist noise. The noise voltage has averaged value of zero, but a nonzero mean-square value.

In a resistor R, thermal noise can be represented by a series of noise voltage source v_n² = 4kTR fΔ or by a shunt noise current source _n^{2 4}kT f

i R

= Δ , where k is

Boltzmann’s constant (about 1.38×10^-23 J/K), T is the absolute temperature in Kelvins, and Δf is the noise bandwidth. However, purely reactive elements generate no thermal noise.

Shot Noise:

Shot noise occurs in PN junctions, and there occurred two conditions for shot noise:

(1) There must be direct current flow.

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(2) There must be energy barrier over which a charge carrier hops.

Charge comes in discrete bundles. The randomness of the arrival time gives rise to the whiteness of shot noise. Therefore the shot noise can be modeled by a shunt noise current source i_n² =2qI_DCΔf , where q is the electronic charge, IDC is the DC current in amperes, and Δf is the noise bandwidth in hertz.

Flicker Noise:

Flicker noise appears as 1/f character and is found in all active devices, as well as in some discrete passive element such as carbon resistors. In diodes, flicker noise is caused by traps associated with contamination and crystal defects in the depletion regions. The traps capture and release carriers in a random fashion and the time constants associated with the process giving rise to the 1/f nature of the noise power density. The flicker noise in diode can be represented as ²_j

j

i K I f

= f A⋅ ⋅ Δ , where K is the process-dependent constant, Aj is the junction area, and I is the bias current. In MOSFET, charge trapping phenomena are invoked in surface, and its type of noise is much greater than that of the bipolartransistor. The flicker noise in MOSFET can be given by:

2

2 2

2 m

n T

ox

g

K K

i f A f

f WLC f ω

= ⋅ ⋅ Δ ≈ ⋅ ⋅ ⋅ Δ (2-3)

where K is the process-dependent constant, and A is the area of the gate.

2.2.2 Noise Model of MOSFET

The dominant noise source in CMOS devices is channel noise, which basically is thermal noise originating from the voltage-controlled resistor mechanism of a MOSFET. This source of noise can be modeled as a shunt current source in the output circuit of the device. The channel noise of MOSFET is given by

f g kT

i_nd² =4 γ _d0Δ (2-4)

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where γ is bias-dependent factor, and g is the zero-bias drain conductance of the _d0 device. Another source of drain noise is flicker noise and is given by Eqn. 2-3. Hence, the total drain noise source is given by

WLC f

At RF frequencies, the thermal agitation of channel charge leads to a noisy gate current because the fluctuations in the channel charge induce a physical current in the gate terminal due to capacitive coupling. This source of noise can be modeled as a shunt current source between gate and source terminal with a shunt conductance gg, and may be expressed as

f

and δ is the gate noise coefficient. This gate noise is partially correlated with the channel thermal noise, because both noise currents stem from thermal fluctuations in the channel and the magnitude of the correlation can be expressed as

j

where the value of 0.395j is exact for long channel devices. Hence, the gate noise can be re-expressed as

where the first term is correlated and the second term is uncorrelated to channel noise.

From previous introduction of MOSFET noise source, a standard MOSFET noise model can be presented in Fig. 2.4, where i_nd² is the drain noise source, i is the _ng²

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gate noise source, and v is thermal noise source of gate parasitic resistor _rg² r_g.

2.2.3 Noise Figure of Cascaded Stages

For a cascade of m stages, the overall noise figure can be characterized by Friis formula

This equation indicates that the noise results from the decrease in each stage as the gain preceding the increase in stage. Hence, the first few stages in a cascade are the most critical for noise figure. But if a stage exhibits attenuation, then the noise figure of the following circuit is amplified when referred to the input of that stage.

2.2.4 Noise Factor of a Two Port Network

Noise factor F is a useful measure of the noise performance of a system. It is defined as the ratio of the available noise power Pno at output divided by the product of the available noise power at input Pni which times the networks’s numeric gain G, or equivalently defined as the ratio of the signal to noise power at the input to the signal to noise power at the output .

o

The noise factor is a measure of the degradation in signal to noise ratio due to the noise from the system itself. Since the noise factor relates to the input noise power, a

C

gs

v

gs

g

m

v

gs

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standardized definition of noise source has been setup: a resistor at 290K. A more general expression of noise factor NF is called noise figure which is just noise factor expressed in decibels:

F

NF =10log (2-12) When several networks are cascaded, each has its own gain Gi and noise factor Fi. The total output of the noise is composed of all the noise from each stage but with different amount of contribution to the noise performance. The noise factor of a cascade networks is given as

1 ... From (2-13), the noise factor of the first stage is most critical and must be keep as low as possible and its gain should be as large as possible to suppress the noise in the following stage. The result is intuitive since there is less interference of noise effect when the signal level is high.

2.2.5 Optimum Source Impedance for Noise Design

The noise factor of a two port network can be given as [5]

2

where Rn is the correlation resistance which showed the relative sensitivity of the noise figure to departures from the optimum conditions, and Zo is the characteristic impedance of the system. This equation expresses that there exists an optimum source reflection coefficient (Γ_opt) or equivalently an optimum source impedance (Z_opt) at the input of the network in order to deliver lowest noise factor (F_min). The value of Γ provides a constant noise factor value forming non-overlapping circles on the s

Smith chart. It is usually the case that the optimum noise performance trades with the maximum power gain.

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2.3 Linearity and Nonlinearity

All electronic circuits are nonlinear: a fundamental truth of electronic engineering. The linear assumption underlying most modern circuit theory is in practice only an approximation. Some circuits, such as small-signal amplifiers, are only very weak nonlinear; however, they are used in systems as if they were linear. In these circuits, nonlinearities are responsible for phenomena that degrade system performance and must be minimized. Other circuits, such as frequency multipliers, exploit the nonlinearities in their circuit elements; these circuits would not be possible if nonlinearities did not exist. Among these, it is often desirable to maximize the effect of the nonlinearities and even to maximize the effects of annoying linear phenomena.

The problem of analyzing and designing such circuits is usually more complicated than for linear circuits, and it is the subject of much special concern.

Linear circuits are defined as those for which the superposition principle holds.

Specifically, if excitations x1 and x2 are applied separately to a circuit having responses y1 and y2 respectively, the response to the excitation ax1+bx2 is ay1+by2, where a and b are arbitrary constants. This criterion can be applied to either circuits or systems.

This definition implies that the response of a linear and time-invariant circuit of system includes only those frequencies present in the excitation waveforms. Thus, linear and time-invariant circuits do not generate new frequencies. As nonlinear circuits usually generate a remarkably large number of new frequency components, this criterion provides an important dividing line between linear and nonlinear circuits.

Nonlinear circuits are often characterized as either strongly nonlinear or weakly nonlinear. Although these terms have no precise definitions, a good working

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distinction is that a weakly nonlinear circuit can be described with adequate accuracy by a Taylor series expansion of its nonlinear current/voltage (I/V), charge/voltage (C/V), or flux/current (Φ/I) characteristic around some bias current or voltage. This definition implies that the characteristic is continuous and has continuous derivatives.

And for most practical purposes, they do not require more than a few terms in its Taylor series. Virtually all transistors and passive components satisfy this definition if the excitation voltages are well within the component’s normal operating ranges; that is, below saturation.

2.4 Nonlinear Phenomena

2.4.1 Harmonic Generation

Assumption of the current nonlinear element is given by the expression:

2 3

I =aV bV+ +cV (2-15)

where a, b, and c are constants, real coefficients. Assuming that Vs is a two-tone excitation of the term:

1 1 2 2

( ) cos( ) cos( )

s s

V =v t =V ωt +V ω t (2-16)

Substituting (2.1) into (2.2) gives, for the first term,

1 1 2 2

( ) ( ) cos( ) cos( )

a s

i t =av t =aV ωt +aV ω t (2-17)

After doing the same with the second term, the quadratic, and applying the well-known trigonometric identities for squares and products of cosines, we obtained:

1 2

2 2 2 2 2

1 1 1 2

1 2 1 2 1 2

( ) ( ) { cos(2 ) cos(2 )

2

2 [cos(( ) ) cos(( ) )]

b s

i t bv t b V V V t V t

V V t t

ω ω

ω ω ω ω

= = + + + +

+ + + −

(2-18)

and the third term, the cubic, gives

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The total current in the nonlinear element is the sum of the current components in (2-17) through (2-19).

One obvious property of a nonlinear system is its generation of harmonics of the excitation frequency or frequencies. These are evident as the terms in (2-17) through (2-19) at mω1 and mω2. The mth harmonic of an excitation frequency is an mth-order mixing frequency. In narrow-band systems, harmonics are not a serious problem because they are far removed in frequency from the signals of interest and inevitably rejected by filters. In other systems, such as transmitters, harmonics may interfere with other communication systems and must be reduced by filters or other means.

2.4.2 Intermodulation Distortion

All the mixing frequencies in (2-17) through (2-19) that arise as linear combination of two or more tones, often called Intermodulation (IM) products. IM products were generated in an amplifier or communications receiver, often come with a serious problem. While the interfered spurious signals can be mistaken for desired signals. IM products are generally much weaker than the generating signals; however, a situation often arises wherein two or more very strong signals, which may be outside the receiver’s passband, generate an IM product that is within the receiver’s passband and obscures a weak and desired signal. Even-order IM products usually occur at frequencies well above or below the generating signals, and consequently are often of little concern because…... The IM products of greatest concern are usually the third-order ones that occur at 2ω1-ω2 and 2ω2-ω1, and because they are the

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strongest of all odd-order products and close to the generating signals, they often cannot be rejected by filters. Thus, intermodulation is a major concern in microwave system.

2.5 Fundamental of the Volterra Series

The nonlinearity of the system often leads to interesting and important phenomena, such as harmonics, gain compression, desensitization, blocking, cross modulation, intermodulation, etc. These distortions will degrade the performance of the system, while Volterra series will be used for distortion computations. It can provide designers some information to derive which circuit parameters or circuit elements they have to modify to obtain the required specifications. Therefore, Volterra series will be introduced in the following section.

In fact, the Volterra series describe a nonlinear system in a way which is equivalent to the Taylor series approximating an analytic function. A nonlinear system excited by a signal with small amplitude can be described by the Volterra series, which can be broken down after the first few terms. The higher the input amplitude, the more terms of that series need to be taken into account to describe the system behavior properly.

For very high amplitudes, the series diverges just as Taylor series. Hence, Volterra series are only suitable for the analysis of weak nonlinear circuits.

The Volterra series approach has been proven to be useful for hand calculations of small transistor networks. Since Volterra kernels retain phase information, they are especially useful for high-frequency analysis.

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The theory of Volterra series can be viewed as an extension of the theory of linear, first-order systems to weakly nonlinear systems. And a system is considered as the combination of different operators of different order in the Volterra series description, as shown in Fig. 2.5. Every block H1, H2, and Hn represents an operator of order 1, 2, …, respectively. The amount of operators must be used depend on the input amplitude. In general, the weakly nonlinear effects can be described accurately by taking into account third-order effects only.

In the time domain, the transformation on an input signal (x(t)) was performed by a nth-order Volterra operator that is given by:

∫

The n-dimension integral can be seen as an nth-order convolution integral. The function )h_n(τ₁,τ₂,",τ_n is an nth-order Volterra kernel. The output of a nonlinear system can represent the sum of the output of a first-order Volterra operator with the output of a second-order one, a third-order one and so on, as shown in Fig. 2.5. The Volterra series of the nonlinear system can be expressed as

)]

In the frequency domain, the nth-order Volterra kernel can be given by

∫

and is called the nth-order nonlinear transfer function or the nth-order kernel transform.

Fig. 2.5 Schematic representation of a system characterized by a Volterra series

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2.6 Nonlinear Performance Parameters in Terms of Volterra Kernels

When a system that is described by a Volterra series up to order three, it is excited by the sum of two sinusoidal excitations A₁cosω and ₁t A₂cosω . Then ₂t the output is given by the sum of the responses listed in Table 2.1. From Table 2.1, the expression for the second and third harmonic distortion in terms of general Volterra

When a system that is described by a Volterra series up to order three, it is excited by the sum of two sinusoidal excitations A₁cosω and ₁t A₂cosω . Then ₂t the output is given by the sum of the responses listed in Table 2.1. From Table 2.1, the expression for the second and third harmonic distortion in terms of general Volterra

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