Communications Integrated Circuit Design by

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(1)Communications Integrated Circuit Design. by. Dr. Steve Tu Rm SF726C E-mail : [email protected]. Department of Electronic Engineering Fu Jen Catholic University Fall 2007 (p1).

(2) Outline – 1.. Wireless receiver architectures.. 2.. RF design.. 3.. Low noise amplifier design.. 4.. Mixer design.. 5.. Oscillators.. 6.. Frequency synthesizers – theory.. 7.. Frequency synthesizers – circuits.. 8.. Power amplifiers.. (p2).

(3) Recommended Books – 1. . 2. . Razavi, B. 1998, “RF Microelectronics”. Reading, Prentice Hall. An introductory book for RFIC. Also see the paper - Razavi, B.“Challenges in portable RF transceiver design,” IEEE Circuits and Devices, pp. 12-26, Sep. 1996 for even more concise introduction.. Lee, T. H.. 1999. “The design of CMOS Radio- frequency Integrated Circuits”, Cambridge University Press. Very rich contents for all aspects.. Grading Policy – 1. Midterm exam ( 40% ). 2.. Final exam ( 40% ). 3.. Misc. ( interaction with the lecturer. ) ( 20% ). (p3).

(4) Lecture 1 – Introduction to Wireless Receiver Architectures 1. Introduction to Communication Systems encoding modulation Information source. Transmitter. distortion. decoding modulation. Channel. Receiver. noise. Destination. distortion. A communication system exists to convey a message from a source to a destination. The form in which the information is sent is highly dependent upon the channel through which it must be conveyed, i.e. the medium ( air, cable, water etc. ), distance, external interference etc. Generally the original information from the source will be unsuitable for immediate transmission. The information must be converted to electrical form firstly, then it is pre-processed ( filtered, encoded etc. ) to improve system performance. The processed signal is then used to modulate a carrier, i.e. it is imposed on a high frequency sine wave. Various types of modulation can be used, e.g. AM, FM, PM…. The information is then transmitted through the atmosphere as an electromagnetic wave. The term ‘channel’ is often used to refer to the frequency range allocated to a particular system.. (p4).

(5) The signal will be inevitably deteriorated during transmission due to the addition of noise, which will have the greatest effect where the signal is weakest. Thus noise at the input to the receiver will be the most noticeable. The receiver performs the reverse process to the transmitter, i.e. demodulation and decoding. Consider the transmission of audio information which occupies the frequency range below 20kHz. If no modulation scheme was employed, then all signals transmitted would interfere with one another, and signals would be mixed up. Modulation is thus employed to convert signals to different parts of the electromagnetic spectrum and hence separate them. Each signal is given its own frequency location, which is known by the receiver. The modulation frequency is also chosen to make transmission feasible. For efficient radiation and reception the antenna should have a length which is comparable to a quarter wavelength of the signal. This is around 75 meters at 1 MHz, but increases to 5 km at 15 kHz.( clearly impractical !) The bandwidth of the channel will depend upon the modulation scheme used and the required data transmission rate.. (p5).

(6) Wireless Radio 1.. Conventional wireless systems include :. . mobile telephone systems satellite communications cordless phones. 2.. Wireless systems are noise limited.. 3.. Base stations transmit on the same carrier frequency, so they must be spaced apart to eliminate co-channel interference.. Cellular Radio. 7 6. 1. 5. 2 3. 4. cluster. (p6).

(7) Each cell has its own base station. Each cell within a cluster is assigned a different carrier frequency. Cells operating with equal carrier frequencies are thus separated by one cluster. A receiver will be able to pick up a signal from an adjacent cluster, thus is designed to lock onto the strongest signal ( i.e. that from the base station within its own cluster ). To increase the system capacity, the number of cells is increased ( by reducing cell size ). The principal attributes of cellular radio are : 1. High capacity. 2. Concept of cells. 3. Frequency re-use. First generation is analog system such as AMPS (Advanced Mobile Phone System) Typically use FDMA ( Frequency Division Multiple Access ) Second generation is digital systems including : GSM ( Group Special Mobile ) NADC ( North American Digital Cellular ) – co-exists with AMPS. Typically use FDMA/TDMA ( Time Division Multiple Access ). (p7).

(8) 2. Receiver Architectures The Superheterodyne Principle – If we are trying to select one particular frequency channel from the complete RF spectrum, then we need a bandpass filter to reject any unwanted frequencies. Generally this filter has to be narrowband, and high-Q filters are difficult to design at high frequencies. This problem is compounded if the input signal frequency is variable ( i.e. the signal is transmitted in one of a number of possible channels, each with the same bandwidth ). A tunable, high-Q bandpass filter with constant bandwidth is now required. The solution is to employ a superhet receiver ( supersonic heterodyne ). This system downconverts the input signal to an intermediate frequency ( IF ), and a bandpass IF filter is then used to select the wanted signal. The design of the bandpass IF filter is easy since it doesn’t have to be tunable, and the IF centre frequency is much lower than the input RF signal.. aerial. mixer. IF signal. RF signal pre-filter. bandpass filter first oscillator. mixer. baseband signal second oscillator. The downconversion is performed by ‘mixing’ ( multiplying ) the RF input signal ( fRF) with a local oscillator signal ( fLO), such that the resulting output is at the required IF frequency ( fIF). (p8).

(9) Received RF signal = 2 A cos[( f RF )t + θ ]. , Local oscillator signal =. cos( f LO )t. Mixer output = 2 A cos( f LO )t cos[( f RF )t + θ ]. = A cos[( f LO − f RF )t − θ ] + A cos[( f LO + f RF )t + θ ]. i.e. sum and difference components The sum components are at a very high frequency and are removed by filtering. The difference frequency component is a replica of the RF component in terms of amplitude and phase, but is shifted down to an intermediate frequency ( IF ), f IF = f LO − f RF The oscillator frequency is fLO is often tunable to ensure that a range of input RF frequencies can be selected. • Image signals – An image channel ( fIM ) is also converted to the intermediate frequency, where fIF = fIM - fLO. wanted. fIF. fRF. image. fLO. fIM. This image signal must be suppressed by filtering before downconversion. The pre-filter can be a simple lowpass filter. The pre-filter should pass RF signals that are at fLO – fIF, but reject image signals at fLO+ fIF. The design of the pre-filter is thus easy if fIF is fairly high.. (p9).

(10) pre-filter. fIF. fRF. fLO. fIM. bandpass filter 2fIF The bandpass filter that selects the required channel is centred at fIF. The design of this filter is easy if fIF is fairly low. Thus there is a trade-off between these two requirements.. fIF. fRF. fLO. fIM. fX. 2fLO. • Oscillator harmonics : Second harmonic 2fLO mixes with fX to produce fIF. The need. to remove fX may further constrain the pre-filter design.. (p10).

(11) Double Superheterodyne Receiver Architecture – antenna 1st mixer. RF amp 1st filter. 2nd filter. 2nd mixer 1st IF stages. 3rd filter. injection filter. injection filter. ~. ~. 1st local oscillator. 2nd local oscillator. In a superhet receiver, the design of the prefilter is easy if the IF is high, while the design of the IF bandpass filter is easy if the IF is low. The double conversion superhet receiver avoids this conflict. The first IF is high which ensures that the image frequency is well separated from the wanted signal. The second IF is low, which enables the design of circuits with sharp selectivity and hence good adjacent channel rejection.. (p11).

(12) Homodyne ( Direct Conversion ) Receiver – Double superhet designs are popular because the first IF can be high ( to give a wide separation between the wanted signal and the image channel ), while the second IF can be low ( making it easier to separate the wanted signal from the adjacent channel ). However, the IF filters generally have to be made with external components, because a 20 % variation in tolerance ( typical of on-chip passive components ) is unacceptable at several hundred MHz. Generally a double superhet receiver is a two-chip solution ( at least ). With additional external filter components, the complete design is bulky and expensive. An alternative is to use direct conversion. In this case, a single local oscillator is used whose frequency is equal to the RF carrier frequency, and thus the IF = 0. No bandpass filtering is required as the signal is converted directly to baseband. In addition, there is no image signal, thus no image filtering is needed. All signal filtering is at baseband frequencies, and therefore can be performed onchip. This means that a single-chip receiver is feasible using direct conversion. The. main. drawback. with. direct. conversion. architectures. is. their. susceptibility to LO ‘re-radiation’ which is picked up by the antenna. After mixing, this leads to DC offsets in the receiver, which will directly corrupt any DC information in the signal.. (p12).

(13) Even-Order Distortion – Even-order nonlinearity is specially important in homodyne downconversion. Refer to the following figure. Suppose there are two strong interferers close to the channel which experiences a nonlinearity in LNA with y(t) = α1x(t) + α2x2(t). Direct Feedthrough. ω. 0 LNA. Interferers. Desired Channel. ω. cosωLOt. If x(t) = A1cosω1t + A2cos ω2t , then y(t) must contain α2A1A2cos(ω1- ω2)t term, which indicates that two high-frequency interferers generate a low-frequency beat in the presence of even-order distortion. The problem is shall the mixer exhibits a finite direct feedthrough from RF input to IF output, then the translated interferers corrupt the downconverted signal since mixers usually suffer from some asymmetry and their operation can be described as vRF(t)(a + Acos ωLot ), where a is a constant. Thus, a fraction of vRF(t) appears at the output with no frequency translation. (ps) Even-order terms – If x(t) = Acos ωt applied to a system with the function y(t) = α1x(t) + α2x2(t) + α3x3(t) , then y(t) = α1 Acos ωt + (α2A2/2)(1+ cos 2ωt ) + (α3A3/4)(3cos ωt + cos 3ωt) = α2A2/2 + (α1 A + 3α3A3/4 ) cos ωt + (α2A2/2) cos 2ωt + (α3A3/4) cos 3ωt ⇒ From the expansion, we know even-order terms result from αi with even i and vanish if the system has odd symmetry eg. differential or complementary circuit topologies.. (p13).

(14) Image-Reject Receivers – Hartley receiver architecture : IF1. RF. LOA. LNA. φ2 IF out. φ1 IF2. LOB. ~ local oscillator. Local oscillator :. LO A = 2 cos(ω L t − φ1 ). LOB = 2 cos ω Lt. RF signal :. A cos(ω At + φ A ) + B cos(ω B t + φ B ) Where ωA=( ωL- ωIF ) is the wanted signal, and ωB=( ωL+ ωIF ) is the image. After mixing : { Recall 2cosXcosY = cos(X-Y) + cos(X+Y) and cos(-X) = cos(X) }. (p14).

(15) IF1 = 2 A cos(ω At + φ A ) cos(ω L t − φ1 ) + 2 B cos(ω B t + φ B ) cos(ω L t − φ1 ) = A cos((ω L − ω A )t − φ1 − φ A ) + A cos((ω L + ω A )t − φ1 + φ A ) + B cos((ω B − ω L )t + φ B + φ1 ) + B cos((ω B + ω L )t + φ B − φ1 ). IF2 = 2 A cos(ω At + φ A ) cos ω L t + 2 B cos(ω B t + φ B ) cos ω L t = A cos((ω L − ω A )t − φ A ) + A cos((ω L + ω A )t + φ A ). + B cos((ω B − ω L )t + φ B ) + B cos((ω B + ω L )t + φ B ) Lowpass filter removes sum components :. IF1 = A cos((ω L − ω A )t − φ A − φ1 ) + B cos((ω B − ω L )t + φ B + φ1 ) = A cos(ω IF t − φ A − φ1 ) + B cos(ω IF t + φ B + φ1 ). IF2 = A cos((ω L − ω A )t − φ A ) + B cos((ω B − ω L )t + φ B ) = A cos(ω IF t − φ A ) + B cos(ω IF t + φ B ). After phase shift -φ2 :. IF1 = A cos(ω IF t − φ A − φ1 − φ2 ) + B cos(ω IF t + φ B + φ1 − φ2 ) IF2 = A cos(ω IF t − φ A ) + B cos(ω IF t + φ B ) Adding signals IF1 and IF2 :. IFout = A cos(ω IF t − φ A − φ1 − φ2 ) + A cos(ω IF t − φ A ). + B cos(ω IF t + φ B + φ1 − φ2 ) + B cos(ω IF t + φ B ). φ +φ ⎞ ⎛φ +φ ⎞ ⎛ = 2 A cos⎜ 1 2 ⎟ cos⎜ ω IF − φ A − 1 2 ⎟ 2 ⎠ ⎝ 2 ⎠ ⎝ φ −φ ⎞ ⎛ φ −φ ⎞ ⎛ + 2 B cos⎜ 1 2 ⎟ cos⎜ ω IF − φ B + 1 2 ⎟ 2 ⎠ ⎝ 2 ⎠ ⎝. (p15).

(16) To avoid signal distortion, we require cos. φ1 + φ2. =1. i.e.. (φ1 + φ2 ) = 4nπ. 2 φ − φ2 To ensure image rejection, we require cos 1 = 0 i.e. (φ1 − φ2 ) = (2n + 1)π 2 Thus. φ1 = 900 , φ2 = −900. Weaver receiver architecture :. Sin(ω1t). Sin(ω2t). A. _ C IF out. RF. LNA. B Cos(ω1t). RF signal :. +. D. Cos(ω2t). x(t ) = ARF cos ω RF t + Aim cos ω imt. The voltage at A :. x A (t ) =. A ARF sin (ω1 − ω RF )t + im sin (ω1 − ω im )t 2 2. The voltage at B :. xB (t ) =. A ARF cos(ω1 − ω RF )t + im cos(ω1 − ω im )t 2 2. (p16).

(17) Therefore, the voltages at C and D are. ARF [cos(ω 2 − ω1 + ω RF )t − cos(ω 2 + ω1 − ω RF )t ] 4 A + im [cos(ω 2 − ω1 + ω im )t − cos(ω 2 + ω1 − ω im )t ] x (-1) 4. xC (t ) =. and. (+). A xD (t ) = RF [cos(ω 2 + ω1 − ω RF )t + cos(ω 2 − ω1 + ω RF )t ] 4 A + im [cos(ω 2 + ω1 − ω im )t + cos(ω 2 − ω1 + ω im )t ] 4. cancel each other !!. respectively.. Digital-IF Receivers – RF input. BPF. cosωLot. ADC. Multiplier. LPF. output. Digital Sinewave Generator. • The first IF signal is digitized via an ADC. Further stages of filtering and mixing are then performed in the digital domain. The main issue in the design of digital-IF receivers is the performance of the A/D converter, which must have adequate bandwidth. Typical performance requirements cannot generally be met by today’s conventional Nyquist rate A/D converters.. (p17).

(18) Subsampling Receivers – X( f ) x(t), continuous-time signal -fn p(t), sampling function. 0. f. fn. …. … t T-3. T-2. T-1. T0. T1. T2. T3. Xs( f ) Xs(f), Fourier transform of the sampled signal. …. … f -2fs. -fs. 0 fn fs. 2fs. • Sampling a continuous-time signal x(t) is to represent x(t) at a discrete number of points t = nT, where T is the sampling period and the sequence we denote T1, T2, T-1, T-2,….. • A bandpass signal centered on f0 is sampled at a rate f0/m ( i.e. well below the Nyquist rate ). Provided that the sampling rate is slightly greater than twice the bandwidth of the RF signal, then aliasing will not occur. • Sampling Theorem –A bandlimited signal x(t) having no frequency components above fn hertz is completely specified by samples that are taken at a uniform rate greater thab 2fn hertz. The frequency 2fn is known as the Nyquist rate.. • The main drawback with this technique is the aliasing of noise, since subsampling by a factor m effectively multiplies the downconverted noise power of the sampling circuit by a factor 2m, since this noise is not bandlimited.. (p18).

(19) 3. System Response and Specifications Receivers are commonly specified in terms of sensitivity, spurious response rejection, intermodulation rejection and intercept point.. Decibels Specifications, calculations… are commonly quoted in dBm. Decibels are power ratios, so it is meaningless to quote a power level in dB unless a reference level ( P1) has already been given. Power ratio ( bels ) = log10(P2/P1) Thus a power ratio of 1 bel means that P2=10P1. Since more modest power ratios will result in fractions of a bel, it is more common to use the decibel scale, where 1 decibel ( dB ) = 1/10 bels : Power ratio ( dB ) = G(dB) = 10log10(P2/P1) or P2 = P1 10(G/10) A power gain of 3dB means that P2 = 2P1 A power gain of 1dB means that P2 = 1.26P1 ( i.e. 26% increase in power ) If we have a power level P2 that we want to convert to dB, we need to define a reference. power. P1.. The. simplest. value. to. choose. is. P1. =. 1W. ( P2(dB)=10log10(P2/1)). In RF circuits, we are usually dealing with power levels much below 1 W, and the standard power reference P1 is 1mW. To indicate that this reference is being used, the power gain is denoted dBm : P2(dBm)=10log10(P2/1x10-3).. (p19).

(20) Noise Figure and Receiver Sensitivity – The sensitivity is a measure of the minimum input signal that can be detected by the system. This depends on the signal-to-noise ratio ( SNR ) required at the detector, and the noise figure ( NF ) of the complete system.. Psig. Pni. System. Detector. Psys. Psig : received signal power Pni : received noise power Psys: equivalent input-referred noise of system Noise factor is defined as the total output ( or input ) noise power of the system divided by the output ( or input ) noise power due to the source alone. Noise factor. SNRin =. Psig Pni. F=. Psys + Pni Pni. SNRout =. Psig Pni + Psys. SNRin Pni + Psys = = Noise factor F SNRout Pni NF (dB) = SNRin (dB) – SNRout (dB). (p20).

(21) We require a certain SNRout = SNRdet at the detector in order to recover the signal. Thus, SNRin ( dB ) = SNRdet ( dB ) + NF SNRin ( dB ) = Psig ( dB ) – Pni ( dB ) We define the sensitivity of the system as the minimum input signal power that we can successfully detect. Sensitivity = Psig ( dB ) = SNRin ( dB ) + Pni ( dB ) = Pni ( dB ) + NF + SNRdet ( dB ) Pni = input ( received ) noise.. Spurious Response Rejection – All superhet receivers have the potential for responding to frequencies other than the desired channel. Spurious response rejection of 70 to 100dB can be practically achieved. Most spurious responses originate in the mixers, especially from harmonic mixing of the RF and LO signals. Any RF frequency that satisfies the following relationship is a potential spurious response,. mf RF − nf LO = ± f IF. ( f RF 1 =. nf LO − f IF , m. f RF 2 =. nf LO + f IF ) m. Common spurious responses ( m, n ) • Image ( -1, 1 ) or ( 1, -1 ) • Half IF ( 2, -2 ) or ( -2, 2 ) ( Half IF interferer experiences 2nd-order distortion and LO contains a significant second harmonic as well, then the IF output exhibits a component at IF band ). (p21).

(22) Effects of Nonlinearity While many analogue and RF systems can be ideally modeled as linear circuits for their small signal response, nonlinearities often lead to some important phenomena. For example we assume. y (t ) ≈ α1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t ) If the input signal is then. y (t ) =. x(t ) = A cos ωt. α 2 A2 ⎛ 2. 3α 3 A3 ⎞ α 2 A2 α 3 A3 ⎟⎟ cos ωt + + ⎜⎜α1 A + cos 2ωt + cos 3ωt 4 2 4 ⎝ ⎠. ⎛ 3α 3 A2 ⎞ ⎟⎟ A cos ωt Due to nonlinearity, we can spot that for fundamental freq. ⎜⎜α1 + 4 ⎝ ⎠. 3α 3 A2 the gain is α1 + 4. which is dependent on A. Notice that the gain here is. a decreasing function of A if α3 < 0. We can therefore define “1-dB compression point”. 20logAout. 1dB. A1-dB. 20logAin. (p22).

(23) In fact, nonlinearity can also be described as “intermodulation distortion” in a “2-tone” test, here we explain the concept of “intercept point”.. Intercept Point – Intercept point is a measure of circuit or system linearity that allows us to calculate distortion or intermodulation product levels from incoming signal amplitudes. The intercept point represents a fictitious input amplitude at which the desired signal components and the undesired signal components are equal in amplitude. The order of the intercept point refers to how fast the amplitudes of the distortion products increase with an increase in input level. The intercept point is a fictitious value, since at high signal levels the active device will generally saturate, and thus the undesired signal amplitude never equals that of the desired signal. Assume then,. x(t ) = A cos ω1t + A cos ω 2t. y (t ) = α1 ( A cos ω1t + A cos ω 2t ) + α 2 ( A cos ω1t + A cos ω 2t ). 2. + α 3 ( A cos ω1t + A cos ω 2t ). 3. This will lead to several combination of freq. say, ω1+ ω2 ,…… however, the point is the components, 2ω1 - ω2 or 2ω2 - ω1 if ω1 and ω2 close each other will damage the carrier spectrum since the magnitude of 2ω1 - ω2 ( or 2ω2 - ω1 ) is ¾ α3A3. We can define “IP3” ( IIP3 / OIP3 ).. α1 A. OIP3 20log(α1A) change to log axis. ⇒. 20log(¾ α3A3). ¾ α3A3 IIP3 20logA A. (p23).

(24) Lecture 2 – RF Design 1. Introduction An emphasis on digital systems ( increased performance ) has meant that analog design has generally been neglected in terms of teaching / textbooks / design tools etc…However radio waves are analog signals, which generally need to be detected, amplified, filtered and converted to a lower frequency before digital processing sections of the circuit can take over. Digital circuits can be designed at an abstract level, without considering the circuit implementation. With RF analog, all aspects of the system implementation must be considered as part of the design. It is important to identify the key difference between lumped and distributed design techniques. Basically when the signal wavelengths are close to the dimensions of the integrated circuit, then characteristic impedances become significant and we essentially need to consider the circuit in terms of transmission lines. (1). At lower frequencies where the signal wavelength is much larger than the dimensions of the circuit, the design can be considered in terms of lumped components, allowing some of the more classical low frequency analog circuit techniques to be applied. (2). At intermediate frequencies we enter the realms of hybrid lumped / distributed design. Many RF designs fall into this category, although every day we see new technologies and circuit techniques developed which increase the frequency range for which lumped approaches are possible.. (p1).

(25) RF IC’s are generally designed without the use of special microwave components, and techniques are very similar to those employed at lower frequencies. However at RF, many parasitic effects become significant, and this results in complex circuit models for analysis. The frequency performance of an RF or broadband circuit will depend on the frequency capability of the devices used, and no amount of good design can compensate for transistors with an inadequate range. As a rule, designs are kept as simple as possible since at high frequencies all components have associated parasitics. However, we also obtain more good news : • No circuit behavior is really unexplainable. Strange behavior generally results from parasitics or interactions which were overlooked during the design phase. • The development of advanced computer simulation tools has greatly simplified the analysis of RF circuits.. (p2).

(26) 2. Terminology RF Spectrum Radio frequencies ( RF ) extend from 10s of kHz to many GHz. Wireless communication frequencies presently range from 150-900 MHz (paging), 900MHz-2.4GHz ( cordless and cellular phones). (p.s.). 900MHz, 2.4GHz : H2O resonance. • In RF circuits, the standard power reference P1 is 1mW. To indicate that this reference is being used, the power is denoted dBm. Similarly, dBµ denotes 1 µW as reference. Thus 7dBm = 5mW. • The power level can be translated into a voltage if the resistance level is known, since V 2 =. P2 • R. In RF circuits, the resistance level is generally assumed. to be 50Ω. Thus 7dBm = 5mW = 500mV. • Alternatively, the signal voltage V2 can be quoted relative to a voltage reference level V1, where it is assumed that V1 and V2 see equal resistance levels :. G (dB) = 10 log10 (V 2 2 / V 12 ) = 20 log10 (V 2 / V 1). • dBV and dBµV denote reference levels of 1V and 1 µV respectively.. Available Noise Power Maximum signal power transfer occurs when the load impedance is matched to the source impedance, so matching is an important concept in RF design.. (p3).

(27) Rs Vns Rl. However if Rl = Rs, then the maximum noise power will also be transferred from the source into the load. The ‘available power’ ( Nt) is defined as the mean square thermal noise voltage ( i.e. power ) transferred from the source to the load, when the load is a noiseless resistance equal to the source resistance. Thus. Vns (rms) Vns(rms ) Vns 2 N t = v(rms ) ⋅ i (rms ) = ⋅ = 2 2 Rs 4 Rs V2/Hz = kT∆f = 4 ×10 − 21 V2/Hz or -174 dBm. where k is Boltzmann’s constant ( 1.38x10-23 J/K). This gives a measure of the minimum input signal power that must be supplied to a system if the signal level is to be greater than the noise level. The output noise power is No(dBm) = -174 +G + 10log10B +NF. (p4).

(28) 3. S-Parameters ( Scattering Parameters ) -. V1. I1. Two-port. I2. V2. Two-port networks can be characterized by various parameter sets, e.g. (i). Z-parameters :. V1 = ZiI1 + ZrI2 V2 = ZfI1 + ZoI2. (ii). Y-parameters :. I1 = YiV1 + YrV2 I2 = YfV1 + YoV2. (iii). h-parameters :. V1 = hiI1 + hrV2 I2 = hfI1 + hoV2. The chosen parameter set depends on how easily the various parameter values can be measured in a given situation. For active circuits, particularly discrete transistors, h-parameters are traditionally employed at fairly low frequencies. This parameter set is measured by open and short circuiting the input and output ports . At 100s of kHz and above it becomes difficult to obtain reliable open circuits, because of parasitic capacitances. Therefore at medium to high frequencies, Yparameters are generally employed to measure circuit performance. These parameters are determined only by short circuiting the input and output ports. At UHF and above, Y-parameters also become unsuitable because (a). It is difficult to realize reliable short circuits. (b). Devices may oscillate under short circuit test. This has led to the concept of scattering parameters ( s-parameters) being employed.. (p5).

(29) S-parameters characterize a network based on the concept of traveling waves. An incident wave traveling down a transmission line will be reflected to some extent when it reaches a load, and the magnitude of the reflected wave will depend on the relative impedance of the transmission line and load. S-parameters are small-signal quantities which enable impedance matching, input and output reflection coefficients, gain and stability to be determined for a given network. The popularity of S-parameters is due to the development of specialized test equipment to characterize and measure these values, and almost all RF devices are characterized by their manufacturers over a wide range of frequencies and biasing conditions.. Vi1 Z0. Two-port. Zi. Vr1. Vi2 Z0. Zr. Vr2. Consider the two-port embedded within an external system with characteristic impedance Z0 . If Zi = Z0, then all the incident power will be dissipated in the input of the two port, and no power will be reflected back to the source ( thus Vr1 = 0 ). Similarly if the two-port output impedance Zr = Z0 then there will be no power reflected at the output port (Vr2 = 0 ). If the two-port is mismatched to the characteristic impedance Z0, then power will be reflected at the both the input and output ports.. (p6).

(30) The reflected voltages at the input and output ports can be defined as. Vr1 = f11 Z0 Vr 2 = f 21 Z0. Vr1 = f11 Vi1 + f12 Vi2 ⇒ Vr2 = f21 Vi1 + f22 Vi2. Vi1 Z0. a2 =. Vi 2 Z0. b1 =. Vr1 Z0. Define. a1 =. and. S11= f11 , S12 = f12 , S21 = f21 , S22 = f22. b2 =. Vi1 V + f12 i 2 Z0 Z0 Vi1 V + f 22 i 2 Z0 Z0. Vr 2 Z0. Therefore b1 = S11 a1 + S12 a2 b2 = S21 a1 + S22 a2 Notice that a1, b1, a2, b2 are the square roots of the incident and reflected ( or scattered ) powers at port1 and port2, respectively. These quantities can be related to the total terminal voltages and currents.. 1 a1 b1. Two-port Network. ⎛ S11S12 ⎞ ⎜⎜ ⎟⎟ ⎝ S 21S 22 ⎠. 2 b2 a2. (p7).

(31) Measurement of S-parameters. ZS VS. a1. a2. Two-port network. ~ b1. ZL b2. If ZL is matched to Z0 then no power is reflected at the load, and a2 = 0 : S11 = ( b1 / a1 )a2 = 0 S21 = ( b2 / a1 ) a2 = 0 If Zs is matched to Z0 then a1= 0 : S12 = ( b1 / a2 )a1 = 0 S22 = ( b2 / a2 )a1 = 0 • S11 is the input reflection coefficient when the output circuit is matched. • S21 is the forward transfer coefficient when the output is matched. • S12 is the reverse transfer coefficient when the input is matched. • S22 is the output reflection coefficient when the input is matched.. (p8).

(32) 4. Why 50-ohm RF standard load ? We can explain the “why” with the following two aspects. 1. Power-Handling Capability – •. Basic Concept - Consider a coaxial cable with an air dielectric. There will be some voltage at which the dielectric breaks down. For a smaller inner conductor, due to the tighter radius of curvature, the breakdown voltage will be lower, which it tends to decrease the cable’s power-handling capability. On the other hand, one can increase the inner diameter to increase the breakdown voltage whereas the characteristic impedance would then decrease, which in turn tends to reduce the power deliverable to a load. Thus, the trade-off leads to a well-defined ratio of conductor diameters, which maximises the powerhandling capability.. •. Now, let us find out the optimum load value in terms of the factor. Here we need an equation for the peak electric field between the conductors, and another for the characteristic impedance of a coaxial cable.. Emax =. V a ln(b / a). Z0 = µ / ε ⋅. ln(b / a ) 60 ≈ ⋅ ln(b / a ) 2π εr. in which a and b are the inner and outer radius, respectively and εr is the relative dielectric constant ( unity for air line case ). • Notice that the maximum power deliverable to a load is proportional to V2/Z0. Therefore,. [. 2 2 ε r Emax ⋅ a 2 ln(b / a) V 2 Emax ⋅ a 2 ln(b / a ) 2 P∝ = = Z 0 (60 / ε r ) ⋅ ln(b / a) 60. ] (p9).

(33) Taking the derivative yields dP = d ⎡a 2 ln⎛⎜ b ⎞⎟⎤ = 0 ⎢ ⎥ which we can obtain. ⎝ a ⎠⎦. da ⎣. da. b = e Substituting the b, a ratio to the characteristic a. impedance equation we obtain Z0= 30 Ω. 2. Attenuation –. • Basic Concept – Remember a general expression for the attenuation constant of a transmission line that account for both dielectric and conductor losses. Look up the fact that the attenuation per length due to dielectric loss is independent of conductor dimensions. Simplifying the equation to account only for the attenuation due to resistive loss gives. α≈. R 2Z 0. In which R is the series resistance per unit length. At high frequencies, R is due mainly to the skin effect. To reduce R, we can increase the diameter of the inner conductor, which it would tend to reduce Z0 at the same time. Again, we might not obtain any benefits. • Now, we’d like to make the trade-off with the following equations,. R≈. ⎡1 1⎤ + 2πδσ ⎢⎣ a b ⎥⎦ 1. where σ is the conductivity of the wire and δ is the skin depth, δ = ⎡1 1⎤ εr + R 2πδσ ⎢⎣ a b ⎥⎦ ≈ Therefore, α = 2Z 0 ⎡ ⎛ b ⎞⎤ 2 ⎢60 ln⎜ ⎟⎥ ⎝ a ⎠⎦ ⎣. 2. µ ⋅σ ⋅ω. 1. dα a ⎛b⎞ = 0 ⇒ ln⎜ ⎟ = 1 + da b ⎝a⎠ Therefore, we obtain a value of about 3.6 for b/a, corresponding to a Z0 of Taking the derivative yields about 77Ω. ⇒. Since 77Ω gives us minimum loss and 30Ω gives us maximum powerhandling capability, a reasonable compromise is the average, 50Ω.. (p10).

(34) 5. Impedance Matching • The maximum power is transferred from the source to the load when the network is purely resistive, and the source and load resistances are matched. If Zs = Rs +jXs, then power transfer is a maximum when Re(Zl) = Re(Zs). Im(Zl) = -Im(Zs). i.e. Zl and Zs should be complex conjugates. • One of the most important aspects of high frequency circuit design is the problem of impedance matching. This involves the design of a network inserted between source and load to provide maximum power transfer between them. • The matching networks must be theoretically lossless, so must comprise inductance and capacitance only. Matching networks are therefore frequencyselective, and provide matching over a certain bandwidth. This can be an advantage if we are trying to select a particular signal from the complete RF spectrum. • Matching networks are thus characterized by center frequency f0 and Q :. Q=. f0 ∆f. where ∆f is the bandwidth f2-f1, where f1 and f2 are the –3dB frequencies either side of f0. (p11).

(35) Transformer Networks Transformers are employed to change the effective source or load resistance. I1. V1. V2. I2. Rs Rl Vs. 1:T. •. ⎛V ⎞ V2 = T 2 ⎜⎜ 1 ⎟⎟ I2 ⎝ I1 ⎠. where T = turns ratio (V2/V1). thus Rl’ = (1/T2) Rl, where Rl’ is the effective load resistance seen by the source Rs. Matching occurs if T = (Rl/Rs )1/2.. • In many cases, the required value of inductance is not practical ; in a sense that coils with the required inductance may not be available with the required high values of Q0.. ⇒ We may use a transformer to effect an impedance change. • A tapped coil , known as an autotransformer can be used as shown in the following figure. Note that the tapped inductor is employed as an impedance transformer to allow using a higher inductance ( L’ ) and a smaller capacitance ( C’ ). (p10) (p12).

(36) • In applications that involve coupling the output of a tuned amplifier to the input of another amplifier, the tapped coil can be used to raise the effective input resistance of the latter amplifier stage as shown in the following figures.. • However the components are fairly bulky, and coils with low losses and winding resistance are expensive. Therefore LC matching networks are often preferred.. Two-element Matching Networks • Two-element matching networks are simple, but f0 and Q cannot be chosen independently. • To calculate the component values required for impedance matching, the following impedance transformations are useful :. (p13).

(37) Series. Parallel. Rs Xs. Rp. Xp. (i). Parallel → Series. Rs =. Rp 1 + Qp 2. Xs =. Xp. Qp =. where. 1 + 1 / Qp 2. Rp Xp. (i). Series → Parallel. (. Rp = Rs 1 + Qs 2. Xp = Xs (1 + 1 / Qs 2 ). ). where. Qs =. Xs Rs. • Notice that Qs and Qp are frequency dependent. Matching network :. R1. jX1. ~. jX2. R2. We must choose jX1 and jX2 such that the equivalent series impedance of ( R2 // jX2 ) = R1 – jX1, i.e. choose :. Qp 2 = (Rp / Rs − 1) = ( R 2 / R1 − 1). Qp =. R2 X2. (p14).

(38) Three-element Matching Networks • Three-element matching networks allow f0 and Q to be determined independently.. jX2. R1. jX1. jX3. R2. π-network jX1. R1. jX3. jX2. R2. T-network • We can consider the network as two 2-element networks back-to-back as follows :. jX2. R1. jX2. jX1. jX3. R2. for π-network. (p15).

(39) jX1. jX3. R1. jX2a. jX2b. R2. for T-network • The two halves of the network are designed to give complex conjugate ‘virtual impedance’ R +/- jX. • If the virtual resistance R < R1, R2 then we choose the π-network :. Qa = ( R1 / R − 1). Qb = ( R 2 / R − 1). The operating Q will be the maximum of the two. The value of Q is thus fixed by choosing the value of R ( or vice versa ! ). jX1, jX2 and jX3 are then determined from Qa and Qb. • If the virtual resistance R > R1, R2 then we choose the T-network :. Qa = ( R / R1 − 1). Qb = ( R / R 2 − 1). Again, the operating Q will be the maximum of these two values.. (p16).

(40) 6. Smith Chart • The Smith chart is an approach to represent series and parallel impedances, including very large and very small values. • The Smith chart also simplifies the process of designing impedance matching networks. • Lines of constant resistance are circles with their centers on the horizontal diameter. • Lines of constant reactance are arcs of circles with their centers on a vertical line running through the right-hand end of the horizontal diameter. • The Smith chart can be used to represent either series impedance or parallel admittances. (i). As a series impedance chart :. increasing series inductance. resistance. increasing series capacitance (ii). As a parallel admittance chart :. increasing parallel capacitance. conductance. increasing parallel inductance. (p17).

(41) Smith Chart Examples 1. Impedance Chart. • Adding a series resistor Z. Z0. • Adding a series inductor L. Z0. (p18).

(42) • Adding a series capacitor C. Z0. (p19).

(43) 2. Admittance Chart. • Adding a parallel resistor. G. Z0. • Adding a parallel inductor. L. Z0. (p20).

(44) • Adding a parallel capacitor. C. Z0. (p21).

(45) 7. Practical Design Issues RF circuit design is very much dependent upon the properties of individual components and circuit layout. Physical layout of devices and components on an PCB or on an IC requires great care if optimum performance is to be achieved. Strays and parasitics affect the performance of RF circuits. Although done almost intuitively by an experienced engineer, many of the problems are open to analysis in a fairly simply way. (i). Conducting track – • Tracks have resistance which increases at higher frequency due to the “skin effect”; most of the current tends to flow near the outer surface of the metal line at high frequency, it increases the effective ohmic resistance of a metal line. A skin depth. δ is given by. δ=. 2. µ ⋅σ ⋅ω. where δ is defined as the equivalent thickness of a hollow conductor shown below. µ is the magnetic permeability of the material. σ is the resistivity , and ω is the frequency of interest. δ. γ. • A loop of conductor has an associated inductance. Two loops will have mutual inductance.. (p22).

(46) • Signals can couple from one track onto another if the two tracks are running parallel. Therefore signal leads should not run parallel to other signal leads. (ii). Resistors – • Resistors have associated parasitics, and should be characterised at the frequency of operation. Resistors generally become reactive above 50 to 100MHz. • Wirewound resistors should never be used in RF circuits ( too inductive !! ). (iii). Capacitors – • A practical capacitor generally looks like a series RLC circuits. At high frequencies, the inductive reactance can dominate. Thus track lengths to capacitors must be kept as short as possible to minimise this effect. • Stray capacitance will occur whenever two conductors are separated by dielectric. (iv). Inductors• Discrete inductors are generally bulky or expensive. • Inductors are generally not available on ICs, because they require too much chip area. Specialised IC processes generally provide a few characterised inductor structures. (v). Packaging – • The effect of interconnect parasitics and cross-coupling are greatly reduced on an IC, where dimensions are much smaller. However at some point the signal must leave the chip. The bond wires to a package, and the leadframe itself, introduce significant parasitics which can often dominate the resulting system performance. • Pad capacitance is significant at high impedance inputs. • Bond wire inductance is significant at low impedance inputs.. (p23).

(47) • Package parasitics should be included in high-frequency simulation.. (p25).

(48) Lecture 3 – Low Noise Amplifier Design Part I - Noise fundamental • Noise can be defined as any undesirable disturbance that obscures or interferes with a desired signal. • In electrical systems, the existence of noise is due to the fact that electrical charge is carried in discrete amounts equal to the electron charge. • The instantaneous value of noise cannot be predicted, but the average ( rms ) value can be measured over a period of time. From the average value, we can predict the probability of a noise signal having a specific amplitude at a specific point in time. Much noise has a Gaussian distribution of amplitude with time.. f(x). x The Gaussian probability density function can be written :. ⎛ 1 f ( x) = ⎜ ⎝ σ 2π where. ⎛ − ( x − µ )2 ⎞ ⎞ ⎟ ⎟ exp⎜⎜ 2 ⎟ ⎠ ⎠ ⎝ 2σ. µ : mean ( average ) value σ : standard deviation. The total area under the curve must equal one, and generally the mean level µ is zero. To an excellent approximation, the amplitude of common electrical noise is within 3σ of the mean value µ.. (p1).

(49) Noise modeling : noise voltages and currents are shown directionless since they are constantly varying in amplitude and phase. noise voltage vn(t). noise current in(t). Since noise is completely random with zero mean value, calculations involve mean square values, which are measurements of the dissipated noise power. The root mean square ( rms ) value of a noise source is also an important value, since it measures the effective output noise voltage ( or current ). However remember that noise is not sinusoidal, so the value measured by a true rms voltmeter will be too low.. ⎛1 vn(rms) = ⎜ ⎝T. ∫. T. 0. 1/ 2. ⎛1 in(rms) = ⎜ ⎝T. ⎞ vn (t )dt ⎟ ⎠ 2. ∫. T. 0. 1/ 2. ⎞ in (t )dt ⎟ ⎠ 2. The rms value of a noise signal indicates the normalized noise power of the signal. IF the random signal vn(t) is applied to a 1Ω resistor, the average power dissipated equals the normalized noise power :. Pdissipated. 2 [ vn(rms )] =. 1Ω. = vn 2. instantaneous. mean square ( noise power ). rms ( noise voltage ). Noise voltage. vn(t). vn2. vn(rms). Noise current. in(t). in2. in(rms). (p2).

(50) Noise power spectral density Describes the noise content in a 1Hz bandwidth. It has units of V2/Hz and is denoted S(f) since it is often frequency-dependent. The spectral density of white noise is constant, since by definition there is equal noise power in every 1Hz of bandwidth.. Svn(f) white noise. vn2. frequency ∆f ( noise bandwidth ) The total mean square noise vn2 is found by integrating the spectral density function over the bandwidth of interest,. vn 2 = ∫ Svn( f )df For white noise, vn2 = Svn(f) fBW , where fBW is the system bandwidth in Hz. Noise bandwidth The equivalent noise bandwidth is defined as the frequency span of a noise power curve width an amplitude equal to the peak value, and with the same total integrated area.. S(f). B.W.. frequency. (p3).

(51) Main Sources of Noise in Electrical Circuits ( I ). Thermal Noise – • Thermal noise arises in any resistance and is due to the thermal fluctuations of free electrons. Thermal noise has been extensively studied, and the mean square value is given by Svn(f) = 4kTR V2/Hz. vn2 = 4kTR ∆f V2. where k is Boltzmann’s constant ( 1.38x10-23 J/K ). • The spectral density of thermal noise is flat and thus thermal noise is ‘white’. • In noise calculations, a resistor can be represented as an ideal noiseless component in series with a thermal noise voltage vn. By Norton’s theorem, this series arrangement can be replaced by an equivalent noise current generator in parallel with a resistance.. R ( noiseless ) in(rms) vn(rms). in2 = vn2 / R2 =4kT∆f /R. in(rms) = vn(rms) / R = ( 4kT∆f /R )1/2. kT/C Noise – • The expression vn2 = 4kTR ∆f predicts that an open circuit ( infinite resistance ) will generate an infinite noise voltage. In practice this is not the case as there is always some parallel capacitance.. (p4).

(52) vns 2 = 4kTRs∆f. Rs C. vno. vno 1 = vns 1 + j 2πfCRs. vns. Total mean square noise power is. vno 2 = ∫. ∞. 0. 2. ∞ 1 4kTRs kT 2 vns df = ∫ df = 0 1 + ( 2πfCRs ) 2 1 + j 2πfCRs C. • The output noise voltage at an open circuit thus depends only on the temperature and the shunt capacitance. • This noise limit is referred to as kT/C noise and is important in applications which use sample-and-hold circuitry.. ( II ).. Shot Noise –. • Shot noise is associated with current flow across a potential barrier such as a pn junction. Since the charge is carried by discrete particles, there is a continuous fluctuation in the instantaneous value of the current. The resulting noise current is Sin(f) = 2qIdc A2/Hz. in2 = 2qIdc∆f A2. where q is the electron charge ( 1.602 x10-19 ) and Idc is the dc current.. (p5).

(53) ( III ). Flicker Noise ( 1/f Noise ) – • Flicker noise is essentially a low-frequency phenomenon exhibited by almost all electronic devices, but its nature is still not properly understood. Flicker noise is associated with a flow of dc current, and in many cases is due to surface traps capturing and releasing electrons in a random fashion. • Flicker noise has the form Sif(f) = k ( Ia/fb ) A2/Hz. if2 = k ( Ia/fb ) ∆f A2. where typically a is between 0.5 and 2 , and b ≅ 1. k is an unknown constant which can vary by orders of magnitude across the same process. However, average values are usually predicted from multiple measurements. • Flicker noise can generally be ignored above 1kHz or so, when other noise contributions within the circuit become dominant.. ( IV ). Burst Noise – • Burst noise occurs in all semiconductor devices, and is seen as sudden changes in the average level of the already existing noise ( the noise of noise !! ). • If a noise signal is amplified and fed to loudspeakers the result is typically a ‘constant’ noise hiss plus a random crackle. • Burst noise is particularly common in monolithic IC amplifiers, although its origins are obscure and not well understood.. (p6).

(54) Noise Definitions, and Equivalent Noise Models Noise summation – • Consider two noise sources vn1(t) and vn2(t) whose mean square values are known. If we add the two noise sources in series, then the combined output is vnt(t) = vn1(t) + vn2(t) Thus the total mean square value is. vnt 2 =. 1 T. 2 ( ) vn 1 ( t ) + vn 2 ( t ) dt ∫0 T. which can be expanded to give. 2 T vnt = vn1 + vn2 + ∫ vn1(t )vn2(t )dt T 0 2. 2. 2. The first two terms are the mean square values of the two noise sources. The last term describes the correlation between the two noise sources. If the two sources are generated independently and there is no relationship between their instantaneous values, then they are said to be uncorrelated, and the third term is zero. In electrical circuits we will be mainly dealing with uncorrelated noise sources from various different components. The principle of superposition applies for uncorrelated sources ; the output response is the sum of the responses for each source acting alone, and with all other voltage sources short circuit and all other current sources open circuit. • For n uncorrelated noise sources, the mean square value of their total sum is vnt2 = vn12 + vn22 + vn32 ……..+vnn2. (p7).

(55) Notice that the rms value is vnt(rms) = ( vn12 + vn22 + vn32……..vnn2 )1/2 ≠ vn1(rms) +vn2(rms) +vn3(rms) ………+vnn(rms). Equivalent Noise Models – • A circuit ( such as an amplifier ) may contain many internal sources of noise. To simplify noise calculations, we transform all these internal noise voltages and currents into two equivalent noise sources at the input ( denoted vn and in ). The system itself is then noiseless. • A noise source is referred to the input by dividing it by the forward gain from the input to the noise source location.. Rs. vn. Rs. G vns. in. G veq. • If we know the source resistance, we can calculate a total equivalent input noise voltage, veq2 = vns2 +vn2 + in2Rs2 where vns2 is the thermal noise contributed by the source resistance Rs. • The total equivalent output noise of the system is vno2 = G veq2 In this example, G is the power gain of the system.. (p8).

(56) Noise Factor / Noise Figure – • Noise factor F is a parameter usually quoted for amplifying devices / systems, and is a measure of how much additional noise will be generated by the system. • Noise factor F is defined as a power ratio. F=. Total _ equivalent _ input _ noise _ power Input _ noise _ power _ due _ to _ the _ source _ only. G ⋅ veq 2 = G ⋅ vns 2 Thus F = ( veq2 / vns2 ) = 1 + ( vn2 + in2 Rs2 ) / vns2 • Ideally F = 1 , and the amplifying device adds no additional noise to the input signal.. Signal to Noise Ratio – • We can define SNRin = Received signal power / Received noise power = vsig2 / vns2 Similarly. SNRout = Output signal power / Output noise power = Gvsig2 / Gveq2 = vsig2 / veq2. Thus, F = SNRin / SNRout = veq2 / vns2. (p9).

(57) Noise Performance of Cascaded Networks -. vns. G1. G2. first stage. second stage. vnt. G1, G2 are power gains • Considering the first stage alone F1 = veq12 / vns2 Thus the equivalent input noise voltage veq12 = F1 vns2. • This equivalent input noise veq12 consists of the received ( source ) input noise and internal noise vn12 contributed by the first stage, therefore veq12 = vns2 +vn12 vn12 = veq12 – vns2 = ( F1 – 1 ) vns2 • Considering the second stage alone, vn22 = ( F2 – 1 ) vns2 • Combining both stages, the total output noise is vnt2 = G1G2vns2 + G1G2vn12 + G2vn22 = [ G1G2F1 + ( F2 –1 )G2] vns2 • Noise factor of the cascaded stages is F = vnt2 / ( G1G2vns2 ) = F1 + ( F2 – 1 ) / G1 • The noise factor of a cascaded network is therefore determined primarily by the first stage provided that the first stage gain is large.. (p10) (p10).

(58) Friis Equation – • For multistage system, we can employ Friis equation to calculate the noise figure of the system. The equation is described as follows For m stages,. NFtotal = 1 + ( NF1 − 1) +. NFm − 1 NF2 − 1 NF3 − 1 + ........ + + G1 G1 ⋅ G2 G1 ⋅ G2 ⋅ ⋅ ⋅ G( m −1). (p11).

(59) Part II – Low-Noise Amplifier ( LNA ) Design 1. Dilemma in LNA design -. Rs. Req Q1. Vin. ~. • We represent the input-referred noise of a bipolar transistor by a series resistor, Req and the noise factor is NF = 1+Req/RS . On the other hand, the input-referred noise voltage per unit bandwidth is given by. ⎛ ⎛ 1 ⎞ V ⎞ ⎟⎟ = 4 KT ⎜⎜ rb + T ⎟⎟ Vn 2 = 4 KT ⎜⎜ rb + 2gm ⎠ 2IC ⎠ ⎝ ⎝ Therefore, we have Req = rb + VT / 2IC. For NF = 2 dB, Req needs to be 29 ohms, which means one can obtain a reasonable noise performance provided 1.. Q1 must be relatively large,. 2.. Q1 must be biased at a high current.. These two requirements actually deteriorate the targets of low-power, small chip-area.. (p12).

(60) • The interface between the antenna and the LNA entails an issue – If we treat LNA is a voltage amplifier, we wish the input impedance is infinite whereas from the viewpoint of noise performance, it could lead to large noise. The solution could be employing a transformer to perform the impedance transformation to obtain minimum NF. However, the integrated transformer can be impractical for the state-of-the-art technology !! • Since the bandpass filter following the antenna is designed for 50-Ω impedance matching, the LNA is thus designed to have a 50-Ω resistive input impedance. If the source and load impedances seen by the filter deviate from 50 Ω significantly, then the characteristics of the filter may exhibit considerable loss and ripples. • As far as the quality of impedance matching, we can firstly consider “return loss”, which is defined as 20log⎪Γ⎪, where Γ is the reflection coefficient with respect to a source impedance R0. That is,. Γ=. Z in − R0 Z in + R0. Γ=. ∆R 2 R0 + ∆R. Let Zin = R0 +∆R yielding. For -15 to –20dB return loss in a 50-Ω system, ∆R ≈ 15 to 9 Ω, which is very common in today’s process. • The reverse isolation of LNA determines the amount of LO signal that leaks from the mixer to the antenna. The leakage is mainly from capacitive paths, substrate coupling, and bond wire coupling.. (p13).

(61) • In the presence of feedback paths from the output to the input, the circuit could become unstable. To explain this, let us check a popular approach to qualify the stability - Stern stability factor, defined as. 1 + ∆ − S11 − S 22 2. K=. 2. 2. 2 S 21 S12. where ∆ = S11S22 – S12S21. Notice that the circuit is unconditionally stable if K > 1 and ∆ < 1, which also means that stability improves as S12 decreases. In other words, if. the. reverse. isolation of the circuit increases, the stability improves. • Two techniques employed in traditional RF circuits to increase the reverse isolation, namely “neutralization” and “cascoding”.. Vcc. Vcc ZL. ZL. L1 Cµ. Vout. C1 Vin. Vout. Q1. Vb. Q2. Vin. Q1. parasitic. Where L1 and Cµ resonate at the frequency of interest. Alternatively, the feedback can be suppressed through the employment of a cascode configuration, but also increases noise.. (p14).

(62) 2. Input Matching • The input impedance match can be a very difficult task in the CMOS LNA design ! Here we explain why ? But first, let us study the present circuit configurations. 1. Resistive termination - a 50-Ω resistor is placed in parallel with the input, and a capacitive part of the input impedance is canceled by an external inductor as shown in the following figure.. The termination resistor, however generates noise as well. The key point here is that the circuit must provide a 50-Ω input resistance for impedance matching without generating the thermal noise from the 50-Ω resistor. 2. 1/gm termination – a common-gate stage designed to provide an input resistance of 50 Ω can also be employed for the impedance matching.. In other words, we let 1/(gm+gmb) = 50 Ω. Once again, the input capacitance may be tuned out by a external inductor.. (p15).

(63) The main drawback of this approach is that the transconductance of the input transistor cannot be arbitrarily high. That is, it imposes a lower bound on the noise figure. For CMOS amplifiers, the noise factor can be described,. NF = 1 +. γ 5 ≥ = 2.2dB α 3. where α = gm / gd0 , γ is the coefficient of the channel thermal noise. gm is the device transconductance. gd0 is the zero-biased drain conductance. γ =2/3 and α = 1 for long-channel devices. In practical case, the noise performance can be worse since short-channel effect leads to α ≤ 1 and γ ≥ 2/3 due to excess thermal noise from the effect of hot electrons. 3. Shunt-series feedback resistive termination – The basic idea here is to employ output voltage signal and input current signal feedback topology to provide a low input impedance, generating a 50-Ω input resistance for impedance matching.. Two problems raise with the topology. Firstly, the feedback signal can contain substantial noise in turn raise the noise figure to high levels. Secondly, since it is a feedback circuit, stability can be a problem, which needs to be carefully design.. (p16).

(64) 4. Inductive degenerative – The idea is to employ L,C elements instead of physical resistors to avoid generating thermal noise. Let us take a look at the following figure.. Here we use two inductors which placed at the gate and source terminals. Note that the required capacitance is provided by the gate-to-source parasitic capacitor of the transistor. The input impedance is. Z in = s ( Ls + Lg ) + (1 / sC gs ) + ( g m1 / C gs ) Ls At the resonance frequency. ω0 =. 1 , Z = ( g / C )L (Ls + Lg )Cgs in m1 gs s. Thus, proper choice of gm1 , Cgs , Ls yields a 50-Ω real part.. (p17).

(65) 3. Bipolar LNA’s • Let us firstly check a simple common-emitter stage shown in the following figure. Where Q2 and I1 define the bias current for Q1 and R1 isolates the signal path from the noise of Q2. R1 and R2 keep the same current flowing through to provide base current for Q1.. Vcc. Vcc. RC Q1. RS Vin. I1 R1. R2. Q2. C1. ~. • From our previous discussion, we know that Q1 must be a large device biased at a relatively high current. If we take into account the parasitic capacitance, the effect is going to be amplifying the noise of Q1. ⇒. For a particular choice of the size and bias current of Q1, the noise figure may reach a minimum.. • The base thermal noise as a current source can be described as : I n 2 = 4kT and the total input-referred noise voltage including Rs is given by. Vtot. 2. IC / β 2VT. 2 ⎛ g m Rs ⎞ 1 ⎟ = 4kT ⎜⎜ Rs + rb + + 2gm 2 β ⎟⎠ ⎝. The noise figure is therefore equal to 2. V r g R 1 NF = tot = 1 + b + + m s Rs 2 g m Rs 2β 4kTRs. (p18).

(66) 4. CMOS LNA’s While MOSFETs were considered noisy devices, scaling technologies have dramatically improved their performance. Basically, MOSFETs exhibit only one primary noise source, that generated in the channel instead of base resistance for the bipolar transistors. Notice that the distributed gate resistance of MOS devices also contribute thermal noise whereas it can be minimised by laying out the transistor as a parallel combination of many narrower devices. 1. Common-source configuration -. VDD VDD. current source Vb. RD. M2. Vout Vin. M1. C1. parasitic. Vout Vin. M1. C1. parasitic. Due to the low transconductance of MOSFETs, the voltage gain of this circuit is relatively low, which makes the noise contribution from RD and the following stage quite significant. An alternative can be the replacement of the load with a current source to increase the voltage gain as shown in the right figure.. (p19).

(67) VDD However, the current source itself also. M2. contributes noise. We may postulate that. Vout Vin. Remember. reduced if M2 also carries out signal amplification. The idea was illustrated as. M1. that. the overall input-referred noise can be. shown on the left figure.. the. VDD. inductive. degeneration input impedance matching. LD. could be an excellent topology since the input resistance is from the combination of. capacitors. and. inductors. Vout. which. Vb. M2. avoids employing resistors. The circuit is. Rs. shown in the right figure. Notice that the inductors can be on-chip or off-chip. L1. Vin. and Ls and Cgs of M1 provide conjugate matching at the input. LD provides. L1 M1 Ls. voltage gain. Note that there will be a large voltage drop if a load resistor is employed. The common-gate transistor M2 increases the reverse isolation of the LNA to lower the LO leakage produced by the following mixer. The first stage gain can be described as G = m. 1 / sC gs1 io v gs1 ⋅ = g m1Qin = g m1 ⋅ v gs1 vin Rs + Z in. = g m1 /(ω0C gs ( Rs + ωt Ls )) = ωt /(2ω0 Rs ). ( s(Ls+L1) = 1/sCgs at resonant frequency ! ). Where Zin = ωt Ls and gm1 / Cgs = ωt and Qin is the effective Q of the input amplifier circuit.. (p20).

(68) Lecture 4 – Mixer Design • Mixer is a kind of circuits in which two signals are “mixed” to produce desired sum or difference frequencies. • Any nonlinear device can serve as a mixer – nonlinearity is required for the production of frequencies not present in the input. Thus mixers may utilize diodes, bipolar transistors, and FETs. • Frequency multipliers rely on the nonlinear characteristic of a device to perform the process : xout = f ( xin ). Since f(x) is a nonlinear function, it can be expanded as a power series, f(x) = a0+a1x+a2x2+a3x3+….. • The n’th harmonic of x is generated by the n’th power of x in the equation. For instance, if x is a single-tone signal represented by (Acosωt), in which A is the amplitude and ω is the fundamental frequency. The square of x generates the second harmonic (cos2ωt) as follow. A2 x = ( A cos ωt ) = (1 + cos 2ωt ) 2 2. 2. • If the index ‘n’ is an even number, the n’th power of x also generates evenorder harmonics ( including the DC term ) with lower order than ‘n’. If the index ‘n’ is an odd number, the n’th power of x generates other odd-order harmonics ( including the fundamental term ) with lower order than n. For instance, the cube of x generates the fundamental (cosωt) and third-order (cos3ωt) harmonics as follow. 3 A3 A3 x = ( A cos ωt ) = cos ωt + cos 3ωt 4 4 3. 3. (p1).

(69) • All the even-power terms in the first equation of the last page generate DC component. For instance, if x is a single-tone signal represented by (Acosωt), the square of x generates the DC component ( A2/2 ) and the fourth power of x generates the DC component ( 3A4/8 ). • All the odd-power terms in the first equation lead to gain compression or expansion. For instance, if x is a single-tone signal represented by (Acosωt), the cube of x generates (3A3/4)cosωt, which is added directly to the fundamental harmonic generated by the first-power term. As a result, the fundamental harmonic in the output signal is given by. 3 a1 A cos ωt + a3 A3 cos ωt 4 If a3 has opposite phase from a1, the (a3x3) term causes gain compression. On the other hand, if a3 has the same phase as a1, the (a3x3) term causes gain expansion. • If the input comprises two signals to be multiplied, say xin = xa + xb, then the square term in the output will generate the product of the two signals. Many other terms are also generated ; these other mixing products must be removed since they can be considered as distortion terms. • The following figure illustrates a simple mixer that consists of a nonlinear device with two input voltages v1(t) and v2(t) of different frequencies f1 and f2, respectively. If the device were perfectly linear, the output voltage or current would be containing frequencies f1 and f2 only.. _. v1(t). ~ +. +. ~. _. vi(t). Nonlinear device. io(t). v2(t). (p2).

(70) • Mixing is usually performed to make the information signal more suited to the transmitting medium. The mixing process must be one which preserves the original information. If the input signal v1(t) = Asin(ω1t + ϕ1) and v2(t) = Bsin (ω2t + ϕ2) , then the output signal contains the sum and difference frequencies,. v1 (t ) ⋅ v2 (t ) = AB sin (ω1t + ϕ1 )sin (ω 2t + ϕ 2 ) =. AB (cos((ω1 + ω 2 )t + (ϕ1 + ϕ 2 )) − cos((ω1 − ω 2 )t + (ϕ1 − ϕ 2 ))) 2. • Usually a filter is employed to select one of these components, while rejecting the other. If the resulting output signal is at a higher frequency than. the. input. signal,. the. process. is. known. as. upconversion.. Downconversion occurs when the output frequency is lower than the input frequency.. • Bipolar mixers -. Ic1 +. Q1. Ic2 Q2. Va _ Iq. (p3).

(71) • The differential pair is an example of a simple multiplier since, Iout = Ic1 – Ic2 = Iq tanh ( Va / 2 Vt ). G=. Multiplier gain. ∂I out = (I q / 2Vt )sech 2 (Va / 2Vt ) ∂Va. • Notice that the gain is DEPENDENT on the input signal magnitude. For a gain linearity of at least 99%, sech2(Va/2Vt ) ≥ 0.99 or Va < ± 5.2mV. • If Va > 5mV, then the output harmonics will cause significant distortion. If we restrict operation to small signals only ; Iout = ( Iq/2Vt ) Va ( tanh(x) ≅ x , when x is very small ) • A second input signal can be introduced through Iq as shown in the figure below,. Ic1 +. Q1. Ic2 Q2. Va _ Vdc Vb. Iq. • Suppose Ic3 = Iq + gmVb, where Iq is a constant bias and Vb is a small signal input voltage, then. I c1 − I c 2 ≅. (I. q. + g mVb )Va 2Vt. =. Iq gm VaVb + Va = KVaVb + GVa 2Vt 2Vt. Where K is the multiplier gain = gm / 2Vt and G = Iq / 2Vt. (p4).

(72) • In general, the amplified output GVa is an order of magnitude larger than the multiplied output KVaVb. If the circuit is being employed to carry out frequency conversion, the amplified signal is generally outside the frequency band of interest and so can be rejected by filtering. However, if this amplified output signal is large it may saturate the multiplier output. • The amplified output ( GVa ) can be rejected by using a balanced configuration as shown below,. Io1. +. Q1. Io2. Q2. Q3. Q4. Va _ Iq1. Iq2. I o1 − I o 2 = (I c1 − I c 2 ) + (I c 3 − I c 4 ). = I q1 tanh (Va / 2Vt ) + I q 2 tanh (− Va / 2Vt ). = (I q1 − I q 2 ) tanh (Va / 2Vt ) = (I q1 − I q 2 )(Va / 2Vt ). ( if Va < 5 mV ). (p5).

(73) • If the lower current sources are driven differentially with a small signal input Vb, then. I q1 = I q + g mVb. and. I q 2 = I q − g mVb. I out = I o1 − I o 2 = 2 g mVb (Va / 2Vt ) = VaVb ( g m / Vt ) • To generate the balanced currents Iq1 and Iq2 ,. Io1. +. Q1. Io2. Q2. Q3. Q4. Va _ + Vb _. Q5. Q6. Iq. I q1 = I c 5 = (1 + x ). Iq 2. I q 2 = I c 6 = (1 − x ). Iq 2. where x = tanh ( Vb / 2Vt ). (p6).

(74) • For small input signals ( Vb << Vt ) then x ≅ Vb / 2Vt Iq1 = ( Iq + gmVb )/2 , Iq2 = ( Iq – gmVb )/2 where gm = Iq / 2Vt Therefore Iout ≅ ( Iq / 4Vt2 )VaVb This kind of configuration is known as the “double-balanced” or “Gilbert” multiplier.. (p7).

(75) Terms to Describe Mixer Performance • Conversion Gain – the ratio of the output ( IF ) signal power to the ( RF ) input signal power. • Noise Figure – the SNR at the input ( RF ) port divided by the SNR at the output ( IF ) port. • Isolation – the amount of “leakage” or “feedthrough” between the mixer ports. e.g. “fLO at RF port isolation” is the amount the fLO signal is attenuated when measured at the RF port. • Conversion Compression – the RF input power level above which the curve of IF output power versus RF input power deviates from linearity. Over this level, further increases in RF input level do not result in proportional increases in output level. Typically, the input level at which the compression is 1 or 3 dB is given in mixer specifications.. IF output level in dBm. 3dB. 3 dB. RF input level in dBm. ( Compression level ). (p8).

(76) • Passive and Active Mixers – “passive” mixers usually refer to the mixers which does not provide any gain. However, “active” mixers ( Gilbert multipliers ) always provide significant gain with proper choice of device size and bias currents. The following figures show these configurations.. VDD VLO. R1. R2 VIF. VRF. M1. VIF RL. M1. M2. VLO VRF. M3. • In addition, according to Friis equation, active mixers can reduce the noise contributed by subsequent stages, thus they are widely used in RF systems. • Passive mixers, however typically achieve a higher linearity and speed and used in higher frequency applications such as microwave circuits. • SSB and DSB Noise Figures – Considering the image-frequency problem of a heterodyne receiver. Basically, we call the signal in the system is “singlesideband” signal since the desired signal spectrum resides on only one side of the LO frequency. Let’s take a look at the noise figure for the system. Here we obtain the thermal noise of Rs ( source resistance ) in both the signal band and the image band. Upon downconversion, both the noise in the signal band and the noise in the image band are translated to the IF band.. (p9).

(77) ωLO. ωLO. ωIF. 0. By contrast, for homodyne receiver the signal band ( double sideband ) is located at the centre of the local oscillator frequency. Upon downconversion, only the noise at the signal band is translated to DC. We therefore, reach a conclusion that the SSB noise figure of a mixer is 3 dB higher than the DSB noise figure. • Port-to-Port Isolation – 1. LO-RF feedthrough – results in LO leakage to the LNA. 2. LO-IF feedthrough – the following stage may be desensitized. 3. RF-IF feedthrough – “unconverted” problem ( RF frequency appears at the IF band which results in even-order problem ). • Single-Balanced and Double-Balanced Mixers – 1. “Single-balanced” – differential LO signal and single-ended RF signal. 2. “Double-balanced” – both differential LO and RF inputs. ( Gilbert cell ) Notice that double-balanced configurations generate less even-order distortion.. (p10).

(78) • Spurious Response – Generally speaking, a mixer can generate various crossproducts of the RF and LO signals as well as their harmonics. The frequency of the resulting components can be expressed as | mωRF ± nωLO | . Thus, in the mixer design, a difficult task is to ensure that, except | ωLO - ωRF |, such components do not fall in the IF band. • Nonlinearity – The main source of distortion in active mixers is the nonlinearity of THE INPUT VOLTAGE-TO-CURRENT CONVERTER. In addition, the nonlinearity also stems from the base-emitter diffusion capacitance of the differential pair since in practice, the differential pair is not close abruptly – the two transistors are simultaneously on for a fraction of the period. Of course, one can increase the slew rate to lower the effect. • Schmook’s technique for linearization – ( This technique is employed for double-balanced mixers ( Gilbert cell )). A bipolar differential pair employs two transistors with different emitter areas, A and nA which is shown on the upper part of the next page. Its input-output characteristic is then shifted by VTlnn as shown on the right figure. Now if two such differential pairs are cross connected as shown on the lower part of the bottom figures of the next page, then the overall characteristic tends to be “smoother”. ( for more detailed, see pp407-411, Dec. 1975, IEEE JSSC ). (p11).

(79) Iout Iout +. A. nA. Vin _. Vin. Iout nA +. Iout A. A. Vin _. nA Vin. • Bipolar Mixers – Single-balanced active bipolar mixers can be implemented as shown on the top of next page. Here the emitter degeneration is employed to achieve higher linearity. The RF input varies the collector current of Q1 and the switching operation of the differential pair multiplies this variation by a square wave.. (p12).

(80) VCC. VCC. RC. RC. RC. RC. VIF +. VIF. Q2. +. Q3. VLO _. Q2. VLO _ RS. Q1 Q1. VRF. Q3. ~. RE. RS VRF. RE. Vb. ~. • The common-emitter configuration has a higher input impedance, which is not suited to heterodyne receivers where the image-reject filter must be terminated with a 50-Ω impedance. • By contrast, for the common-base configuration, RE+1/gm is chosen to be about 50- Ω. • Main source of distortion here – (1). The nonlinearity of the input voltage-to-current conversion as happened in the network consisting of Q1, RE, and RS in both figures. (2). The variation of the base-emitter diffusion capacitance of the differential pair to the collector of Q1. This stems from the simultaneously on for a fraction of the period. During the period, the base-emitter capacitance of the differential pair acts as a nonlinear shunt for the collector current of Q1, which distorts the collector currents of the differential pair. ⇒ increase the LO slew rate or decrease the base resistance of the differential pair.. (p13).

(81) • CMOS Mixers – Active CMOS mixers can also be implemented as the single-balanced and double-balanced. configurations.. In. addition,. most. of. the. design. considerations for bipolar mixers apply to the CMOS mixers as well. However, for CMOS switching pair, typically we need greater swings to experience complete switching than does a bipolar counterpart. Therefore, we shall know that some of the RF current generated by the current source is wasted, which lowers the conversion gain. CMOS mixers can also be realized in passive form as shown in the following figure. Where the switching devices M1 and M2 are driven by complementary phases of the LO. Advantages over the active form 1. Achieve higher linearity if M1 and M2. VLO. experience overdrive voltage in the on state. 2. No power from the supply voltage needed. M1 Drawbacks – 1. Since the gain is less than unity, the stage. VRF. VIF M2. cannot reduce system noise. 2. Large devices needed for M1 and M2 ( to reduce on-resistance ), which increases the. VLO. capacitive feedthrough from LO to IF.. (p14).

(82) Lecture 5 – Oscillator Design The requirements for an oscillator – 1. Low phase noise 2. Wide tuning range ( to allow detection of more channels ) However, these two requirements are contradictory !!. Phase Noise – . Consider an oscillator producing an ideal sinusoid at a frequency ω0= 2πf0 = 2π/T0 . The output waveform can be expressed as Vout = V0cos ω0t and the spectrum consists of two impulses at ω = ±ω0 as shown in figure(a).. . Also, the period is constant and the zero-crossing points occur at EXACTLY integer multiples of T0.. . In actual, however oscillator circuit and system noise varies the period of oscillation randomly as shown in figure(b), which reveals the signal carries a finite energy at ω0+∆ω.. (p1).