Circuits, Devices, Networks, and Microelectronics CHAPTER 20. NOISE ANALYSIS and LOW NOISE DESIGN 20.1 THE ORIGINS OF NOISE Electrical noise is a background “grass” of unwanted signals, usually due to thermal origins. It has a nearly constant amplitude density across the frequency spectrum that tends to mask and obscure the waveforms and information which we wish for our circuits to process. Noise is an inescapable fact of circuits and signals. It is generated in most part by thermal fluctuations in the motion and flow of charge. It is an important factor in the design and analysis of communication circuits, and therefore most treatments of noise are developed in the context of communications electronics. But electrical noise, and its companion problem, distortion, are an important and necessary consideration of any circuit, in which a signal, whether of linear or logic form, is to be processed. A figure of merit that defines a circuit in terms of its signal transfer properties is the dynamic range (DR) given by The smallest usable signal level is defined by the noise limit. The largest usable signal is defined by the distortion limit, which is usually a consequence of the compliance (±VS) limits of the circuit. In matters of electrical noise, components and devices are defined by thermal kinetics. Thermal statistical fluctuations will produce a random set of signals within an electrical component. Thermal effects manifest themselves as fluctuations in electrical currents. The basic unit of thermal energy (fluctuation) is given by w = kT (defined by the fugacity for electrons), where k = Boltzmann’s constant and T = absolute temperature. Translate that into the fluctuations in power and p w B kT B (20.1-2a) where B = bandwidth of the frequencies passed by the circuit. Since power usually relates to signal through resistance then the (rms) noise thermal power is 2 2 p en R in G (20.1-2b) where R = resistance and G = conductance. These give noise signals in terms of en = noise voltage and in = noise current. The maximum signal that can be passed from the source, = the resistance itself, (according to maximum power transfer) is then 511 Circuits, Devices, Networks, and Microelectronics 2 en 4kTRB (20.1-3a) or 2 in 4kTGB (20.1-3a) Equation (20.1-3) is called Johnson-Nyquist’s theorem. Note that the measure is in terms of a mean square noise voltage (or current). Consider the following example: EXAMPLE 20.1-1: What is the amplitude en of the thermal noise associated with a resistance R = 1k at a noise bandwidth B = 1 MHz? kT SOLUTION: e 2 4kTRB 4 q R B n q = 4 × .025 × (1.602 × 10 -19) × 103 × 106 s-1 = (1.602 × 10-20) × 109 V2 = 16 × 10-12 V2 The calculation assumes that the circuit is at room temperature (~300K) for which kT/q = VT (= thermal voltage) .025V. The 4kT term has value 4kT = 1.6 × 10-20 (room temperature) and this value may be adopted as a shortcut default for most situations involving noise analysis. Taking the square root gives noise amplitude en = 4V Equations (20.1-3a) and (20.1-3b) may be represented by the network equivalents of figure 20.1-1 Figure 20.1-1: Equivalent circuits for thermal noise. This type of noise is also called Johnson noise, or white (full spectrum) noise since it is spread uniformly across all frequencies of the spectrum. It is also called thermal noise since it is a result of thermal statistics. 512 Circuits, Devices, Networks, and Microelectronics Even though each resistance contributes noise as its own little noise source and the network can be analyzed as if it were a collection of noise sources and resistances, this would be a challenging and totally useless exercise. The only place that it finds merit is in circuit simulation, where a part-by-part analysis might be appropriate. Use of Thevenin theorem to reduce the network to a single source and resistance works just fine. The only reason that the separated analysis might be of benefit is for sensitivity analysis of each resistance, and that is a task that is more appropriate for circuit simulation software. Assessment of a network with both resistances and reactive components implies the need for an expanded form of equation (20.1-3). This expanded form is the Nyquist formula and is given by e 2 2kT R( f )df 4kT R( f )df (20.1-4) n 0 where R( f ) is the real part of the impedance Z( f ), as seen across the output nodes. The Nyquist formula can also be written as e 2 4kTR G( f )df 4kTRB n 0 2 2 for which G( f ) is the (frequency dependent) power gain = vout /vin , assuming the resistance term R is separable. And it gives a clear definition of the noise bandwidth. Consider the following example: EXAMPLE 20.1-2: Consider the single-time constant RC network shown by figure 20.1-2. Using the 2 Nyquist formula find (a) noise signal en and (b) noise bandwidth B. Figure 20.1-2: Single-time constant RC network. 1 R SOLUTION: The impedance looking into the output is Z G sC 1 jRC R and its real component is ReZ( f ) R( f ) 1 2 R 2C 2 513 Circuits, Devices, Networks, and Microelectronics The Nyquist formula then gives R 4kT RCd 4kTR 1 1 e 2 4kT df 4kTR n 2 2 2 2C 2 2 2 2 RC 2 2RC 2 0 1 R C 0 1 R C and the noise bandwidth is B f3dB 2 Note that the noise bandwidth is greater than the (STC) 3dB bandwidth by the factor /2. Extending this outcome as an approximation to all circuit networks is not unreasonable and so noise bandwidth will then be expected to relate to the circuit (3dB) bandwidth as: 1 Noise bandwidth = B f (20.1-5) 2 3dB 4 3dB 20.2 SYSTEM NOISE ANALYSIS Noise is a form of background signal. In this respect it should be expected that the detector/transducer at the front end of the circuit is the major source of the noise, in terms of some sort of ‘dark current’ signal, with characteristics as might be represented by the source resistance RS. But it is also true that the components within the amplifier/interface circuit also contribute noise, and it is the assessment of the system noise factors that is necessary to identifying the noise characteristics of the system. The quantitative comparison between carrier signal and noise signal is defined by the signal-noise ratio (20.2-1) Both signal S and noise N are amplified by the gain factor G of the amplifier. But the noise NO at the output is greater than the amplified input noise = G × Ni, courtesy of the additional noise contributed by the components within the amplifier, i.e. N O GN i N Oa GN i GN a G(N i N a ) (20.2-2) 514 Circuits, Devices, Networks, and Microelectronics where NOa = additional noise at the output = GNa. It is as if an additional noise source Na exists at the input. The sum (Ni + Na) is then an equivalent noise input, and it gives a means to define the principal noise characteristic for the system ≡ the noise factor F, identified by the ratio Ni N a N a F 1 (20.2-3) Ni Ni Take note that F is always greater than unity. Equation (20.2-3) can be restated in terms of the power gain G = SO/Si, where SO and Si represent the signal levels at output and input, respectively of the system, i.e. NO NO Si Ni F (20.2-4) GNi SO Si Ni SO NO i.e. the noise factor also represents the degradation of the S/N due to the system. It is important to note that the signal-to-noise and noise factor are ratio relationships of signal power. The maximum signal power transferred and consequently the maximum available noise power per bandwidth occurs when Rin = RS, for which 2 en 4kTRS B N i kTB (20.2-5) 4RS 4RS and N kT B T F 1 a 1 e 1 e (20.2-6) kTB kTB T Equation (20.2-6) shows that the noise character of a system has a corollary parameter in the form of an equivalent noise temperature Te for which Te (F 1) T (20.2-7) EXAMPLE 20.2-1: Noise factor F = 2.5 corresponds to Te (2.5 1) 300 = 450K (The nominal circuit temperature T is assumed to be = 300K.) 515 Circuits, Devices, Networks, and Microelectronics Since the ultimate interest is the dynamic range of the system, a lower limit, or smallest usable signal, must be identified. This limit is defined as the noise floor Nf = Si(min) needed to achieve a given output S/N. A corollary to this definition is the minimum detectible signal, which is the input signal voltage 2 vi(min) associated with the noise floor. Assuming that Si vi RS , the minimum detectible signal is then vi (min) 4RS Si (min) (20.2-8) From the definition (20.2-4) of noise factor in terms of S/N ratios, and assuming the systems are matched, (for which Rin = RS) then SO SO N f Si (min) Ni F kTB F (20.2-9) NO NO EXAMPLE 20.2-2: What is the noise floor and minimum detectible signal for a system with bandwidth 20kHz, noise factor 8dB, input resistance Rin = 50 if the output signal/noise requirement is 10dB.