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4.3: Noise

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    46118
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    Fluctuations of voltage and current arise from several different physical processes yielding noise with various statistical properties. Various types of noise are important in electronic circuits and these types range from noise, such as thermal noise, that is very well understood, to noise that has been observed and seriously affect the performance of RF circuits but not well understood. An example of the latter is phase noise on oscillators which manifests itself as random fluctuations of the phase or frequency of an oscillation signal. One of the problems in understanding noise is that it can be difficult to describe a particular type of noise in both the time and frequency domains. Not all types of noise can be described in a straightforward way and the sources of some types of noise are not understood. The impact of noise on an RF system is described in the frequency domain whereas the physical origins of noise must necessarily be in the real world (i.e., the time domain). The impact of noise is also in the time domain resulting in bit errors of a received, demodulated, and processed digitally modulated signal. Noise originating from a noise source is shaped by the characteristics of a circuit before it is observed externally and thus the true nature of noise is further obscured.

    4.2.1 Observations of Noise Spectra

    Noise in electronics is attributed to the random movement of carriers and most types of noise have a power spectral density that is flat with respect to frequency. Such noise is called white noise and if it is filtered, lowpass filtered or bandpass filtered, it is called additive white Gaussian noise (AWGN) as the noise then has an asumed Gaussian statistical distribution. The best understood noise is thermal noise and is attributed to the random movement of electrons due to the random vibration of the lattice of a conducting material. The theory of thermal noise is based on the fluctuation-dissipation theorem [1] which can be used for most materials in thermal equilibrium. This theorem applies both to classical and quantum mechanical systems and describes the classical noise encountered up to several terahertz as well as the quantum-mechanical effects that shape noise above a few terahertz at room temperature or at much lower frequencies as temperature approaches absolute zero [2].

    The fluctuation-dissipation theorem relates thermally-induced fluctuations in a material to the resistance of the material. The physical origin of thermal noise is the net effect of the rapidly-fluctuating currents resulting from thermal fluctuations of free electrons in a resistive (or conductive) material. The fluctuations of the electrons result from the vibrations of the atoms in a material’s lattice. These vibrations relate directly to temperature and temperature is regarded as a direct measure of the state of entropy or random vibration of the lattice. So the fluctuation-dissipation theorem describes how vibration (i.e. temperature) of the lattice induces small current fluctuations in a material and thus describes how heat energy is converted into the electrical energy known as Johnson noise. This is the opposite of the effect of resistance which converts electrical energy into heat energy when the movement of electrons (as current) causes the lattice to vibrate more. So it is not surprising that the noise current is directly related to resistance.

    Key results of the fluctuation-dissipation theorem are that the available noise power from a resistor or group of resistors is linearly proportional to temperature, the available noise power is independent of the resistor value, and that the power spectral density (in watts per hertz) is independent of frequency, i.e. it is white, up to a few terahertz at room temperature.

    Consider the block of resistive material in Figure \(\PageIndex{1}\)(a). Lattice vibrations cause random movement of electrons and thus there is a myriad of tiny current sources. The block of resistive material has terminals and at those terminals a resistance can be measured. Application of the fluctuation-dissipation theorem determines that the net effect of the little noise-current sources in the material is the same as that of a current source in parallel

    clipboard_e0c04aefbc6bf284cc2e01f6f552a8bce.png

    Figure \(\PageIndex{1}\): Noisy resistive networks

    with the effective resistance as shown in Figure \(\PageIndex{1}\)(a). The noise current source, \(i_{n}(t)\), has statistical properties such that the available noise power is proportional to the bandwidth over which the noise power is measured and to the temperature of the material in kelvin. If the resistive material is at a temperature \(T_{0}\), then the noise temperature of the material \(T_{0}\). Going one step further, it can be imagined that the resistive material can be described by a network of resistors as shown in Figure \(\PageIndex{1}\)(b). If \(R_{S}\) and each resistor in the network is at the same temperature \(T_{0}\), then the whole network, including \(R_{S}\), is equivalent both in terms of resistance and noise to a single noisy resistance \(R_{x}\), that is at \(T_{0}\). So the noise temperature of the resistive network is equal to that of \(R_{x}\) both being \(T_{0}\). This is true provided that there are no non-thermal sources of noise inside the two-port network. For example, if there were transistors then an additional source of noise is shot noise. Then the noise temperature looking into port \(2\) of the network would be greater than the noise temperature of the source resistance at port \(1\).

    In Figure \(\PageIndex{1}\)(c) a noisy resistance \(R_{S}\) has an available noise power and all of that available noise power is delivered to the load \(R_{L}\) provided that \(R_{L} = R_{S}\). (All of the available noise power from \(R_{L}\) will also be delivered to \(R_{S}\) under matched conditions.) Even if a resistor is not loaded, so \(R_{S}\) is on its own, there will be noise power in the resistor which will be constantly generated and reabsorbed (through resistive heating) so that the resistor is in thermal equilibrium.

    A noisy resistor is equivalent both electrically and from a noise perspective to a noise-free resistor with a shunt noise current source as in Figure \(\PageIndex{1}\)(a) or equivalently as the same value of resistance with a noise voltage source (which will be introduced latter). The noise voltage and current sources are random and if the noise is lowpass or bandpass filtered as in Figure \(\PageIndex{1}\)(d), the resulting noise voltage \(v_{nL}(t)\) across the noise-free load will have Gaussian statistics. Since it is inevitable that noise will be filtered in a circuit, e.g. there will be at least parasitic capacitances, thermal noise is often treated as being additive white Gaussian noise (AWGN) as its statistical properties will be Gaussian. This is indeed fortuitous as it is possible to greatly simplify the treatment of noise if it can be consider to be random with Gaussian statistics. This is exploited in the development of the mathematics of random processes in Appendix 1.A of [3] as it applies to both noise and

    clipboard_ec324d1c34f1ae9a8be815a984609abad.png

    Figure \(\PageIndex{2}\): Attenuator example.

    digitally modulated signals. The key result is that noise can be described in the frequency domain and this understanding and characterization can be translated to the real world, i.e. the time domain.

    As an example, it is possible to undertake a frequency domain analysis of the circuit shown in Figure \(\PageIndex{2}\)(a) where a source \(R_{S}\) is connected to a resistive attenuator and then to a noise-free load \(R_{L}\). As usual the matched resistances external to the two-port are the same here so that \(R_{S} = R_{L}\). The network of Figure \(\PageIndex{2}\)(a) can be replaced by a network of noise-free resistors with noise voltage sources \(v_{nS},\: v_{n1},\: v_{n2},\) and \(v_{n3}\). The noise voltage sources are random and independent and so are uncorrelated. Thus to evaluate the noise current \(i_{nL}\) in \(R_{L}\) the powers of the individual contributions to \(i_{nL}\) need to be summed first. This can be shown to be identical to calculating \(i_{nL}\) in Figure \(\PageIndex{2}\)(c) where the equivalent resistance \(R_{x}\) of the network is found and then the noise voltage source, \(v_{nx}\) for that resistor used. Of course since the two-port is an attenuator \(R_{x} = R_{S} = R_{L}\).

    4.2.2 Characterization of Thermal Noise

    While the impact of noise on RF circuits is measured and categorized in the frequency domain, the physical sources of noise are in the real world. Noise in a conductor is manifested as random fluctuations in time of voltage and current. While random, the noise can have different statistics depending on how it originates. The three major physical sources of noise affecting electronic circuits are thermal, shot, and flicker.

    Thermal noise is more formally known as Johnson-Nyquist noise and is also referred to as Johnson noise or Nyquist noise. The noise is due to random fluctuations of charge carriers inside a conductor occurring with or without applied voltage or current. A resistor at room temperature (\(290– 298\text{ K}\) or \(19–25^{\circ}\text{C}\)) has an available noise power of \(−174\text{ dBm}\) in \(1\) hertz of bandwidth. The noise is uncorrelated so that the noise power in a second hertz of bandwidth will add. So in \(2\) hertz of bandwidth the available noise power is \((−174 + 3)\text{ dBm} = −171\text{ dBm}\).

    The extent of fluctuation is linearly proportional to absolute temperature. Also the noise power generated is independent of the resistance of the conductor. The original derivation of thermal noise is due to Nyquist [4], who showed that the power spectral density (PSD) of the available noise power from a resistor (of any value) is

    \[\label{eq:1}S_{t}(f)=kT \]

    where \(k\) is the Boltzmann constant and \(T\) is the temperature in kelvin. The subscript \(t\) here is used to indicate thermal noise. The SI units of \(S_{t}\) are watts per hertz but is more commonly expressed as \(\text{dBm}\) per hertz. The available noise power in a bandwidth \(B\) is (in units of watts)

    \[\label{eq:2}P_{t}(f)=kTB \]

    When quantum effects are important, Equation \(\eqref{eq:1}\) is modified and the thermal noise PSD is

    \[\label{eq:3}S_{t}(f)=\frac{hf}{e^{hf/kT}-1} \]

    where \(h\) is Plank’s constant. In Equation \(\eqref{eq:3}\) \(hf\) is the energy of a photon of frequency \(f\) and \(kT\) is the average thermal energy (i.e. vibrational kinetic energy) of the material. To determine the frequency at which the simpler form of \(S_{t}\) can be used consider the following. At low to moderate frequencies Equation \(\eqref{eq:3}\) can be expanded as

    \[\label{eq:4}S_{t}(f)=\frac{hf}{1+(hf/kT)+\frac{1}{2}(hf/kT)^{2}+\ldots -1}\approx\frac{kT}{1+\frac{1}{2}(hf/kT)} \]

    Thus the thermal noise power available will drop off as frequency increases, and is at one-half its low-frequency value at a critical frequency \(f_{c} = 2kT/h\). At room temperature this is approximately \(12\text{ THz}\). So quantum effects on thermal noise are not of concern at room temperature at frequencies below a few terahertz.

    Extensive modern treatments of thermal noise are available in [5] and [6].

    Example \(\PageIndex{1}\): Available Noise Power

    What is the available noise power from a resistor in a \(50\text{ MHz}\) bandwidth and at \(20^{\circ}\text{C}\).

    Solution

    The PSD is, from Equation \(\eqref{eq:1}\),

    \[\begin{align}S_{t}(f)&= (1.381\cdot 10^{−23}\text{ J/K})\cdot ((273 + 20)\text{ K})\nonumber \\ \label{eq:5}&= 4.046\cdot 10^{−21}\text{ J}=4.046\text{ zJ} = 4.046\text{ zW/Hz} = −173.9\text{ dBm/Hz}\end{align} \]

    That is, \(4.046\text{ zJ}\) (zepto joules). The thermal noise at room temperature is usually taken as \(−174\text{ dBm/Hz}\). This noise power is equally divided between amplitude noise and phase noise (each is \(−177\text{ dBm/Hz}\)) [7]. \(S_{t}(f)\) is multiplied by the bandwidth to obtain the total available thermal noise power, so for a \(50\text{ MHz}\) bandwidth, the thermal noise power is

    \[\begin{align}P_{t}(f) &= (4.046\cdot 10^{−21}\text{ J})\times (50\cdot 10^{6}\text{ Hz})\nonumber \\ \label{eq:6}&= 2.023\cdot 10^{−13}\text{ J}\cdot\text{s}^{-1}= 202.3\text{ fW} = −100.1\text{ dBm}\end{align} \]

    This is an appreciable power given that cell phones can operate with receive signals smaller than \(−90\text{ dBm}\). So the lesson here is to use the smallest bandwidth possible in designs.

    4.2.3 Environmental Noise

    Noise in RF and microwave systems includes noise from the environment as well as noise generated within the circuitry itself. Noise from the environment can have galactic origins, when it is known as cosmic background noise, from black-body radiation, or can be artificially generated noise. In cellular communication systems the major source of interference is from other phones and base stations in the cellular system. Provided this is uniformly random over the communication band, it can be treated as random noise. Uniformly random noise (i.e., white noise) can be modeled by a resistor held at what is called the noise temperature.

    A noisy resistor generates white noise that has a flat PSD, i.e. noise spectral density is independent of frequency. A noisy resistor can be modeled by a noise-free resistor and a random voltage or current source denoted by \(v_{n}\) and \(i_{n}\), respectively (see Figure \(\PageIndex{3}\)). The sources \(v_{n}\) and \(i_{n}\) are random and their spectral densities, \(S_{vn}\) and \(S_{in}\) respectively, are related by

    \[\label{eq:7}S_{vn}(f)=R^{2}S_{in}(f) \]

    The noise voltage spectral density of a resistor \(R\) at temperature \(T\) is [8]

    \[\label{eq:8}S_{vn}(f)=\frac{v_{n}^{2}}{B}=4kTR \]

    so that

    \[\label{eq:9}v_{n}^{2}=4kTBR \]

    Here \(k\) is the Boltzmann constant, \(T\) is the temperature in kelvins, and \(B\) is the bandwidth in hertz.

    It is possible for the noise temperature looking into a two-port to be less than the ambient temperature and then the effective noise temperature of the structure is used as a measure of available noise power. A typical situation is specifying the noise presented by an antenna to a receiver. The noise captured by an antenna is from the environment with usually only a small portion of the noise coming from the antenna itself. In the absence of antenna loss, an antenna pointed into space will have a noise temperature corresponding to the cosmic microwave background radiation with an effective noise temperature of about \(3\text{ K}\).

    Two noise voltage sources, \(v_{n1}\) and \(v_{n2}\), in series can be partly correlated. Then the two noise sources can be replaced by a single source \(v_{n}\), where

    \[\label{eq:10}v_{n}^{2}=v_{n1}^{2}+v_{n2}^{2}+2C_{n1, n2}v_{n1}v_{n2} \]

    Here \(C_{n1,n2}\) is the correlation coefficient, and \(−1\leq C_{n1,n2}\leq 1\). If the sources are uncorrelated, as they would be for two resistors, \(C_{n1,n2} = 0\) and

    \[\label{eq:11}v_{n}^{2}=v_{n1}^{2}+v_{n2}^{2} \]

    clipboard_ef82036367dfdf26bb7c62fb15f9ec9da.png

    Figure \(\PageIndex{3}\): Thermal noise equivalent circuits: (a) noisy resistor modeled as a noise-free resistor in series with a random noise voltage source \(v_{n}\); and (b) noisy resistor modeled as a noise free resistor in parallel with a random noise current source \(i_{n}\).

    Correlation of noise sources is important to modeling noise in transistors, as there can be a common physical origin for noise that is modeled as two noise sources in a circuit model.

    4.2.4 Thermal Noise and Capacitors

    Thermal noise is also seen with reactive components such as a capacitor where it is known as \(kT/C\) (read as k-T-C) noise. The series resistance, \(R\), of a capacitor, \(C\), contributes the thermal noise but the RC combination also filters the noise. This noise can be treated the same way as the thermal noise analysis above, but there is a short-hand way of looking at the noise. Sarpeshkar et al. [9] showed that the mean-square noise voltage on a capacitor of value \(C\) within the noise bandwidth of the RC circuit (derived as \(1/(4RC\)) in hertz) is

    \[\label{eq:12}\overline{v}_{n}^{2}=\frac{kT}{C} \]

    At room temperature (i.e., \(T = 20^{\circ}\text{C} = 293\text{ K}\)), the root mean square noise voltage, \(\sqrt{\overline{v}_{n}^{2}}\), on a \(10\text{ pF}\) capacitor with a \(1\:\Omega\) series resistance is \(20\:\mu\text{V}\) in a \(25\text{ GHz}\) bandwidth.

    4.2.5 Physical Source of Shot Noise

    Shot noise is due to current being carried by discrete charge carriers. It is important when there is a region that is scarce of free carriers. Shot noise is particularly important with semiconductor devices but was first observed by Schottky in 1926 in vacuum tubes. In a semiconductor the charge carriers, electrons and holes, are discrete and independent. As such the current fluctuates as the number of carriers varies in discrete steps. On average there is a net velocity of carriers passing a point per time interval. For shot noise to be observed above thermal noise, the carriers should be constrained to pass in just one direction. This is the situation in many semiconductor devices where the depletion region formed at the interface of pn junctions forces the current to flow in just one direction. Shot noise is more significant when the number of charge carriers is small, a situation that also exists in semiconductors. However, even in a semiconductor, there are enough carriers for shot noise to have a Gaussian distribution so that statistically it looks like thermal noise [10]. The shorter the critical time scale, and for microwave and RF circuits this is the period of the waveform, the fewer the number of carriers that will pass a point and the greater the fractional contributions of fluctuations in carrier numbers.

    The RMS current fluctuation due to shot noise is

    \[\label{eq:13}\sigma_{i}=\sqrt{2eIB} \]

    where \(e\) is the elementary charge, \(B\) is the bandwidth in hertz, and \(I\) is the current. For a DC current of \(1\text{ mA}\) and in a \(1\text{ Hz}\) bandwidth, the RMS current fluctuation due to shot noise is \(\sigma_{i} = 18\text{ pA}\). If this current flows through a resistance \(R\), the spectral noise density in the resistor is

    \[\label{eq:14}P_{s}=2e|I| \]

    clipboard_e2211d15c545dbafa06552517e96bc3b5.png

    Figure \(\PageIndex{4}\): Noise and two-ports: (a) amplifier; (b) amplifier with excess noise; and (c) noisy two-port network.

    Note that \(P_{s}\) is independent of temperature and frequency, although eventually quantum effects become important [11]. In contrast, thermal noise is proportional to temperature. The current in Equation \(\eqref{eq:14}\) is the instantaneous current. So shot noise varies during an RF cycle as the current flow varies. If the DC current is much larger than the time-varying current, then the noise that is created by the separable pulses has a flat frequency spectrum and can be modeled by a white noise source.

    So the lesson here is that in active circuit designs, bias currents should be minimized. Also note that the noise power is a function of the resistance value but active circuits with high resistance values also tend to have low current levels.

    4.2.6 Physical Source of Flicker Noise

    The third type of noise that is of concern with RF and microwave circuits is flicker noise, sometimes called \(1/f\) (one-over-f) noise because of its power spectral density shape. Flicker noise is due to diffusion, traps in a semiconductor, and surface traps. A free carrier is immobilized or trapped when it falls into a trap, that is, a recombination center. When several such carriers are trapped, it means that they are not available for conduction and as a result, the resistance of the semiconductor is modulated. These fluctuations have multiple relaxation times. Flicker noise is considered again in Section 6.4 in regards to the characterization of local oscillator modules. A complete physical understanding of flicker noise is not available, and flicker noise is a major concern with oscillators.


    4.3: Noise is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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