Noise In Analogue Circuits PDF

Check in: 339 670 Section 10 Noise Acknowledgement: Some material here is from content developed by John Dawson and Andy Marvin for a previous module. Content in small font is to support revision. RNG...

Check in: 339 670 Section 10 Noise Acknowledgement: Some material here is from content developed by John Dawson and Andy Marvin for a previous module. Content in small font is to support revision. RNG, ver 1.0 1 What you will learn.. Sources of noise Quantifying and Modelling noise Circuit and System noise 2 Noise Noise in general means unwanted signals We concentrate on Electrical Noise Electrical Noise is a current or voltage signal that is unwanted in an electrical circuit In general, real signals are the summation of the unwanted noise and desired signal Noise is often the limiting factor in a circuit’s or system’s performance. E.g. distance of a radio link 3 Sources of noise Fig: Taxonomy of noise In this part of the course, we will focus mostly on the natural sources of noise intrinsic to electronic circuits due to the behaviour of electronics (and other charge carriers) in circuits and components. 4 Random Noise in the time domain (1Vrms) White https://en.wikipedia.org/w/index.php?ti tle=White_noise&oldid=801232060 Pink https://en.wikipedia.org/w/index.php?ti tle=Pink_noise&oldid=800407601 Red https://en.wikipedia.org/w/index.php?ti tle=Brownian_noise&oldid=772796459 See also https://en.wikipedia.org/wiki/Colors_of_noise 5 Properties of noise Magnitude in time domain Mean RMS/Power Peak value Probability distribution of amplitude (probability distribution function) Distribution of energy in frequency domain (spectrum) Randomness Correlation or non-correlation with other sources 6 Natural Sources of noise (intrinsic noise) Thermal noise in conductors/resistors (Gaussian white noise) Flicker noise Shot noise Avalanche Noise 7 Thermal noise Thermal noise is the result of random (Brownian) motion of electrons (charge carriers) in a conductor/resistor due to thermal effects. Thermal noise is also called Johnson noise or Johnson-Nyquist noise after the scientist who observed/determined the characteristics of noise. The maximum noise power that can be delivered by a resistor is: 𝑃𝑁 = 𝑘𝑇𝐵 (watts) where 𝑘 ≈ 1.38×10−23 J⋅K−1 is the Boltzmann constant, 𝑇 is the absolute temperature in kelvin, and B is the bandwidth Noise power available can be can be describes in terms of power per unit bandwidth: 𝑃𝑛 = 𝑘𝑇 (watts/Hz) A capitalised suffix indicates the noise power, voltage or current in a defined bandwidth (𝐵 Hz) and a lower case suffix denotes power, current or voltage in a 1 Hz bandwidth. 8 Thermal noise characteristics 1. In the frequency domain: White (in the same way as white light carries all colours) Constant power spectral density (PSD) at all frequencies of interest up to terahertz frequencies. Any two samples are independent and uncorrelated 2. In the time domain: Amplitude of a noise sample is a random variable With zero-mean Gaussian probability density function (pdf) Since thermal noise is due to the summation of the effects of many randomly moving electronics and the central limit theorem states that a value that depends on the sum of many independent events has a normal (Gaussian) distribution of values. 9 Noise models for a resistor (1) 𝑃𝑛 = 𝑉𝑛2 𝑓 , 𝑉 2 /𝐻𝑧 4𝑘𝑇𝑅𝑁 f Thévenin equivalent model For a resistor: The power spectral density is: volts 2 𝑃𝑛 = 4𝑘𝑇𝑅𝑁 Hz The open-circuit rms noise voltage is: 𝑉𝑁 = 4𝑘𝑇𝐵𝑅𝑁 (volts) In a 1 Hz bandwidth the rms noise voltage is: 𝑉𝑛 = 4𝑘𝑇𝑅𝑁 (volts/ Hz) 10 Noise models for a resistor (2) 𝑃𝑛 = 𝐼𝑛2 𝑓 , 𝐴2 /𝐻𝑧 4𝑘𝑇 𝑅𝑁 f Norton equivalent model For a resistor: The power spectral density is: 4𝑘𝑇 amps 2 𝑃𝑛 = 𝑅𝑁 Hz The short-circuit rms noise current for the Norton equivalent circuit is: 𝐼𝑁 = 4𝑘𝑇𝐵/𝑅𝑁 (amps) or 𝐼𝑛 = 4𝑘𝑇/𝑅𝑁 (amps/ Hz) 11  Real inductor and capacitor with noise sources All circuit elements with resistive components will have thermal noise and so appropriate noise sources should be introduced. 𝑉𝑁 𝐿 𝑅𝑆 Inductor 𝑉𝑁𝑃 𝑅𝑃 𝑉𝑁𝑆 𝑅𝑆 Capacitor 𝐶 Fig: Equivalent circuits for real inductor and capacitor showing noise sources. 12 Noise equivalent bandwidth In real circuits the signals and noise are bandlimited In the simple case of a low-pass system such as an operational amplifier, we can define its bandwidth by the upper 3dB cutoff (break) frequency. P(0): low frequency gain Brick-wall filter f-3dB: -3 dB cutoff 𝑃(𝑓) BN : noise equivalent bandwidth Fig: Noise bandwidth versus 3dB bandwidth for a low pass filter (first-order) 13 Noise equivalent bandwidth The noise bandwidth BN of a system is defined from the figure in the previous slide and as: 𝐼 ∞ 𝐵𝑁 = ‫׬‬0 𝑃 𝑓 𝑑𝑓 (Hz) 𝑃 0 where 𝑃(0) is the power gain in the pass-band. For a first order low-pass filter with a 3 dB frequency 𝑓0 the noise bandwidth, 𝜋 𝐵𝑁 = 𝑓0 2 14 Gaussian Probability Distribution (normal distribution)(1) The probability of a voltage having a particular value follows a normal or Gaussian distribution in some natural noise sources: thermal, shot and flicker noise Fig: Gaussian probability density function. The Gaussian distribution is expressed as a probability density function (pdf): 1 𝑣 − 𝑣ҧ 2 𝑝 𝑣 = 𝑒𝑥𝑝 − 2 2𝜋 𝜎𝑣 2𝜎𝑣 15 Gaussian Probability Distribution (normal distribution)(2) Noise voltage has positive and negative values that cancel out to result a zero mean value. The standard division of the distribution is the rms value of the voltage, 𝜎 = 𝑉𝑟𝑚𝑠 So the zero mean Gaussian distribution for the amplitude of noise: 1 𝑣 2 𝑝 𝑣 = 𝑒𝑥𝑝 − 2𝜋 𝑉𝑟𝑚𝑠 2 𝑉𝑟𝑚𝑠 2 The area under the pdf curve between the limits represents the probability of measuring a voltage in that range. The peak value is often taken as approximately 𝑉෠ ≈ 4𝜎 = 4𝑉𝑟𝑚𝑠 16 Flicker noise (1) Flicker noise is a low-frequency noise (usually considered up to around 1 kHz) Flicker noise is also known as Pink noise – reddish colour in the lower range of visible spectrum 1/f noise (one over f) because of its power spectral density which is inversely proportional to frequency Its source is associated with defects in the crystal structure of the resistive materials and effects at grain boundaries on the movement of charge carriers as they are trapped and released. 17 Flicker noise (2) Flicker noise has a Gaussian amplitude probability density function. 𝐾𝑓 2 Spectral density: 𝐼𝑛2 (𝑓) = 𝐼𝐷𝐶 A /Hz 𝑓 where 𝐾𝑓 - magnitude parameter of the source and 𝐼𝐷𝐶 is the DC current flow. 18 Flicker noise (3) 𝑉𝑁 𝑉𝐹 𝑅 Fig: Equivalent circuit for a resistor with thermal and flicker noise An equivalent circuit of a resistor with both thermal and flicker noise sources is shown in the above figure. The flicker noise current 𝐼𝐹 is represented by its Thévenin equivalent 𝑉𝐹 = 𝐼𝐹 𝑅. 19 Flicker noise - example Flicker noise 1/f noise corner Thermal noise Fig: Equivalent input noise spectral density for the TL071 op-amp (taken from the TL071 datasheet) 20 Shot noise (1) Shot noise arises from the random arrival of charge carriers across a potential boundary (e.g. pn junction of a diode or a thermionic valve.) The total current across the boundary is the sum of all the charges arriving per second. However the arrival of each charge is an individual event and the random unevenness in their arrivals constitutes a noise current. 21 Shot noise (2) Shot noise is strictly a Poisson process, but as usually a large number of charge carriers are considered in electronic circuits, it properly approximated by a Gaussian pdf. (Central limit theorem) Spectral density: 𝐼𝑛 = 2𝑞𝐼𝐷𝐶 A2 /Hz 2 Here q is the electronic charge (1.60 x 10-19C), 𝐼𝐷𝐶 is the whole current across the junction and B is the bandwidth (Hz) of the measurement of 𝐼𝑁. Shot noise has a white power spectral density. 22  Equivalent circuit for a diode with shot noise 𝐼𝑁 𝐼𝐷𝐶 𝑉𝐷 Fig: Equivalent circuit for a diode with shot noise The diode current for an applied voltage (see the diagram for notation) at temperature T: 𝑞𝑉𝐷 𝐼𝐷𝐶 = 𝐼𝑆 𝑒𝑥𝑝 −1 𝑘𝑇 Shot noise when the diode is forward biased: 𝑞𝑉𝐷 2 𝐼𝑁 = 2𝑞𝐼𝑆 𝑒𝑥𝑝 𝐵 A2 𝑘𝑇 Shot noise when the diode is reverse biased: 𝐼 2 = 2𝑞𝐼 𝐵 𝑁 A2 𝑆 23  Adding noise sources Fig: Nosie simulation of a voltage divider @300 K (note the contribution by 10k to the total output noise is smaller compared to 5k due to the voltage divider arrangement). 𝑃𝑁_𝑡𝑜𝑡𝑎𝑙 = 𝑃𝑁_5𝑘 + 𝑃𝑁_10𝑘 (W) OR 𝑃𝑛_𝑡𝑜𝑡𝑎𝑙 = 𝑃𝑛_5𝑘 + 𝑃𝑛_10𝑘 (W/Hz) (𝑉𝑁_𝑡𝑜𝑡𝑎𝑙 )2 = (𝑉𝑁_5𝑘 )2 + (𝑉𝑁_10𝑘 )2 (V 2 ) OR (𝑉𝑛_𝑡𝑜𝑡𝑎𝑙 )2 = (𝑉𝑛_5𝑘 )2 + (𝑉𝑛_10𝑘 )2 (V 2 /Hz) For uncorrelated noise sources, the resultant noise power (or squared rms voltage) is the summation of individual noise powers (squared rms voltages). The same applies to the spectral densities. 24 Signal-to-Noise ratio (reminder) Signal-to-noise ration (SNR) can be defined as 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑠𝑖𝑔𝑛𝑎𝑙 𝑝𝑜𝑤𝑒𝑟, 𝑃𝑆 𝑆𝑁𝑅 = 𝑢𝑛𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑠𝑖𝑔𝑛𝑎𝑙 𝑝𝑜𝑤𝑒𝑟 𝑛𝑜𝑖𝑠𝑒 , 𝑃𝑛𝑜𝑖𝑠𝑒 𝑃𝑆 SNR in dB: 𝑆𝑁𝑅𝑑𝐵 = 10𝑙𝑜𝑔 𝑃𝑛𝑜𝑖𝑠𝑒 If power is normalised to 1 Ω load: 𝑃𝑆 𝑃𝑆 𝑉𝑆,𝑟𝑚𝑠 𝑆𝑁𝑅𝑑𝐵 = 10𝑙𝑜𝑔 = 20𝑙𝑜𝑔 = 20𝑙𝑜𝑔 𝑃𝑛𝑜𝑖𝑠𝑒 𝑃𝑛𝑜𝑖𝑠𝑒 𝑉𝑛𝑜𝑖𝑠𝑒,𝑟𝑚𝑠 25  System noise performance SNR1 SNR2 SNR3 SNRout Noisy Noisy Noisy Block 1 Block 2 Block 3 Fig: System with cascaded (noisy) functional blocks. We will now look at how to analyse the performance of a system. Each noisy block will add noise to its input i.e SNR is always worst (if no gain in a block) What we aim to do is to minimise the degradation of the SNR as the signal passes through the system. 26 Summary Intrinsic noise sources Thermal: White power spectrum; Gaussian amplitude distribution Shot: White power spectrum; Gaussian amplitude distribution Flicker: 1/f power spectrum; Gaussian amplitude distribution Noise equivalent bandwidth 27 Noise in analogue circuits 28  Noise in amplifiers At the design stage of the Op-Amp each component’s noise contribution (thermal, flicker and shot noise) is taken into consideration to determine the total output noise of the amplifier. Such complex calculations is unnecessary for the user of an Op- Amp Thus we need to model the output noise of an analogue circuit that represents its noise characteristics. Fig: Simplified circuit of TL071 Op-Amp 29 Equivalent noise model for an amplifier Input-referred noise 𝑉𝑛𝑖 Typical values of 𝑉𝑛𝑖 are 1 nV/Hz1/2 to 20 nV/Hz1/2 and typical values of 𝐼𝑛𝑖 are 100 attoamp/Hz1/2 to 10 picoamp/Hz1/2. Noiseless Output 𝐼𝑛𝑖 amplifier Fig: Equivalent noise model for an amplifier. Noise is always measured at the output of a circuit. It can be referred to the input of the circuit such that the same noise is presented at the output of the noiseless circuit. For an operational amplifier we use two fictitious input noise sources, a current source and a voltage source as shown in the diagram above. 30 Noise Factor (F) and Noise Figure (NF or FdB) Noise factor of a system is defined as 𝑇𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡 𝑛𝑜𝑖𝑠𝑒 𝑝𝑜𝑤𝑒𝑟,𝑃𝑂𝑈𝑇 𝑃𝑂𝑈𝑇 𝐹= = 𝑂𝑢𝑡𝑝𝑢𝑡 𝑛𝑜𝑖𝑠𝑒 𝑝𝑜𝑤𝑒𝑟 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑛𝑜𝑖𝑠𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑜𝑢𝑟𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡 𝐺𝑃 𝑃𝐼𝑁 𝐺𝑃 − 𝑝𝑜𝑤𝑒𝑟 𝑔𝑎𝑖𝑛; 𝑃𝐼𝑁 − 𝑛𝑜𝑖𝑠𝑒 𝑝𝑜𝑤𝑒𝑟 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡 Noise Figure 𝑁𝐹 = 𝐹𝑑𝐵 = 10𝑙𝑜𝑔 𝐹 For a noise free system the Noise Factor is unity (1) and the Noise Figure is 0dB. For real systems noise factor is greater than 1 and NF grater than 0 dB. 31  Noise factor of an amplifier (1) For simplicity, only noise sources are 𝑉𝑁𝑖 shown in the figure. 𝑉𝑁 𝑉𝑁𝑇 = 𝑉𝑁2 + 𝑉𝑁𝑖 2 2 2 + 𝐼𝑁𝑖 𝑅𝑆 𝐼𝑁𝑖 𝐺 2 2 Note that:𝑉𝑁2 = 𝑉𝑛2 𝐵𝑁 , 𝑉𝑁𝑖 = 𝑉𝑛𝑖 𝐵𝑁 and 𝑅𝑆 2 2 𝐼𝑁𝑖 = 𝐼𝑛𝑖 𝐵𝑁 𝑉𝑁𝑇 The noise voltage from the Thévenin equivalent source is reduced by the factor: 𝑍in 𝐺 𝑍in + 𝑅s but this reduction is often neglected for 𝑅𝑆 amplifiers with high input impedance. 𝐺 − 𝑝𝑜𝑤𝑒𝑟 𝑔𝑎𝑖𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑟 32 Fig: Simplified amplifier with the Thévenin equivalent source  Noise factor of an amplifier (2) Using the definition of Noise Factor: 𝑉𝑁𝑖 2 𝐺𝑉𝑁𝑇 𝐹= 𝐺𝑉𝑁2 𝑉𝑁 2 2 2 𝐺 𝐺 𝑉𝑁2 +𝑉𝑁𝑖 +𝐼𝑁𝑖 𝑅𝑆 𝐼𝑁𝑖 𝐹= 𝐺𝑉𝑁2 𝑅𝑆 2 2 2 𝑉𝑁𝑖 +𝐼𝑁𝑖 𝑅𝑆 𝑉𝑁𝑇 𝐹 =1+ 𝑉𝑁2 𝑉𝑁2 = 4𝑘𝑇𝐵𝑅𝑆 𝑉𝑁𝑇 For high input-Z 𝐺 2 2 𝑅𝑆 𝑉𝑁𝑖 𝐼𝑁𝑖 𝑅𝑆 𝐹 =1+ + 4𝑘𝑇𝐵𝑅𝑆 4𝑘𝑇𝐵 33 Fig: Simplified amplifier with the Thévenin equivalent source  For small 𝑅𝑆 , noise factor is Noise factor of an amplifier (3) dominated by the noise voltage 𝑉𝑁𝑖 source 2 2 𝑉𝑁𝑖 𝐼𝑁𝑖 𝑅𝑆 𝑉𝑁 𝐹 =1+ + 4𝑘𝑇𝐵𝑅𝑆 4𝑘𝑇𝐵 𝐼𝑁𝑖 𝐺 For large 𝑅𝑆 , noise factor is 𝑅𝑆 dominated by the noise current source 𝑉𝑁𝑇 Noise Factor approaches 1 as the values of equivalent noise sources go zero. 𝐺 𝑅𝑆 34 Fig: Simplified amplifier with the Thévenin equivalent source  Noise factor of an amplifier (4) 𝑉𝑁𝑖 2 2 𝑉𝑁𝑖 𝐼𝑁𝑖 𝑅𝑆 𝐹 =1+ + 𝑉𝑁 4𝑘𝑇𝐵𝑅𝑆 4𝑘𝑇𝐵 𝐼𝑁𝑖 𝐺 𝑅𝑆 Analytically, you will be able to find the optimum source resistance minimises the noise factor. 𝑉𝑁𝑇 And that is when 𝑉𝑁𝑖 𝑅𝑆 = 𝐼𝑁𝑖 𝐺 𝑅𝑆 35 Fig: Simplified amplifier with the Thévenin equivalent source Noise in Communication Systems (Radio Frequency Systems) Here we consider the performance of radio frequency (RF) communication systems ( few MHz to THz region) We can still use the Noise Figure for analysis Modern communication systems are built with sub-systems with standardised input and output impedance (50 Ω) Matches with the characteristic impedance of a coaxial transmission line No requirement to optimise for different impedances. 36 Motivation A typical 4G mobile phone receiver has a specified minimum sensitivity of -87 dBm and a bandwidth of 5 MHz at frequencies around 2 GHz To achieve a data rate of 20 Mb/s it is required to have a 12 dB signal to noise ratio. What is the noise level possible? Compare this with the available power of a resistor at 290 K. 37 Noise Factor (F) and Noise Figure (NF) of radio frequency sub- systems The definition of Noise Factor of a system depends on the noise from the source at the input For a RF systems, source noise power is from a resistive source at the Standard Noise Temperature (T0) defined as 290 K. G F T0=290 K 50 Ω 50 Ω Fig: An amplifier in a 50 ohm system with a power gain G and noise factor F. 38 Excess Noise 𝑃𝐼𝑁 𝑃𝑂𝑈𝑇 𝑃𝑂𝑈𝑇 𝑃𝑂𝑈𝑇 Noise Factor: 𝐹 = = 𝐺𝑃𝐼𝑁 𝐺𝑘𝑇0 𝐵 The total noise at the output: 𝑃𝑂𝑈𝑇 = 𝐹𝐺𝑘𝑇0 𝐵 The noise power at the output due to the source: 𝐺𝑃 𝑃𝐼𝑁 = 𝐺𝑘𝑇0 𝐵 The extra noise power at the output originating in the sub-systems is called the Excess noise 𝐸𝑥𝑐𝑒𝑠𝑠 𝑁𝑜𝑖𝑠𝑒 = 𝐹 − 1 𝑘𝑇0 𝐵𝐺 39 Noise factor of a cascaded system (1) G1 G2 G3 F1 F2 F3 T0 50 Ω 50 Ω Output Output Output stage 1 stage 2 stage 3 Fig: Three amplifiers in cascade in a 50 ohm system. Now we need to find the overall Noise Factor of the system. 40 Noise power Output stage 1 Output stage 2 Output stage 3 contribution Input noise source 𝐺1 𝑘𝑇0 𝐵 𝐺1 𝐺2 𝑘𝑇0 𝐵 𝐺1 𝐺2 𝐺3 𝑘𝑇0 𝐵 + + + Stage 1 Excess 𝐹1 − 1 𝐺1 𝑘𝑇0 𝐵 𝐹1 − 1 𝐺1 𝐺2 𝑘𝑇0 𝐵 𝐹1 − 1 𝐺1 𝐺2 𝐺3 𝑘𝑇0 𝐵 + + Stage 2 Excess 𝐹2 − 1 𝐺2 𝑘𝑇0 𝐵 𝐹2 − 1 𝐺2 𝐺3 𝑘𝑇0 𝐵 + Stage 3 Excess 𝐹3 − 1 𝐺3 𝑘𝑇0 𝐵 Total output noise power 41 Noise factor of a cascaded system (2) Overall Noise Factor 𝐺1 𝐺2 𝐺3 + 𝐹1 − 1 𝐺1 𝐺2 𝐺3 + 𝐹2 − 1 𝐺2 𝐺3 + 𝐹3 − 1 𝐺3 𝐹= 𝐺1 𝐺2 𝐺3 𝐹2 − 1 𝐹3 − 1 𝐹2 − 1 𝐹3 − 1 𝐹 = 1 + 𝐹1 − 1 + + = 𝐹1 + + 𝐺1 𝐺1 𝐺2 𝐺1 𝐺1 𝐺2 For N stages: 𝐹2 − 1 𝐹3 − 1 𝐹𝑁 − 1 𝐹 = 𝐹1 + + + ⋯+ 𝐺1 𝐺1 𝐺2 𝐺1 𝐺2 ⋯ 𝐺𝑁−1 42 Noise factor of a cascaded system (3) For N stages: 𝐹2 − 1 𝐹3 − 1 𝐹𝑁 − 1 𝐹 = 𝐹1 + + + ⋯+ 𝐺1 𝐺1 𝐺2 𝐺1 𝐺2 ⋯ 𝐺𝑁−1 The above is called the Friis formula for combining noise outputs of cascaded stages. Note that the largest contribution to the overall Noise Factor comes from the Noise Factor of the first stage. Low noise design is much more important for the first two stages. 43 Noise factor of a cascaded system (3) For N stages: 𝐹2 − 1 𝐹3 − 1 𝐹𝑁 − 1 𝐹 = 𝐹1 + + + ⋯+ 𝐺1 𝐺1 𝐺2 𝐺1 𝐺2 ⋯ 𝐺𝑁−1 Low noise design is much more important for the first two stages. This is the reason why systems such as satellite TV receivers always have a LNB (low noise block) or LNA (low noise amplifier) right after the antenna. A typical low-noise amplifier used in say a GPS receiver has a Noise Figure of 0.4dB and its equivalent Noise Factor is 1.1. 44 Noise Factor of passive components In most cases, for passive components, the noise factor is the inverse of the gain (usually a loss or an attenuation). E.g. a passive filter with a gain of 0.5 would have a noise factor of 2; or in decibels, a passive filter with a gain of –3 dB has a noise figure of 3 dB. Examples of lossy stages: Frequency selection filters Transmission lines linking stages Frequency changing mixers 45 Noise Temperature (1) Sometimes the term Noise Temperature is used to characterise the noise performance of a system (or amplifier). This approach is more useful in many ways particularly in radio receivers where the external noise received into the system is from the antenna. Let’s use an amplifier to explain the noise temperature. 46 Noise Temperature (2) Consider a stage with gain 𝐺. The stage is assumed to be completely noise free and its excess noise at its output is replicated by introducing an equivalent fictitious noise source at its input defined by a noise temperature 𝑇𝑁. Noiseless amplifier 𝑘𝑇0 𝐵 Output noise power 𝐺𝑘 𝑇𝑁 + 𝑇0 𝐵 𝐺 𝑇0 𝐹 𝑘𝑇𝑁 𝐵 Fig. Determining the noise temperature of an amplifier. 47 Noise Temperature and Noise Factor Noiseless amplifier 𝑘𝑇0 𝐵 𝐺𝑘 𝑇𝑁 + 𝑇0 𝐵 𝐺 𝑇0 𝐹 𝑘𝑇𝑁 𝐵 Fig. Determining the noise temperature of an amplifier. Noise Factor 𝐺𝐾 𝑇𝑁 + 𝑇0 𝐵 𝑇𝑁 𝐹= =1+ 𝐺𝐾 𝑇0 𝐵 𝑇0 48  Noise temperature of a multi-stage system Consider now the three stage system again, this time using Noise Temperature. The input Noise Temperatures at each stage can all be referred back to the input by reducing their magnitudes by the gain product of the preceding stages. Thus the total input Noise Temperature 𝑇𝐼𝑁 is: 𝑇2 𝑇3 𝑇𝐼𝑁 = 𝑇1 + + 𝐺1 𝐺1 𝐺2 49 Example: Antenna noise temperature (1) If there is an antenna connected to a system, then the output noise of the antenna which is also characterised by an equivalent antenna noise temperature TANT can be added. Note that this is not a thermodynamic temperature. The antenna Noise Temperature is representative of all the external noise sources contributing to the noise from the antenna at the frequency being received. External noise sources include, in varying amounts according to the frequency, the ground, buildings, people, the atmosphere, the ionosphere, the sun, other stars, and intergalactic gases such as hydrogen. 50 Example: Antenna noise temperature (2) Check in code: 566 355 If the antenna receives a signal power 𝑃𝑟𝑒𝑐 , then the signal-to-noise ration is: 𝑃𝑟𝑒𝑐 𝑆𝑁𝑅 = 𝑘 𝑇𝐴𝑁𝑇 + 𝑇𝐼𝑁 𝐵 51 Summary Properties on noise Probability density function (pdf), correlation, noise power spectral density Natural sources of noise Thermal Shot 1/f noise Noise in Analogue circuits Noise Figure/Factor: 𝐹 = 𝑃𝑂𝑈𝑇 /(𝐺 𝑃𝐼𝑁 ) Minimum noise, source impedance: 𝑅𝑠𝑚𝑖𝑛 = 𝑉𝑁 /𝐼𝑁 Cascaded stages: NF of first stage has most effect Noise temperature Good for systems with an antenna 52 Noise temperature of a multi-stage system Consider now the three stage system again, this time using Noise Temperature. 𝑘𝑇0 𝐵 𝑇0 𝐺1 𝐺2 𝐺3 𝑇1 𝑇2 𝑇3 𝑘𝑇0 𝐵 𝑇0 𝐺1 𝐺2 𝐺3 𝑇𝐼𝑁 53  LNA Coaxial Receiver Gain (dB) Gain Gi Noise Figure (dB) Boise Factor F 54