ELEC3106 Electronics. Lecture notes: non-linearity and noise. Objective. Non-linearity. Non-linearity measures

Transcription

1 ELEC316 Electronics Lecture notes: non-linearity and noise Objective The objective o these brie notes is to supplement the textbooks used in the course on the topic o non-linearity and electrical noise. Non-linearity Linearity in electronic circuits are oten o paramount importance. Small non-linearities can cause undesired distortion o signals or even cause system mal-unction. Audio equipment, or instance, need to be reasonably linear, or unpleasant distortion is heard. Scientiic measurements equipment where very accurate measurements are needed need to be linear. Radio-transceivers, particularly the ones used in mobile phones, need to be extraordinarily linear, or channel intererence occur. Figure 1 shows examples o distorted sinusoidal signals that are oten encountered: clipping o the signal due to limited power supplies, cross-over distortion oten seen in some classes o power ampliiers, and a slew-rate limited signal caused by the inite speed o the electronic circuit. V ( ) (c) (d) (e) Figure 1: Typical distortions o sinusoidal waveorm: undistorted, clipped, cross-over (c), slew-rate (d), typical requency spectrum (e). Non-linearity measures The are several measures o non-linearity used in various electronics disciplines. In RF electronics, or instance, the third-order inter-modulation intercept point or the 1dB compression point is oten quoted. Which measure is used is oten dictated by what is easily measurable. The non-linearity measure we adopt in this course is the popular Total Harmonic Distortion (THD). For this we need the requency spectrum o a periodic waveorm which is easily obtained at low requencies using the FFT unction on a digital CRO. To calculate the THD, we need the magnitude o the irst harmonic (or undamental), V 1 and also the magnitudes o all higher harmonics V, V 3, V 4,... The THD is then calculated thus (typically quoted in percent): V THD = +V 3 +V 4 +. V 1 ELEC316/notes-noise-nl p. 1/6 School o Electrical Engineering and Telecommunication

2 Noise All circuit elements that exhibit loss generate electrical noise. Ideal reactive components such as inductors and capacitors are noiseless, whereas resistors, transistors and diodes are noisy. There are several dierent noise mechanisms; some well understood, and some still a bit o a mystery to science. Common to most noise processes encountered in electronics is that it is random movements o electrons that generate the noise; thermal noise, or instance is caused by the random movements o electrons due to their thermal energy and increase proportionally with the absolute temperature. Figure shows a typical noisy signal in the time domain and requency domain; you can observe such a signal on your CRO by just turning the gain setting up high. While distortion limits how large signals an electronic system can process, noise limits how small signals the system can process. The dynamic range o is the ratio between the largest and smallest signal that the system can process. Finite dynamic range is what ultimately limits the coverage o mobile phone networks, or instance. v( t ) V ( ) t Figure : Electronic noise. Time domain signal, requency domain signal. Noise signals and noise in resistors Noise signals are (zero mean) random signals. They are described by their variance estimates or requency spectres rather than their time-domain signals. The noise in resistors is usually modelled as a series noise voltage source (v N (t)) as shown in Figure 3. The variance, VN o v N (t) can be estimates thus: V N = 1 T T v N(t)dt. The variance is oten denoted the noise power 1. Note that noise voltage sources are not normally shown with a polarity, as the polarity is always unimportant. Noise in dierent components are uncorrelated. Thus, to ind the total noise power o the series connection o the two resistors in Figure 3, we calculate: V Ntot = 1 T T (v N1 (t) + v N (t)) dt = V N1 +V N + T T v N1 (t)v N (t)dt = V N1 +V N, where the co-variance term ( v N1 v N dt) evaluates to zero or uncorrelated signals. Note here, that when inding the total noise one should thus always add the noise powers o all noise sources! 1 This notation is derived rom signal processing where signals do not have units. You can think o the noise power as the power that would be dissipated in a 1Ω resistor. While the instantaneous noise voltages obviously add, the interesting measure is the power o the signal, not the instantaneous value. ELEC316/notes-noise-nl p. /6

3 Noise signals are better described in the requency domain than in the time domain. One can think o the requency domain signal described as: V n ( ) = F (v N(t)) = e jωt v N(t)dt, and we thus ind the noise power by integration in the requency domain: V N = V n ( )d, where V n ( ) is the noise power spectral density (in V /Hz; datasheets oten quote the noise spectral density V n ( ) in V/ Hz ). 3 All resistors are subject to thermal noise. The thermal noise power spectral density is independent o requency (white noise), as shown in Figure 3(c), and have a value o: V n ( ) = 4kT R (resistor), where k is Boltmann s constant, T is the absolute temperature and R is the value o the resistor. v N R R v N1 R 1 v N V n ( ) 4kTR (c) Figure 3: Noise model or resistor. Single resistor with series noise voltage, series connections o resistors, requency spectrum o thermal noise in resistors (c). Shaped noise To ind the total noise in a circuit, you ind the noise power rom each noise source at the circuit output and add them all up. To illustrate this, let s look at the simple circuit in Figure 4 and ind the noise power across the capacitor. The transer unction rom the noise source V n ( ) to the capacitor voltage V c ( ) (in the requency domain) is H( jω), and we ind using normal circuit analysis: 1 V c ( ) = H( jω)v n ( ) = V n ( ) 1 + jωrc. Thus, we ind the total noise power at the capacitor as: V C = V c ( )d = H( jω) V n ( )d = 4kT R 1 (ωrc) d = 4kT R π/ πrc = kt C Interestingly, even though the capacitor is a noiseless component, the total noise power depends on the capacitor only, and not on the resistor! We can generalise the integral to inding the 3 The math is not strictly correct here; however V n ( ) is usually given as a single-sided spectrum such that the integral should be taken over the requencies to as stated.. ELEC316/notes-noise-nl p. 3/6

4 total output noise power in a circuit with N noise sources V ni ( ) each having individual transer unctions to the output H i ( jω): V Ntot = i H i ( jω) V ni( )d. Note that in the common special case where the noise source is white (Vn ( ) = VnW ), and the transer unction H( jω) is a single-pole lowpass ilter (i.e. in the capacitor example above), one deines the noise bandwidth, B N as B N = π 3dB, where 3dB is the ilter 3 db cut-o requency, and one can simply calculate: V N = V nwb N. R V c ( ) V n ( ) C V c ( ) Figure 4: Noise on capacitor connected to resistor. Schematic, requency domain signal. Shot noise and noise in diodes Diodes and other devices (e.g. vacuum tubes) where electrons cross an energy boundary are subject to shot noise. Shot noise is white and is modelled as a parallel noise current source as shown in Figure 5. The noise power spectral density is: I n( ) = qi D (diode), where q is the electron charge and I D is the device current. I n ( ) i N D qi D Figure 5: Shot noise current in diode. Model, requency domain signal. ELEC316/notes-noise-nl p. 4/6

5 Pink noise and noise in MOSFETs The resistive channel in a MOSFET is subject to thermal noise. Traditionally, this is modelled as a noise current source in parallel with the drain-source path. The spectral density o this is: I n( ) = 4kT g m 3 (MOSFET), where g m is the transistor transconductance. MOSFETs are also subject to pink noise (also known as 1/ noise) which is associated with charge traps in the channel region. This is modelled as a noise voltage source in series with the gate terminal: V n ( ) = K (MOSFET), where K is a suitable constant that depends on device area and other abrication parameters. I n ( ) ( ) V n v N i N 4 ktg m /3 K/ (c) Figure 6: Noise in MOSFET. Model, requency domain signal o channel noise current, requency domain signal o gate noise voltage (c). Signal to noise ratios I signals are too small, they will be swamped by the inherent electrical noise in any electrical system. The signal-to-noise ratio is a measure o how well clear o the noise a signal is. With a signal-to-noise ratio o 1, say, there is usually little point trying digitise that signal with more than about seven bits (18 levels) as the least signiicant bits will just sample the noise and be random 4. To ind the signal-to-noise ratio (SNR), we need the total noise power VN ound above R v N v O V o( ) v S C v O t (c) Figure 7: Signal in noise. Voltage source with output resistance R and parasitic load capacitance C, time domain output voltage, requency domain output voltage (c). 4 Sometimes is is possible to do post-processing that can average out some o the noise, in which case it makes sense to use a ew extra bits in the digitisation. Digital CROs, or instance, normally have this capability. ELEC316/notes-noise-nl p. 5/6

6 as well the signal power (i.e. the square o the RMS value o the signal, VSrms ); we then ind: SNR = V Srms V N, oten quoted in db (i.e. 1log(VSrms /V N ) ). Figure 7 shows an example o a signal source with inite output resistance and capacitance, and time and requency domain output signals. Dynamic range The dynamic range (DR) o a system is the ratio between the largest and the smallest signal that the system can process. In digital systems, or instance, the dynamic range relate to the number o bits used to represent signals (DR = N B, where N B is the number o bits). In analogue systems (or subsystems), the largest signal is limited by distortion while the smallest signal is limited by noise. The exact deinition o what the largest and smallest signals are varies with applications and traditions in dierent disciplines. One reasonable measure is: DN = V Srms,max V N, where V Srms,max is the largest undistorted signal this assumes that distortion sets in suddenly (or instance due to clipping o the signal) rather than steadily increasing with the signal amplitude, and that an SNR=1 is an acceptable minimum signal. ELEC316/notes-noise-nl p. 6/6