[0:18]Hello and welcome to lecture 25 of Analog Integrated Circuit Design. In the previous lecture, we looked at the random noise in a resistor, which happens due to random thermal motion of charge carriers. So, in addition to the current that is predicted by Ohm's law, we'll have some noise. It can be represented as an equivalent noise current across the resistor, or an equivalent noise voltage in series with the resistor. Now, both of these will be described by their noise spectral densities, and in case of resistor's thermal noise, the noise spectral density is white, that means it has a constant magnitude at all frequencies and it is proportional to the absolute temperature. The voltage noise spectral density is 4kT times R volt square per hertz. And the current noise spectral density is 4kT divided by R, amp square per amp square per hertz. The two are equivalent descriptions, and either one can be used in any in any circuit for calculating the effect of resistor's noise. Now, the noise is a random phenomenon, and it can be described by quantities such as the mean square value or the root mean square value. And to do that, there has to be some band limiting, since the resistor's noise is white and uniform over all frequencies. Within some bandwidth, the variance or the mean square value is given by the integration of the spectral density over that bandwidth. Okay? So, the spectral density is nothing but if you pass the noise through an ideal 1 hertz bandpass filter, the variance will be equal to the value of the spectral density at that frequency. Okay? We also elaborated this by calculating the spectral density and the integrated noise in an RC filter. If you have a capacitor across a resistor, it will lead to a finite mean square value of noise, and that is equal to kT by c, which is a fundamental result that appears repeatedly in circuit design. Okay? And as I mentioned in the previous lecture, the MOSFET also has noise, just like a resistor's noise, and that's what we're going to take up in today's lecture. In today's lecture, we'll deal with the noise of the MOS transistor. When we have a MOS transistor, either an N MOS transistor or a P MOS transistor, we have a current that is related to VGS and VDS, and in case of a P MOS transistor, VSG and VSD.
[3:10]Okay? Now, the basic model that we have assumes that ID is related by square law to the VGS. But even if the law is more complicated, there is some formula that gives you the current based on VGS and VDS. Okay? Now, what what does it mean to have noise in a MOS transistor? The drain current will be ID, which is the function of VGS and VDS, as calculated from the formula, plus some noise. Okay? That is, just like we had in a resistor, current predicted by Ohm's law, plus some noise. Here we have current predicted by square law plus some noise. And as with the resistor, we will describe the spectral density of this noise. And from that spectral density, we can calculate mean square value of noise in different circuits and so on. Okay? So, I'll take the N MOS example. So, as usual, we represent the noise by assuming that there is a noiseless transistor, which gives you the current predicted by the formula, plus some noise, IN.
[4:41]Okay? Now, what does the spectral density of the noise current look like? The transistor, as you know, operates in different regions of operation, in the linear region or the triode region and in the saturation region. Okay? Now it's easiest first to think of what happens in the triode region. Let's imagine that the transistor is operating with a certain VGS and with VDS equal to zero. Okay? That means to say it is operating symmetrically. And if we look at the picture of the device, under these conditions, there will be a uniform channel going from the drain to the source. Okay? This is the gate. And we of course have the substrate, which is connected to some potential. The mechanism of conduction in a MOSFET is exactly same as the mechanism in a resistor. It is due to drift of carriers. Okay? And when VDS equal to zero, the MOSFET is simply a resistor. Now, the charge carriers are due to the channel that is formed by inversion, but the operation is very similar to that of a resistor. Okay? So, it turns out that the noise is also exactly the same as what we would have in a resistor. Okay? In the deep triode region, the transistor is nothing but a conductance, which I'll call GMOS. Okay? And the conductance itself is given by the equation in the triode region, which is mu Cox W by L, VGS minus VT. The conductance happens to be controllable by VGS, but it's a conductance, nonetheless. Okay? Now, if we had a resistance whose conductance was GMOS, the noise spectral density would have been 4kT times GMOS. This would be the noise spectral density of the current. Okay? In the saturation region, the noise turns out to be 8/3 kT times GM.
[8:05]Okay? At the operating point, whatever operating point the transistor is at. Okay? Just like all the small signal parameters of a transistor depend on the operating point, the noise also depends on the operating point. And it happens to be 8/3 kT times GM. And you can quickly verify that the current noise in the triode region or the current noise of a resistor is given by some formula which looks like kT times a conductance. And here, it happens to be kT times some transconductance. Okay? And if you assume that the transistor obeys square law, GM is given by mu Cox W by L, VGS minus VT. Okay? And from this expression and that expression, you can see that the noise formula is, in fact, quite similar in the triode region and the saturation region.
[10:04]Okay? The transition as usual is never discontinuous, but it is continuous. If you manage to find the charge, a formula for the charge in all the regions, you can use this formula to accurately predict the noise spectral density.
[10:25]Now, most of the times, in amplifiers, we'll operate in the saturation region, and this is the formula that we use. Okay?
[10:39]The spectral density, SID, it is white. Okay? The noise in a MOS FET is also due to thermal fluctuations of carriers. Okay? This is the thermal noise of the MOS FET. Just like we had the thermal noise of the resistor, and that happens to be white with a spectral density equal to 8/3 kT times GM in the saturation region. Okay? Now, how do we use this in any circuit? In parallel with every MOSFET, we'll add a current source representing the noise of that MOS transistor. Okay? And the spectral density of this is given by that expression. Okay? Now if you have a circuit with a number of MOS transistors, and you have a certain output voltage, let's say V naught, you find the transfer function from every noise current source to the output. Okay? And you will have the output to be a superposition of all these noise current sources. Okay? And because the noise from different devices are uncorrelated from each other, the output will be simply the noise spectral density of first source, times the transfer function magnitude square, plus noise spectral density of the second source, times the transfer function magnitude square, and so on. Okay? Exactly what we did with resistors. So, if you have a number of components, including resistors, you have to do this for every noise source from every MOSFET and every resistor. Okay? We'll soon see an example of how to do these noise calculations when we have MOSFETs and resistors. Okay? Now, it turns out that in addition to thermal noise, there is also flicker noise in a MOS transistor. And this flicker noise happens to have a different spectral density. Okay? Now, like thermal noise is caused by random fluctuations in the motion of charges due to some non-zero temperature, flicker noise is caused by a completely different reason. When you have a MOS transistor, the charge in the inversion layer is what is available for conduction.
[13:13]And the charge in the inversion layer equals the charge on the gate. Okay? That's assuming there are no fixed charges in the dielectric. It turns out that at the interface of the dielectric and the semiconductor, there are trapped charges. And these trapped charges are randomly trapped and released with differing time constants. Okay? There are different theories about flicker noise. One of the theories is that the charges are trapped and de-trapped with different time constants, and the sum of all those things gives you some noise, which is flicker noise. Okay? Because the amount of charge in the inversion layer is balanced by the charge on the gate and also all the fixed charges that we might have in between the two plates, we will have some randomness in the number of charges, and consequently, a randomness in the current as well. And this phenomenon is known as flicker noise.
[14:12]The flicker noise, as thermal noise is described by some spectral density. We can think of the MOS transistor as having thermal noise and also additionally, a flicker noise.
[14:35]In practice, of course, we represent it by a single source, which represents the sum of these two noise currents. Okay? Now, it turns out that the flicker noise is related to some constant, some proportionality constant, and the operating point current divided by L square.
[14:58]And also, the spectral density is not constant with frequency. Okay? This is an important distinction between white noise and flicker noise. A white noise spectral density is constant with frequency, whereas the flicker noise density is not constant with frequency. It is inversely proportional to the frequency. Okay? So, the sum of thermal noise and flicker noise will be the total noise of a MOSFET. Now, as you can expect, this flicker noise spectral density increases when the frequencies are low, because of the inverse proportionality to frequency. So, it is particularly troublesome in low frequency circuits. Okay?
[15:37]So, the MOSFET, if I plot the total power spectral density, total spectral density of the noise current, what I will have is the thermal noise part, okay, which depends on the operating point, as 8/3 kT GM.
[16:04]And the flicker noise part, which I will represent it as a straight line, it is inversely proportional to frequency. But I'll assume that the spectral density is plotted on a log scale, and the frequency is also on a log scale. Okay? So, this is the flicker noise part of it. It is like that.
