C IRCU IT INTU ITIONS

Ali Sheikholeslami

“Noise and Distortion, Part II”

W
Welcome to the 40th article in the “Cir-
cuit Intuitions” column series. As the
title suggests, each article provides
insights and intuitions into circuit
design and analysis. These articles are
sion instrumentation. In this article, I
explain how distortion is created and
how it can be confused with noise as
both are treated as an unwanted added
signal to the desired signal. In the next
the negative swing. This is a direct
consequence of distortion, where dif-
ferent portions of the signal experi-
ence varying degrees of amplification,
similar to how a low-quality magnify-
aimed at undergraduate students but article, we will review simple methods ing glass may create a distorted image
may serve the interests of other read- to reduce distortion and the improve- of the objects it magnifies. To cap-
ers as well. If you read this article, I ment they bring to the desired signal. ture this effect, Figure 1(d) shows the
would appreciate your comments and small-signal voltage gain around each
feedback as well as your requests and What Causes Distortion? value of v in, confirming that different
suggestions for future articles in this Let us review the basic operation of a portions of the input waveform are
series. Please e-mail me your com- simple common-source (CS) amplifier, amplified with different voltage gains.
ments at ali@ece.utoronto.ca. as shown in Figure 1(a), to see what To intuitively see how the input signal
In Part I of this article [1], we may cause distortion in this amplifier. in Figure 1(c) creates a distorted out-
reviewed the concept of signal and The amplifier consists of an NMOS put, imagine dividing the input sinu-
noise in electronic circuits. We said transistor, biased in the saturation soid into 10 intervals along the voltage
that while we use transistors to con- region with a dc gate-to-source voltage axis, with each interval spanning a
trol electrons [2] for the purpose of ^VGSh, and a dc drain-to-source voltage range of 20 mV. According to Figure
signal amplification (or processing in ^VDSh . If we superimpose a small-­signal 1(d), the interval near the negative
general), the same electrons, through ac voltage v in on VGS, we observe a peak of the input experiences a gain
their random movements, create an voltage deviation from the VDS at the of - 2 V/V, whereas the interval near
unwanted signal called noise. This output node, which we define as v out . its zero crossing experiences a voltage
noise appears at the output of the For simplicity, we assume v in ^ t h is a gain of - 6 V/V. If we allow the input
circuit, degrading the amplified sig- low-frequency signal such that the to have a larger swing, say 400 mV,
nal’s quality. Another mechanism by output follows the input without delay. then the intervals near the positive
which transistors may degrade sig- The simulated relationship between and negative peaks of v in will expe-
nal quality is nonlinearity. Through v in and v out is plotted in Figure 1(b). rience a negligible gain close to zero.
this mechanism, the desired signal is We can clearly see a nonlinear relation- What has been discussed so far
altered and distorted, losing some of ship between the two voltages, and can be summarized as follows: the
its original waveshape. We call the dif- the larger the input signal, the more relationship between the input and
ference between the distorted signal this nonlinearity manifests itself. For a output of a typical voltage amplifier
and the original desired signal distor- sinusoidal input with a 100-mV swing, can be expressed as
tion. Minimizing distortion, along with as shown in Figure 1(c), the output volt-
v out = f ^v inh (1)
minimizing noise, is a critical goal in age waveform exhibits a larger down-
transistor circuit design, especially ward swing ^, 600 mVh compared to where f is a nonlinear function. The
in applications that require high sig- its upward swing ^, 400 mVh, which degree of nonlinearity depends on the
nal fidelity, such as audio amplifiers, is consistent with the relationship signal amplitude; there is less nonlin-
communication systems, and preci- depicted in Figure 1(b). earity for very low signal amplitudes
Additionally, due to this nonlinear- (i.e., in the vicinity of v in = 0 V), and
Digital Object Identifier 10.1109/MSSC.2024.3473730 ity, the positive swing at the output more nonlinearity as we increase
Date of current version: 15 November 2024 appears wider (in time) compared to the input amplitude. But how can we

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 VDD = 1.2 V
vout (V)
1

RD = 2 kΩ
0.5

VDS + vout

vin (V)
M1 –0.5
vin
–1
VGS
–0.6 –0.4 –0.2 0 0.2 0.4 0.6

(a) (b)
0.6
vout
0.4 dvout/dvin (V/V)
0
0.2 vin
vin (V)
–2
(V)

0
–0.2 –4

–0.4 –6

–0.6 –8
0 5 10 15 20 25 30 –0.6 –0.4 –0.2 0 0.2 0.4 0.6
Time (ns)
(c) (d)

FIGURE 1: (a) A single-ended CS amplifier with dc bias voltages of VGS between its gate and source, and VDS between its drain and source.
Adding a small-signal voltage oin at the input results in a small-signal oout at the output. (b) The voltage transfer curve of the CS amplifier. (c) A
sinusoidal input waveform of a 100-mV peak in black and its corresponding output waveform in red. (d) The small-signal voltage gain of the
CS amplifier as a function of its input amplitude.

characterize and quantify nonlinear- The Taylor expansion allows us Once we add up all the terms,
ity so that we can speak about the to break the output voltage into the we can rewrite y ^ t h for a sinusoidal
“amount” of nonlinearity? Here, the sum of a linear term ^a 1 x h, second- input as follows:
reader is invited to pause and ponder order nonlinear term ^a 2 x 2h, and
before reading further. higher-order nonlinear terms (such y ^ t h = h 0 + h 1 sin ^~t + { 1h
as a 3 x 3) . We can now see what por- + h 2 sin ^2~t + { 2h
Characterizing Nonlinearity tion of the signal power comes + h 3 sin ^3~t + { 3h + g(5)
We know from basic calculus that any from the linear term, and what
nonlinear function can be expanded portion from the nonlinear terms, where h 0, h 1, and h 2, are determined
into its Taylor series. (Note that, enabling us to quantify the amount by the properties of f( ∙ ) (such as
strictly speaking, the function must be of nonlinearity. Yet, it remains dif- f ^0h, f l ^0h, f m ^0h, and so on) and the
infinitely differentiable for the Taylor ficult to quantify nonlinearity for input amplitude A. This equation tells
expansion to exist.) That is, we can write a general input waveform. For this us that the response of an amplifier
f m^0h 2
reason, we resort to a sinusoidal to a single sinusoidal input with fre-
y = f ^x h = f ^0h + f l^0h x + x +g input, where we can easily measure quency ~ (also called a single tone)
2
(2) the nonlinearity. can be expressed as a sum of sinu-
where we use x to represent v in and Assuming x ^ t h = A sin ^~t h, after soids with frequencies that are integer
y to represent v out . If we further some trigonometric manipulations, multiples of ~ (also called harmon-
assume that the output is zero for a we can write ics), each with a different amplitude
zero input, as we have assumed by and phase. Interestingly, this sum also
a 1 x ^ t h = a 1 A sin ^~t h
the definition of v in and v out in this includes a dc component, specified by
a 2 x 2 ^ t h = a 2 A ` 1 + sin ` 2~t - r jj
2
article, and use a 1, a 2, g to represent 2 2 h 0, which arises by all the even terms
the constant coefficients, we can write in (4). Can you explain this intuitively?
a 3 x 3 ^ t h = 3 ^3 sin ^~t h
3
a A
4 As the sinusoids of different
y = f ^x h = a 1 x + a 2 x 2 + g .  (3) + sin ^3~t + rhh . (4) frequencies are orthogonal to each

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other, the power of their sum is fier under study and feed its output calculated as 13.2%. This relative-
equal to the sum of their powers. In to a spectrum analyzer. Figure 2 ly high distortion is often unac-
other words, shows what we would observe if we ceptable as we thrive to achieve a
feed the output of our CS amplifier, THD well below 1%. But how can
y 2 ^ t h = h 20 + 1 ^h 21 + h 22 + gh (6) shown in red in Figure 1(c), to a spec- we achieve this? What techniques
2
trum analyzer (or to an FFT routine are available to circuit designers
where $
denotes the time-average in a programming language, such as to reduce this distortion? It turns
operator. Knowing that the dc power Python or MATLAB). The spectrum out that there are several, and, by
can easily be filtered (e.g., through a of the output consists of a dc com- combining them, we can easily de-
capacitor), the output power consists ponent, the input tone (at 100 MHz), sign amplifiers with distortion lev-
of the power of the desired signal and and its harmonics at integer multiples els well below 1%. We will review
the power of the undesired harmon- of 100 MHz. We make the following two of these techniques in Part III
ics. Accordingly, we can define the observations: of this article.
total harmonic distortion (THD) as ■ The output includes a dc component ■ The spectrum shown in Figure 2

valued at −24.1 dB, which is consis- also includes the thermal noise of
tent with the dc value of the output the amplifier. Unlike the harmonic
THD = 10 # log e o (7)
h 22 + h 23 + g
h 21 waveform, - 62 mV, as calculated distortions that appear as spikes
from the output waveform in Fig- or spurs in the spectrum, thermal
expressed in decibels. Alternatively, ure 1(c) in the time domain (note that noise, as discussed in the previous
THD can be expressed as the ratio of 10 # log ^- 62 mh2 = - 24.1 dB) . article, is white, with a flat PSD. The
the rms of the sum of the harmon- ■ The desired signal (at 0.1 GHz) has noise PSD for the CS amplifier in
ics to the rms of the desired tone. In an amplitude of - 8.6 dB, which Figure 1 is simulated to be approxi-
other words, is equivalent to an rms value of mately 1.5 # 10 -16 V 2 /Hz. If we in-
370 mV. Given that the rms of tegrate this noise over 3.125-MHz
h 22 + h 23 + g the input is 100 mV/ 2 = 71 mV, intervals, which we have assumed
THD = 100%.(8)
h1
this represents a linear gain of as the noise bandwidth (NBW) in our
- 5.2 V/V. simulations, we will arrive at – 93.3
Measuring THD ■ The second harmonic has an dB/NBW, as indicated by the orange
Despite the lengthy procedure to amplitude of - 28.2 dB, which dashed line in Figure 2. Accordingly,
derive the formula for THD, measur- is equivalent to an rms value of the total noise in the amplifier’s
ing it in the lab (or simulating it using 39 mV. This harmonic alone re- bandwidth (which is simulated to
tools such as SPICE) is quite straight- sults in a harmonic distortion that be 11 GHz) is roughly 1.6 mV rms ,
forward. All we need to do is to apply exceeds 10%. If we add the contri- far lower than the distortion pow-
a sinusoidal signal with a fixed ampli- butions from all other harmonics, er in this case. In other words, the
tude and frequency to the ampli- according to (8), the THD can be thermal noise adds less than 1% of
the desired signal rms to its deg-
radation, compared with 13.2% by
distortion.
0
−8.6 In general, distortion worsens (i.e.,
THD = 13.2% the THD increases) as we increase the
–20 −28.2 NBW = 3.125 MHz input amplitude. This makes intuitive
sense as larger inputs tend to saturate
–40 the output voltage. To demonstrate
PSD (dB/NBW)

this, we increase the amplitude of

our input signal to 200 mV (from an
–60
initial value of 100 mV) and observe
its corresponding output waveform,
–80 as shown in Figure 3(a). The positive
and negative peaks of the output
–100 waveform are noticeably flattened,
indicating a more distorted sinusoi-
dal shape. To quantify this distor-
–120
tion, we plot the PSD of the output
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
f (GHz) in Figure 3(b) and measure a THD of
22.3%, significantly higher than the
FIGURE 2: The PSD (in decibels per NBW, where NBW=3.125 MHz) of the single-ended CS 13.2% THD observed with the 100-mV
amplifier in Figure 1. input amplitude.

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 0.6 0
vout −6.6
−20.3 THD = 22.3%
0.4
–20
vin −34.7 NBW = 3.125 MHz
−37
0.2

PSD (dB/NBW)
–40
0
(V)

–60
–0.2
–80
–0.4

–0.6 –100

–0.8 –120
0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (ns) f (GHz)
(a) (b)

FIGURE 3: (a) The output voltage waveform of the CS amplifier in Figure 1(a) in response to a 200-mV peak sinusoidal input waveform.
(b) The PSD of the output (in decibels per NBW, where NBW=3.125 MHz).

In summary, although both noise State Circuits Mag., vol. 16, no. 3, pp. Solid State Circuits Mag., vol. 14, no. 2,
15–26, Summer 2024, doi: 10.1109/MSSC. pp. 11–17, Spring 2022, doi: 10.1109/
and distortion degrade the quality of 2024.3419508 MSSC.2022.3164811
the desired signal, they are created [2] A. Sheikholeslami, “The art of control- 
ling electrons [Circuit Intuitions],” IEEE
by different mechanisms and exhibit
vastly different properties and charac-
teristics. Noise (especially the thermal
noise) is usually the result of random
movement of electrons in electronic
devices and adds a random, unwanted
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Distortion can be easily measured standards, and professional
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number and amplitude of the har- Visit www.ieee.org.
monics generated in the output. THD
increases with input amplitude, mean-
ing that larger input signals result in
greater distortion.

Acknowledgment
I would like to thank my Ph.D. stu-
dent, Jhoan Salinas, for his assis-
tance with preparing simulation
results for this article.

References Publications / IEEE Xplore ® / Standards / Membership / Conferences / Education

[1] A. Sheikholeslami, “Noise and ­
distortion,
part I [Circuit Intuitions],” IEEE ­S olid

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