ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo - - 1 Chapter II Circuit Analysis Fundamentals From a design engineer’s perspective, it is more relevant to understand a circuit’s operation and limitations than to find exact mathematical expressions or exact numerical solutions. Precise results can always be obtained through proper circuit simulations and through numerical analysis with software programs such as MATLAB or Maple. However, these tools cannot design the circuit for you, especially if you are dealing with analog circuits. Even though design automation programs can synthesize complete digital circuits, and progress is being made toward automating analog circuit design, engineers will still have to maintain knowledge about the functionality of circuits and systems. Hence, this chapter will review and introduce some basic circuit analysis methodologies that can be applied to gain insights into performance characteristics and design trade-offs. Generally, the approach will be to identify and utilize appropriate simplifications that enable approximate analysis, which is an essential skill when making initial component parameter selections for complex circuits and when performing optimizations. Circuit analysis techniques are of fundamental importance when determining the main parameters of an electronic system. Even though software simulators are extremely useful as complementary tools, they cannot compensate for a lack of theoretical understanding on the part of the designer. Therefore, an effective system design procedure begins with selecting the most suitable active devices (transistors, diodes, etc.) and passive devices (e.g. resistors, capacitors, inductors) along with power supplies, and signal sources. Next, the proper component interconnections have to be defined, and the device parameters have to be chosen together with the operating point for each of the active devices in order to realize the desired electronic operation. These steps require theoretical knowledge as well as experience. For this reason, fundamentals of circuit analysis are revisited in this chapter before focusing more on practical examples throughout the book. Special attention is devoted to the physical interpretation of the results as well as the main properties of the basic circuit components and topologies that reoccur frequently in complex circuits. It is important to investigate circuits with the proper analysis techniques, but for an engineer it is even more significant to learn how to apply the results during the design of efficient and reliable electronic systems. Finally, designs should be verified and completed by optimizing component values through accurate simulations with more realistic and detailed models of the components. Before we discuss the properties of basic electronic networks, it is necessary to introduce some essential definitions for the analysis of signal components, system transfer functions, decibel notations, and magnitude/phase responses. II.i. DC and AC signals A typical data acquisition system, such as the one shown in Figure 2.1, consists of a sensor (transducer), a preamplifier, an analog filter, and the signal processor that typically includes an analog-to-digital converter. The transducer detects the physical quantities to be measured and processed (temperature, pressure, glucose level, frequency, wireless signal, etc.). Usually, the sensor’s output is a small signal (in the range of microvolts to millivolts, i.e., in the range of 10 -6 to 10 -3 volts) and it must be amplified to fit within the linear input amplitude range of the analog-to-digital converter. Since the desired signal is typically accompanied by undesired information and random noise, the signal may be “cleaned” through a frequency-selective filter that removes most of the out-of- band contents (at undesired frequencies) before the sensed signal is converted into a digital format and further processed through dedicated software. Transducer Filter (noise removal) Preamp Analog Processing Blocks Electrical Signal Digital Signal Processor Digital Processing Blocks Biological System Analog-to- Digital Converter Biological Signal Fig. 2.1. Front-end of a typical bio-electronics system.
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ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo
- - 1
Chapter II
Circuit Analysis Fundamentals
From a design engineer’s perspective, it is more relevant to understand a circuit’s operation and limitations than
to find exact mathematical expressions or exact numerical solutions. Precise results can always be obtained
through proper circuit simulations and through numerical analysis with software programs such as MATLAB or
Maple. However, these tools cannot design the circuit for you, especially if you are dealing with analog circuits.
Even though design automation programs can synthesize complete digital circuits, and progress is being made
toward automating analog circuit design, engineers will still have to maintain knowledge about the functionality
of circuits and systems. Hence, this chapter will review and introduce some basic circuit analysis methodologies
that can be applied to gain insights into performance characteristics and design trade-offs. Generally, the
approach will be to identify and utilize appropriate simplifications that enable approximate analysis, which is an
essential skill when making initial component parameter selections for complex circuits and when performing
optimizations.
Circuit analysis techniques are of fundamental importance when determining the main parameters of an electronic
system. Even though software simulators are extremely useful as complementary tools, they cannot compensate for
a lack of theoretical understanding on the part of the designer. Therefore, an effective system design procedure
begins with selecting the most suitable active devices (transistors, diodes, etc.) and passive devices (e.g. resistors,
capacitors, inductors) along with power supplies, and signal sources. Next, the proper component interconnections
have to be defined, and the device parameters have to be chosen together with the operating point for each of the
active devices in order to realize the desired electronic operation. These steps require theoretical knowledge as well
as experience. For this reason, fundamentals of circuit analysis are revisited in this chapter before focusing more on
practical examples throughout the book.
Special attention is devoted to the physical interpretation of the results as well as the main properties of the basic
circuit components and topologies that reoccur frequently in complex circuits. It is important to investigate circuits
with the proper analysis techniques, but for an engineer it is even more significant to learn how to apply the results
during the design of efficient and reliable electronic systems. Finally, designs should be verified and completed by
optimizing component values through accurate simulations with more realistic and detailed models of the
components. Before we discuss the properties of basic electronic networks, it is necessary to introduce some
essential definitions for the analysis of signal components, system transfer functions, decibel notations, and
magnitude/phase responses.
II.i. DC and AC signals
A typical data acquisition system, such as the one shown in Figure 2.1, consists of a sensor (transducer), a
preamplifier, an analog filter, and the signal processor that typically includes an analog-to-digital converter. The
transducer detects the physical quantities to be measured and processed (temperature, pressure, glucose level,
frequency, wireless signal, etc.). Usually, the sensor’s output is a small signal (in the range of microvolts to
millivolts, i.e., in the range of 10-6 to 10-3 volts) and it must be amplified to fit within the linear input amplitude
range of the analog-to-digital converter. Since the desired signal is typically accompanied by undesired information
and random noise, the signal may be “cleaned” through a frequency-selective filter that removes most of the out-of-
band contents (at undesired frequencies) before the sensed signal is converted into a digital format and further
processed through dedicated software.
TransducerFilter (noise
removal)Preamp
Analog Processing Blocks
Electrical Signal
Digital
Signal
Processor
Digital Processing BlocksBiological
System
Analog-to-
Digital
Converter
Biological Signal
Fig. 2.1. Front-end of a typical bio-electronics system.
ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo
- - 2
The electrical signals at the output of the transducer are normally composed of two components: the DC (direct
current) and AC (alternating current) signals. As shown in Figure 2.2, the DC component is a time-invariant
quantity, while the AC component is a time-variant quantity and usually this AC component contains the relevant
information to be processed.
You will learn that the signals processed in the amplifiers also contain both DC and AC components. The general
expression for signals such as the one that appears at the output of the transducer in Figure 2.1 can be denoted as:
sAC(t) = SDD + sac(t) (2.0)
where SDD and sac denote the DC and AC signal components, respectively. Before we discuss the manipulation
techniques of these signals, let us define some of the nomenclature based on common conventions in electronics
literature. The following notations are used in this text to identify the nature of the signal components:
CAPITALCAPITAL labels represent only the DC component; e.g. SDD = 10V; IX = 2A.
lowercaselowercase stands for the AC component only; e.g. sac(t) = 2∙sin(t + ) V; isignal(t) = 2∙ej(t + ) A.
lowercaseCAPITAL stands for the combination of DC and AC components; e.g. sAC = SDD + sac(t); iSIGNAL = IX + isignal .
An example of individual signals and their combination is illustrated in Figure 2.2.
Time
DC component
AC component
Amplitude
Time
AmplitudeDC + AC
SDD
sac(t)
sAC(t) = SDD+sac(t)
0 0
(a) (b)
Fig. 2.2. Time domain signals: a) standalone DC and AC signals, b) combination of AC and DC signals.
Although signals found in most real world scenarios are not periodic, the analysis and design of electronic circuits is
conducted based on periodic signals (sinusoidal waveforms, pulse train, triangular waveforms, modulated
waveforms, etc.) because many waveforms can be approximated with summations of periodic signals. The case of
sinusoidal waveforms is especially interesting because periodic waveforms can be represented by a Fourier series
using sinusoids as a basis of functions. It is well known that many practical signals are continuous-time, periodic
with period T, and real functions f(t) are defined for all t; which allows representation with a simplified Fourier
series in the following form1:
T
tncosC)t(fn
n
200
, (2.1)
1 This is a simplified form of the Fourier series; the general form is
T;eC)t(f
n
tnj
n
2
00
ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo
- - 3
where 0 (2/T = 2f) is the fundamental frequency component in radians/sec used for series expansion. In practice,
the f(t) has to be expressed as a linear combination of sine and cosine waveforms, but to simplify our discussion let
us ignore the sine functions. Cn is the nth Fourier coefficient, and it is computed as follows:
dtetfT
C
Tt
t
tjn
n
0
0
01
. (2.2)
In this introduction, Equation 2.1 is a simplification of real signal representation with the intention to emphasize that
the signal’s spectrum often has sinusoidal components at frequencies which are multiples of 0. Furthermore, the
analytical expressions for the signals processed by the electronic circuits during the evaluation of relevant
performance characteristics typically have the same form as in Eq. 2.1. Examples on the use of Fourier series can be
found in a number of textbooks that deal with signal processing, circuit analysis, and circuit realizations.
Notice that C0 (n = 0) in 2.1 represents the DC component (average) of f(t) obtained with dttfT
C
Tt
t
0
0
10
.
Depending on the applications, this DC component may or may not contain desired information. The coefficient C1
(n = 1) is the fundamental (AC) component of the signal, which for practical purposes carries the most relevant
information to be processed in communication applications. The amplitude of the other terms, determined by Cn (n >
1) are known as the harmonic components and are rarely studied in introductory courses. However, they are relevant
and an important subject matter in advanced studies because they decide the quality of the signal. Specifically, the
high-order terms indicate the amount of signal distortion, and they generate undesired frequency components when
processed by nonlinear circuits, which limits performance.
Eqs 2.1 and 2.2 are beneficial because they allow one to avoid analyzing circuits for all possible input signals.
Instead, it is preferred to analyze them for the case of sinusoidal inputs and, from those results, infer system
behavior for any other kind of input signal. This is the so-called frequency domain analysis. In the frequency
domain, we usually extend the analysis of the signals from DC (= 0) up to very high frequencies. Even though the
frequency response of a system can be accurately computed and predicted, it is often not easy to find the exact
transfer function of a system, especially if the mathematical representation is complex. In other words, exact
mathematical derivations often imply too much effort for very little insight since a real electronic system may
contain millions of transistors! In these cases, it is important to have a good understanding of a circuit’s behavior
from a top level rather than being concerned with minor details that can be obtained from simulations. Few
observations about the properties of frequency responses and logarithmic functions lead to useful tools for
evaluating many complex analog circuits from a top-level perspective. Note, in some areas such as power
electronics, the time-domain analysis is more commonly used than the frequency domain analysis. Pulse and
impulse system responses are employed in those cases, and these topics will also be partially covered in the
following subsections.
II.2. Decibels and Bode Plots
i) Magnitude response. In electronics, the magnitude response is usually plotted in decibels (logarithmic scale),
making it easier to compare strong and weak signals. In general, it is often very convenient to use logarithmic scales
in cases where it is hard to visualize signal differences using linear scales. The magnitude in decibels (dB) of a
complex function f(x) is defined as:
)(log20)(log10 10
2
10 xfxfxfdB
, (2.3)
where |f(x)| stands for the magnitude of f(x). The expression 10∙log10 (|f(x)|2) is commonly used in communication
systems where the power of the signal (|f(x)|2) is employed, while the second definition 20∙log10 (|f(x)|) is more
popular in baseband electronic applications (e.g. biomedical, audio, and video frequencies) where the main
quantities of interest are either voltage or current. In any case, the use of decibel notation (logarithmic scale) is very
convenient when dealing with intricate transfer functions. Some relevant properties of the logarithmic function are
listed in Table 2.1. As an engineering student, you most likely already have a good background in logarithm
operations, but be sure to master these properties as they will be used extensively throughout your study of
electronics.
ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo
- - 4
Table 2.1 Fundamental properties of the logarithmic function.
Notice that:
10∙log10(1) = 0 dB shows that 0dB with logarithmic scale corresponds to unity gain with linear scale.
The magnitude of 2-1/2 (=0.7071) corresponds to - 3dB. The frequency at which the gain of a circuit is reduced
by 3dB is often referred to as 3dB-freuncy (f3dB), cutoff frequency, or corner frequency.
The multiplication of functions in linear scale is equivalent to addition of logarithmic functions. In many cases,
it is a lot easier to manipulate logarithmic equations as well as to get a better intuition about the characteristics
of a system through plots of the transfer function in logarithmic scale.
The ratio of two quantities maps into the subtraction of two logarithms.
Exponents in linear expressions correspond to scaling factors in front of logarithms.
To appreciate the benefits of the logarithmic scale, let us consider the following complex function:
)/(1
1
22
1
2
1
AxjA
A
jxA
Axf
, (2.4)
where A1 and A2 are real numbers, and jx is the imaginary part of the denominator. Notice that j2 = -1 and that x is a
real variable which can take values in the range of {0 < x < }. A2 in the above equation defines the frequency of the
pole, which is a significant parameter because it corresponds to the frequency at which the gain (A1/A2) of this first-
order transfer function is reduced by 3dB. The squared magnitude (power) of this function can be expressed as:
22
2
2
12)(*)()(
xA
Axfxfxf
, (2.5)
where f*(x) is the complex conjugate of f(x). The magnitude response (plot of the magnitude of f(x) versus x) for
this function with A1 = A2 = 10 is depicted in Figure 2.3a using linear scale. You can observe that it is very difficult
to extract information from this plot. When x is small, the value of f(x) is close to unity, and it decreases rapidly as x
approaches 10. For example, it is very difficult to accurately predict the value of f(x) for x >> A2 = 10 based on this
plot. Of course, you can see that the function has a tendency to go to zero, but it would be hard to quantitatively
estimate how small the values are when x is large. The same transfer function is plotted with dB-scale on the y-axis
in Figure 2.3b, where the x-axis is in log10 scale. To obtain this dB-log plot, Eq. 2.5 is substituted into Eq. 2.3 prior
to plotting. Clearly, we can extract significantly more information by visual inspection of this plot, especially under
consideration of the properties in Table 2.1. For instance, the function is approximately flat from very small x-values
up to x = 5 [note: f(0) = 1 = 0dB], and the -3dB value occurs at x = A2 = 10 [note: f(10) = 0.707]. The gain of the
Function Properties
log10(1) 0
log10(10) 1
log10(xN) N∙log10(x)
log10(1/xN) -N∙log10(x)
10∙log10(1/2) = 20∙log10(2-1/2) -3.01
10∙log10(2) = 20∙log10(21/2) +3.01
log10(f∙g) log10(f) + log10(g)
log10(f/g) log10(f) - log10(g)
log10(10N) N
log10(AfX/BgX) log10(AfX) - log10(BgX)
= log10(A) + X∙log10(f) - log10(B) - X∙log10(g)
ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo
- - 5
transfer function at x = 1000 is -40 dB, which corresponds to a magnitude equal to 0.01. You can check this value
using your calculator and Equation 2.3.
0E+0 2E+2 4E+2 6E+2 8E+2 1E+3
x
0.00
0.20
0.40
0.60
0.80
1.00
Mag
nit
ude
.1E+0 1E+1 1E+2 1E+3
x
-40
-30
-20
-10
0
Mag
nit
ud
e (d
B)
(a) (b)
Figure 2.3. Magnitude response of the function 10/(10 + jx) using: a) linear-linear scale, b) dB-log scale.
To get more insight, let us more fully examine the previous transfer function. Taking advantage of the properties of
the logarithmic function, the magnitude of the transfer function given in Eq. 2.5 can be expressed in dB:
)(log*10)(log10 22
210
2
110 xAAxfdB
(2.6)
Notice that:
For x << A2 this equation can be approximated as 20∙log10(A1) - 20∙log10(A2) = 20∙log10(A1/A2), which is equal
to 0dB for the example case with A1 = A2 = 10.
For x = A2 = 10, the magnitude of f(x) becomes
|f(x)| = 10∙log10(A12) - 10∙log10(A2
2 ∙ 2)
= 20∙log10(10) - 10∙[log10(102) + log10(2)]
= 20∙log10(10) - 20∙log10(10) - 10∙log10(2)
= -10∙log10(2) = -3.01dB.
For x >> A2 equation 2.6 can be approximated as 10∙log10(A12) - 10∙log10(x2) = 20∙log10(10) - 20∙log10(x) dB.
Notice that the expression consists of a constant term and a negative term that decreases proportional to the -
20∙log10 function of x. If we use a log10 scale on the x-axis, this corresponds to a straight line with a slope of -
20, which agrees well with the plot shown in Figure 2.3b for x >> A2 = 10. If x = 1000, then |f(1000)| ≈ 20 -
20∙log10(103) = -40 dB, which again fits well with the visual inspection that can be made in Figure 2.3b.
The simple analysis above can be performed for any x value.
The transfer functions can be easily plotted in log10-dB scale after some additional observations. For frequencies
beyond the pole frequency (A2 in the previous example) |f(x)| is approximately equal to 20∙log10(A1) - 20∙log10(x)
dB; some values for the previous case are provided in Table 2.2. Beyond the frequency of the pole, a one-decade
increase on the x-axis corresponds to a gain change of -20 dB (equivalently, this can be referred to as 20dB
attenuation). Hence, the slope of the function with log10-scale on the X-axis is -20dB/decade. It can also be shown
that an increase by a factor of 2 (1 octave) on the x-axis corresponds to an attenuation of 6dB for |f(x)| with x >>
A2, leading to a slope of –6dB/octave. Scales in octaves are frequently used in music and medical equipment, and
they provide more resolution (change by a factor of 2 per division on the x-axis) compared to log10-scale on the x-
axis. In both cases, the scales of the axes make it easier to plot complex transfer functions and to identify relevant
properties through visual inspection of their magnitude responses.
ELEN-325. Introduction to Electronic Circuits: A Design-Oriented Approach Jose Silva-Martinez and Marvin Onabajo
- - 6
Table 2.2 Example values for the magnitude of the 20∙log10(x) (x < 1) function in dB.
x
10-1 -20
10-2 -40
10-3 -60
10-4 -80
10-5 -100
ii) The phase response of the first-order transfer function (solid line) displayed in Figure 2.4 requires a bit more
detailed analysis than for the magnitude response. A brief analysis is presented in Appendix A (to be added). It is
important to recognize that a complex expression can always be written in polar form: f(x) = |f(x)|ej, where is the
phase of f(x) defined as = tan-1[Imaginary(f)/Real(f)]. Due to the properties of the exponential function, it is
evident that the phase properties listed in the Table 2.2 hold; where R and Im represent the real and imaginary parts,
respectively.
Table 2.2 Relevant characteristics of complex numbers.
Function Evaluation
ImjRxf
RIm/tan
ImRxf
exfxf
f
j f
1
22
xgxf gfjexgxf
xg
xf
gfje
xg
xf
xgxgxg
xfxfxf
m
n
...
...
21
21
m
k
gk
n
i
fij
m
ne
xgxgxg
xfxfxf11
...
...
21
21
To plot the phase of a transfer function by hand, you must first evaluate the phase of each individual term and add
the phases of the functions in the numerator while subtracting the phase of the functions in the denominator. For
instance, the phase response of Eq. 2.4 is depicted (solid line) in Figure 2.4 (A1 = A2 = 10). Correspondingly, the
phase shift of the complex function can be assessed from the following expression:
)/(tan0][][
][][)]([
2
1
22 AxjxAPhaseAPhase
rDenominatoPhaseNumeratorPhasexfPhaseradians
. (2.7a)
This equation is giving the phase in radians. To convert the phase to degrees, you must multiply the value in radians