IEEE Solid-State Circuits Magazine - Spring 2016 - 81

noThing in engineering iS iMAginAry

Reactance
Axis

Quadrature-Phase
Axis

Using the same word to mean different things is an inherent source of
These are unambiguous names that are perfectly valid axis labels, as
ambiguity in discourse, leading toward communication confusion. It is
shown in Figure S5.
far better to use descriptive words that avoid both ambiguity and confuBy saying "reactance" instead of "the imaginary part," it is immedision in the mind of the listener. Something that is clear to a professor will
ately known that the topic is impedance and specifically effects from
likely not be clear to a student, particularly in early exposure. Similarly,
capacitance or inductance. Similarly, by saying "quadrature-phase"
something understood by an engineering leader will often not be initially
instead of "the imaginary part," it is immediately known that the topic
grasped by a new member of a project team with little or no experience
is signal modulation using the quadrature model. There is no need for
in a specific field. In all of these situations, it is particularly important to
additional context, and communication is much more effective.
not be ambiguous in presenting material allowing
communication to be effective and timely.
My personal experience with this problem
largely involves the use of the word "imaginary."
Originally used in mathematics for sqrt(-1) = i,
the imaginary number gained some physical significance when Gauss showed that i = j = ejr/2,
signifying a quadrature relationship. In engiZL
Q
neering, we have many instances of quadrature
relationships that appear in very different situ-R
+R Resistance
I
In-Phase
ations. Experience shows that using the name
ZC
Axis
Axis
"imaginary" for all of them does produce significant confusion.
Fortunately we already have unique names for
Impedance "Space"
Signal "Space"
these major quadrature relationships. For circuit
impedance we have the names resistive and reactive. For the signal quadrature relationship we FIGURE s5: unambiguous axis labels for impedance and signal modulation planes
have the names in-phase and quadrature-phase. using Cartesian coordinates.

descriptive terms and not using the
same word in multiple and different
situations. Fortunately such unambiguous descriptive names already
exist, and it is best that these descriptive names be widely used. Relevant
examples include the following:
■ for signals: in-phase and quadrature-phase components to the signal modulation
■ for impedances: resistive and reactive components to the impedance.

Why It Matters
Use of identical mathematical labels
when describing multiple engineering signals and systems is fundamentally ambiguous and causes
significant confusion among both
engineering students and practicing engineers. Even though John von
Neumann once said "... in mathematics you don't understand things. You
just get used to them," we can, and
should, do much better.

Firm Foundations to Rely On
All engineers are well trained in
mathematics, a trait that gets deeper
throughout graduate school. This
skill certainly is valuable, but it is
also important to have an equally
excellent grasp of the physical principles that the mathematics are
developed to describe. Looking back
on my nearly 40 years of postgraduate electronics experience, there are
four key physical principles that I
find particularly useful in keeping
me honest within this field:
■ Ohm's law
■ Fourier transform (FT)
■ conservation of energy
■ Shannon's capacity limit.

Ohm's Law (Maxwell's Equations)
Ohm's law is derivable from Maxwell's equations. This is not just a
trivial relationship that is only seen
in elementary course work. Rather, it
is a guiding light that we must con-

tinuously use. This foundation to all
of circuit design remains extremely
valuable, even 190 years after its
publication by Georg Ohm.
It is Ohm's law that results in
the separation of amplifier linearity and amplifier energy efficiency.
Ohm's law requires that maximizing
one necessarily minimizes the other
[12]. This tradeoff is not simply an
inconvenience. It means that any
effort to improve amplifier energy
efficiency beyond class-A (the only
linear amplifier at the transistor)
must involve use of nonlinearity at
the transistor. For example, classAB, -B, and -C amplifiers all use transistor cut-off operation, which is
clearly a nonlinear operating mode.

Fourier Transform (FT)
Relating the physical time domain to
the mythical (but extremely useful)
frequency domain is done through
the FT. The key points to remember

IEEE SOLID-STATE CIRCUITS MAGAZINE

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Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Spring 2016