IEEE Solid-State Circuits Magazine - Spring 2016 - 80

MApping The ModelS To hArdwAre
Any mathematical model of a circuit or system is considered a physically accurate model when each component of the mathematics corresponds directly to a
current flow in the circuit or system. Otherwise the
model is just representative of the end result. One result is illustrated Figure S4.
For the quadrature modulator, the quadrature signal
model is physically accurate because each term in this
model is directly represented in circuit functions and
current flows as shown in Figure S4. Once the final signal is generated and passed through an amplifier (here
a linear amplifier with gain g), the physical accuracy
of the quadrature model is lost. Specifically, the amplifier is sensitive to the magnitude of its input signal,
while the input frequency and any phase shift is passed
through unchanged. None of the terms in the quadrature signal model directly and individually map onto
amplifier current flows.

s (t ) = A (t ) cos (ωt + θ (t ))

+

s (t) = I (t) cos (~t) - Q (t) sin (~t) . (3)
Several important things are lost in
this transformation. The first thing
lost is a direct mapping to underlying physics of signal current flow in
an amplifier (see "Mapping the Models
to Hardware"). In particular, what electron within an amplifier follows this
I(t) waveform? Or this Q(t) waveform?
The answer is none. The quadrature
model (3) accurately describes the net
result of what we observe but does
not always follow the physics underlying the process of interest. Further,

80

S P R I N G 2 0 16

g

-
?

?

?

?

I (t ) cos (ωt ) - Q (t ) sin(ωt ) = s (t )
I (t ) cos (ωt ) - Q (t ) sin(ωt ) = s (t )

FIGURE s4: Signals in the quadrature modulator directly correspond to each
term in the quadrature model. This correspondence does not hold for a signal
passing through an amplifier.

Our descriptions of circuit activity are best
when they are unambiguously descriptive
of the underlying physical mechanisms.
the mathematics of polar numbers
is inherently nonlinear [22]. In general, engineers and mathematicians
do not like to deal with mathematical
nonlinearities, so as soon as a nonlinearity is encountered the first motivation is to linearize it. By applying
a projection linearization technique,
the (polar) phasor is converted to the
quadrature modulation format

s (t ) = A (t ) cos (ωt + θ (t ))

we are doing and in our ability to
take full advantage of the physics
available to us to possibly do more.

Misunderstanding #5

the quadrature (Cartesian) to polar
conversion inherently restricts phase
due to its use of the arctangent (see
"Inverting the Tangent")
i = arc tan c

Q
m.
I

(4)

This gets us immediately back to Misunderstanding #1, something that
physics definitely does not restrict.
The entire inverse tangent is physically real (4) even though it is not a
function. Mathematical convenience
is nice, but physical reality is far more
useful and important.

Why It Matters
History shows that progress is best
achieved when the models used
directly relate to the underlying
physics of the situation modeled.
When using models that do not
accurately describe the underlying
physics, we are not being complete
in both our understanding of what

IEEE SOLID-STATE CIRCUITS MAGAZINE

We can and should describe signals
and impedances as having real and
imaginary parts:
Z = R + jX ; S = I + jQ .

(5)

In mathematics, the imaginary number sqrt(-1) leads to the concept
of complex numbers when combined with real numbers. This same
nomenclature is appropriate in all
applications that use complex numbers in their analysis.

Clarification
Again, the words we use matter -
descriptions and names are important (see "Nothing in Engineering Is
Imaginary").
Nothing in the engineering practices of signal processing and circuit
design is imaginary. Our purpose in
engineering communication is to succinctly and unambiguously describe
what it is that we are doing. This is
most appropriately done by using



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

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