IEEE Solid-State Circuits Magazine - Spring 2016 - 79

inverTing The TAngenT

5
4
3
2
1
0
-1
-2
-3
-4
-5
-9

invertible function. This is the arctangent, shown as the solid curve in
Figure S3. By the nature of the tangent, the range of the arctangent is restricted to the interval [-r/2, r/2]. By requiring invertibility in our mathematical model, we have caused a restriction on the range of the inverse
tangent. Physics has no such restriction on the inverse tangent [10].
In communications we really desire that signal phase extend over the
interval [-r, r]. The arctangent (atan) does not do this, so the function
atan2 is defined that includes parts of the adjacent curves of the complete
inverse tangent. Atan2 is evaluated in a piecewise fashion by first identifying which of the inverse tangent curves needs to be evaluated and then
resolving the desired angle. In all cases, the use of atan or atan2 incurs a
restriction on the range of output phase angle values unless an additional
memory operation is used to access any of the inverse tangent curves.

Inverse Tangent
8
Arctangent
6
4
2
2π Arctangent
0
(atan2)
-2 Arctangent (atan)
-4
-6
-8
7
9
-5
-3
-1
1
3
5
Tangent Value
To Infinity To -Infinity
To Infinity
Inverse Tangent and Subsets
Phase (Radians)

Tangent Value

It is well known that the primary trigonometric functions (sine, cosine,
and tangent) are defined for any angle in the domain (-∞, ∞). The range
of values for the sine and cosine are restricted within [-1, 1], but the
tangent extends across the entire range of (-∞, ∞). All trigonometric
functions are periodic across the domain of phase, which causes their
inverses to be multivalued and therefore not functions. Invertibility of
trigonometric functions is achieved by accepting restrictions in the domain and/or range of the inverse.
Of particular importance to the rectangular-to-polar coordinate
transformation necessary for the quadrature signal model is the inverse
tangent (4). Because we require that the coordinate transformation relationships be mathematically invertible, a multivalued inverse cannot
be used. One instance of the inverse tangent is defined to provide this

-7

To -Infinity

-5

-3 -1 1
3
Phase (Radians)
Tangent

FIGURE s3: details of the tangent and its inverse options.

circuits, there is no physical mechanism to dissipate this RF input
power. This leads to a corollary
question: is there any actual value to
having an "input impedance match"
for a CS amplifier?
It is physically more accurate to
change (1) into a form that I call total
amplifier efficiency:
TAE =

POUT
.
P DC + P IN

(2)

Here it is clear that the input RF
input power adds to the supply
dc power within the CS amplifier
and is dissipated. This is actually
what is happening. It is interesting
to note that when amplifier gain is
high, PIN 11 POUT and with sufficiently high gain the RF input power
can be neglected. It is similarly true

that having the RF input signal
work into the natural high reflection
from a CMOS (here NMOS) gate's reactive impedance, the input power
in (2) also goes toward zero. In
this case both (1) and (2) reduce to
the same value, which is the stage
drain efficiency.

Why It Matters
Our descriptions of circuit activity
are best when they are unambiguously descriptive of the underlying
physical mechanisms. Only then
can we move forward with useful
predictions of what should happen
when specific changes are made.
If our circuit descriptions are not
physically accurate, any extrapolations are risky. This leads to the next
misunderstandings, which originate
from mathematics.

Misunderstanding #4:
Quadrature Signal Representation
Is Physically Accurate
Quadrature signal processing is so
widely used that it has to be correct
and complete.

Clarification
Quadrature signal processing is extremely useful and is correct in as
far as it goes [21], but it is definitely not physically complete as
used. The quadrature signal model
is primarily limited by its use of the
arctangent, which allows the rectangular-polar coordinate transformation to be invertible (see "Inverting
the Tangent").
The solution to Maxwell's equation of the propagating electromagnetic wave is a phasor, of the form
AE ji . This is a polar solution, and

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