IEEE Solid-State Circuits Magazine - Winter 2016 - 9
Notably, Smith also predicted the
3-dB bandwidth of the resulting bandpass response to be 1/ (rNR S C),
where R S denotes the source resistance. This equation proves useful for estimating the bandwidth in
terms of the total commutated capacitance. Smith extended the idea
to a first-order high-pass response
as well, arriving at the notch filter illustrated in Figure 3(b).
The foregoing topologies exemplify time-variant circuits, lending
themselves to a general model proposed by Franks and Sandberg in
1960 [4] and shown in Figure 4. Each
path mixes the input signal with
different phases of the local oscillator (LO), translates the spectrum
to baseband, subjects the downconverted signals to a desired transfer function, and mixes the results
with the LO phases again so as to
return the (shaped) spectrum to its
original center frequency. The term
"N-path filters" was evidently coined
by Franks and Sandberg to refer to
these circuits. An important difference
between this abstraction and Smith's
circuits is that the former assumes
unilateral stages whereas the latter
entail "transparency" between the
input and the output, a useful property for impedance translation.
Impedance Translation by
Partial Commutation
Translational circuits can shift an impedance to a well-defined center frequency. For example, the impedance
of a capacitor, 1/ (j~C), can be translated to a center frequency of ~ LO,
taking on the form 1/ [j (~ - ~ LO) C] .
In other words, the impedance of
the new network goes to infinity
at ~ = ~ LO rather than at ~ = 0. In
this section, we investigate how the
translation occurs. Due to switching
activities, commutated networks can
produce a nonsinusoidal voltage in
response to a sinusoidal current. To
find the impedance, therefore, we
must compute the first harmonic of
the voltage.
Consider the parallel RC branch
shown in Figure 5(a), assuming I in (t) =
I 0 sin ~ in t and R 1 C 1 & Tin = (2r/~ in) .
(a)
(b)
Figure 3: Smith's (a) band-pass and (b) notch filter implementations.
Input
x1(t )
u(t ), U(s)
h(t )
y1(t )
p [t ]
v1(t )
∑
Output
v(t ), V(s)
q [t ]
xn(t )
h(t )
p [t - (n-1)T ]
xN (t )
yn(t )
vn(t )
q [t - (n-1)T ]
h(t )
p [t - (N-1)T ]
yN (t )
vN (t )
q [t - (N-1)T ]
Figure 4: A general model of translational circuits proposed by Franks and Sandberg.
In this case, most of the input current
prefers to flow through C 1, generating VAB . (1/C 1) # I in (t) dt. Now, let
us switch C 1 periodically and study a
special case where the input frequency
The use of timevariant systems
(e.g., mixers) for
the translation of
transfer functions
dates back to the
late 1940s and
early 1950s.
is equal to the LO frequency [Figure 5(b)].
We wish to study the impedance seen
between A and B, Z AB . With the
phase relationship shown here, we
observe that the positive half cycles of
I in flow primarily through C 1, and the
negative half cycles entirely through
R 1 . That is, C 1 receives a half-wave
rectified version of I in, thereby accumulating positive charge. The circuit reaches the steady stage when,
during each positive half cycle, the
charge delivered by I in to C 1 is equal
to the charge drawn by R 1 from C 1 .
The key point here is that the output
voltage swing can become arbitrarily large if R 1 has an arbitrarily high
value-even though C 1 periodically
switches into the circuit.
This analysis culminates in the following observation: a sinusoidal current of frequency fin = ~ in / (2r) = fLO
can produce a very large voltage
swing at fin between nodes A and
B in Figure 5(b), revealing that the
equivalent impedance of the switched
IEEE SOLID-STATE CIRCUITS MAGAZINE
W I N T E R 2 0 16
9
Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Winter 2016
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