IEEE Solid-State Circuits Magazine - Winter 2016 - 12

RS

Vout

+

Vin

Vout

RSW

Vin

-

C1

Ideal
Network

CN

RSW
RSW + RS
0

fLO

2fLO

3fLO

f

Figure 9: The effect of switch resistance on the transfer function.

A
+
X

VAB R1

Iin

C1

LO

LO

Y

C2

-
B
LO

Iin

V1

VAB

t

Figure 10: Steady-state waveforms in a commutated circuit.

∆T

LO

∆T
Iin

LO
t

A
+
Iin

Rsw
X

VAB R
1
-
B

Rsw
Y

Iin

A
+
VAB R1
-
B
VAB
A
+
VAB R1
-
B

(a)

LO

LO

Req
X
C1

Y
C2

Second-Order Effects

(b)

X
Req
2

C1

LO

LO
Y

C2

Req

(c)

Figure 11: (a) The overlap time between LO and LO , (b) the equivalent resistance for
differential discharge, and (c) a single-ended equivalent.

12

W I N T E R 2 0 16

IEEE SOLID-STATE CIRCUITS MAGAZINE

replacing continuous-time capacitive branches in a network with
their commutated counterparts permits such a translation.
Applying this procedure to a
first-order RC filter yields the circuit in Figure 8(a) and a band-pass
response around fLO (and its harmonics). Similarly, as depicted in
Figure 8(b), a high-pass section
can be converted to a notch filter
centered at fLO. In both cases, R S C
must be much greater than TLO .
Recall from Figure 3 that Smith predicts the bandwidth of the response
in Figure 8(a) to be 1/ (NrR S C) . We
can thus identify three critical attributes of this circuit: 1) a band-pass
response with an arbitrarily narrow
bandwidth centered around an arbitrarily high LO frequency, and this
attribute represents an RF filter with
an arbitrarily high quality factor Q;
2) a response with a well-defined,
precise center frequency, fLO; and
3) a precisely programmable center
frequency. Absent in time-invariant
RF filters, this unique combination
enables accurate channel selection
in RF receivers.
Smith's expression, 1/ (NrR S C),
is intriguing in that it implies a reduction in the bandwidth as the
number of commutated capacitors
increases even though only one capacitor is tied to the output at a given
point in time. This puzzle can be
solved if we recognize in Figure 8(a)
that the "dwell" time of the capacitors, i.e., the pulsewidth of the LO
phases, decreases as N increases.
Consequently, each capacitor has
less time to interact with I in and R S
and experiences a smaller voltage
change-as if its value were larger.

2

A number of phenomena degrade
the performance of commutated networks and require attention. These
include the on-resistance of the
switches, R sw, the translations of
the source impedance, and the finite
rise and fall times of the LO.
The on-resistance of the switches
in Figure 8(a) can be factored out as
illustrated in Figure 9(a), suggesting



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

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