IEEE Solid-State Circuits Magazine - Winter 2016 - 10

A

A
+
Iin

+

VAB

R1

C1

Iin

VAB

-
B

S1

R1

C1

-
B

Iin

LO

LO
Iin

VAB
t

(a)

VAB

1

1

Iin

I inR

I inR

LO

(b)

Magnitude (dB)

Iin R

1

R1
I in

VAB

t

(c)
A
+
Iin

t

S1

VAB

LO

R1

-10
-20
-30
-40
1.99 1.995 2 2.005 2.01
Frequency (GHz)

-
B
(d)

(e)

Figure 5: (a) A simple RC circuit driven by a sinusoidal current, (b) waveforms when C 1
is commutated, (c) waveforms when fin ! fLO, (d) a simplified circuit if | fin - fLO | is relatively
large, and (e) the magnitude of impedance versus frequency.

A

VAB

Iin

X
R1

-
B

C1

LO

Y

LO
-10

C2
Magnitude (dB)

+

LO
Iin

-20

-30

-40
1.99

VAB
t

2
2.005
1.995
Frequency (GHz)
(b)

2.01

(a)
Figure 6: (a) An RC circuit using two commutated capacitors and (b) the magnitude of
impedance versus frequency.

10

W I N T E R 2 0 16

IEEE SOLID-STATE CIRCUITS MAGAZINE

capacitor is very high at this frequency. This situation stands in contrast
to that in Figure 5(a), where C 1 yields
a low impedance at fin . We therefore
conclude that the impedance of the
capacitor is translated (upconverted)
to a center frequency of fLO, emerging as 1/ [j (~ - ~ LO) C 1] . Of course,
this property arises from the bilateral nature of the switch: the voltage across the capacitor is mixed
with the LO and up-converted as it
manifests itself in VAB . We call this
circuit's operation "partial commutation" to emphasize that I in does
not see a capacitance for half of the
LO period.
What happens if fin in Figure 5(b)
departs from fLO ? The synchronicity
between I in and the LO no longer
holds, C 1 receives both positive and
negative charge from I in, and the
capacitor voltage grows to a lesser
extent [Figure 5(c)]. If | fin - fLO | is
large enough, we can say that, when
S 1 is on, the swing in VAB is approximately proportional to 1/ (C 1 ~ in) (by
virtue of integration), and when S 1
is off, the swing is proportional to
R 1, a much greater value because
R 1 C 1 & Tin . We can thus approximate
C 1 by a short circuit and reduce the
circuit to that in Figure 5(d). Here,
VAB (t) is simply equal to I in (t) S (t),
where S (t) denotes a square wave
toggling between zero and one. Of
interest to us is the input impedance in the vicinity of fin; so we
must seek the component of VAB
at this frequency. Since S (t) has a
dc value of 0.5, the amplitude of
VAB at fin is given by 0.5 I 0 R 1 . We
conclude that the input impedance falls from (approximately) R 1
at fin = fLO to 0.5R 1 for fin somewhat far from fLO . The simulated
plot in Figure 5(e), where fLO = 2
GHz, confirms this result, revealing very little selectivity in this
input impedance.

Impedance Translation by
Full Commutation
How can we improve the selectivity of
the impedance plotted in Figure 5(e)?
As noted above, the component resulting from I in R 1 overwhelms the



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