IEEE Solid-State Circuits Magazine - Winter 2016 - 11

Translation of Transfer Functions
Equipped with the commutated
capacitors of Figure 6(a), we can
now explore methods of translating
transfer functions. We surmise that

1.5

VAB (V)

1
0.5
0
-0.5
-1
-1.5

0

2

4

6

8

10

12

14

14

18

20

12

14

14

18

20

Time (ns)
(a)
200
100
VAB (V)

up-converted capacitance as fin substantially departs from fLO . This
means that we must not permit much
of I in to flow through R 1 even when C 1
is switched out. This is accomplished
by adding one more capacitive branch
that is controlled by LO [Figure 6(a)],
thus creating "full commutation." Assuming that C 1 = C 2 = C, R 1 C 1 & TLO,
and fin = fLO, we observe that VAB is
equal to VX for one half cycle and to
VY for the other half, growing in amplitude in both positive and negative
directions. That is, VAB is defined
by capacitive dynamics on top and
bottom, with both its positive and
negative amplitudes decreasing as
| fin - fLO | increases. Plotted in Figure
6(b), the impedance continues to fall,
exhibiting greater selectivity than that
seen in Figure 5(e).
We should make two remarks.
First, the condition R 1 C & TLO is
necessary for significant up-conversion of the capacitive impedance.
Each time a switch turns on, the
corresponding capacitor impresses
its voltage upon the output; the
longer this voltage lasts, the more
pronounced is the effect of the
capacitor. Plotted in Figure 7 is the
output voltage for the extreme cases
R 1 C & TLO and R 1 C % TLO, revealing a spectrally rich signal in the
former that stems from substantial up-conversion of the capacitor
voltages. In the latter, on the other
hand, each capacitor rapidly loses
its charge when it is switched in,
allowing VAB to assume the shape
of I in . The key point here is that the
commutated network need not be
driven by a "high" source resistance
so long as R 1 C & TLO .
Second, the translation of the
capacitive impedance in Figure 6(a)
also occurs at the higher harmonics
of the LO. For example, an input of
the form I in = I 0 sin (3~ LO t) leads to
accumulation of charge on C 1 and
C 2 and a tall waveform for VAB .

0
-100
-200

0

2

4

6

8

10
Time (ns)
(b)

Figure 7: The voltage across a commutated network with (a) R 1 C & Tin and (b) R 1 C % Tin .

RS
Vin

Vin

Vout

RS

Vout

C1
C1
Vout

Vout

Vin

Vin

0

f

CN

fLO

0

f

(a)
C1

Vin

C1
Vout

Vin

Vout
RS

RS
CN
Vout

Vout

Vin

Vin

0

f
(b)

0

fLO

f

Figure 8: (a) Low-pass to band-pass and (b) high-pass to notch transformations.

IEEE SOLID-STATE CIRCUITS MAGAZINE

W I N T E R 2 0 16

11



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