IEEE Solid-State Circuits Magazine - Winter 2015 - 10

A C ircu it for All Season s

Behzad Razavi

The Cross-Coupled Pair-Part III

I

In this article, we study applications
of the cross-coupled pair (XCP) in
analog and RF circuits. The XCP can
serve as a negative resistance or a
negative impedance converter in
small-signal operation, or a regenerative circuit in large-signal operation.

VCC

VCC

VDD

XCP as Negative Resistance
The XCP finds wide application as a
negative resistance in the design of
oscillators [1]-[6]. It is interesting to
note how Abraham and Bloch's multivibrator [Figure 1(a)] evolved to the
"relaxation oscillator" [Figure 1(b)]
and to today's differential LC oscillator [Figure 1(c)]. While the first two
operate with hysteresis, the last one
need not.
As a negative resistance, the XCP
serves other purposes as well. For
example, as shown in Figure 2(a),
it can increase the voltage gain of
a differential pair. In this case, the
loop gain (g m3, 4 R D) 2, is chosen less
than unity to avoid latch-up. Note
that the XCP also provides a path
for the bias current of M 1 and M 2,
allowing a higher value for R D, but
it does consume substantial voltage
headroom ^| VGS3, 4 |h.
In some applications, the XCP nonlinearity may be of concern. We can
quantify this effect with the aid of the
equivalent circuit in Figure 2(b), where
the input pair is assumed linear. We
can study the output compression
by finding the condition under which
Vout /DI falls by 1 dB (≈12%). Writing Vout = DI D [(2R D) | | (- 2/g m3, 4)] =
DI D [2R D / (1 - g m3, 4 R D)], we set the
Digital Object Identifier 10.1109/MSSC.2014.2369332
Date of publication: 11 February 2015

10

w i n t e r 2 0 15

(a)

(b)

(c)

Figure 1: The evolution of oscillators from (a) multivibrator to (b) relaxation oscillator to
(c) today's LC oscillator.

gain, Vout /DI D, equal to 1.12 times the
gain under equilibrium (when DI D = 0) .
It follows that
g m3, 4
0.12 ,
= 1.12 g m3, 4, eq R D
g m3, 4, eq

(1)

where g m3, 4, eq denotes the equilibrium
transconductance of the XCP devices.
This expression gives the drop in
g m3, 4 as the output approaches its
1-dB compression point. For example, if g m3, 4, eq R D = 0.75, then g m3, 4 /
g m3, 4, eq = 0.96, i.e., only a 4% reduction
in g m leads to a 1-dB compression in

M3

VDD

VDD

M4

RD

RD
RD

Vout
Vin

the gain. The key point here is that
the closer g m3, 4 R D is to unity, the
less transconductance reduction can
be tolerated and the more nonlinear
the circuit.
The cross-coupling loop of two
transistors need not traverse their
gates and drains; the bulk can supplant the gate as the controlling terminal. Illustrated in Figure 3(a) [7],
the idea is to return the drain voltage of one transistor to the bulk of
the other so as to create a negative
resistance given by R eq = - 2/g mb,

Iss
(a)

RD

Vout

M2

M1

−2
gm 3,4

+∆I

-∆I

(b)

Figure 2: (a) The addition of XCP to raise the gain and (b) the equivalent circuit for smallsignal analysis.

IEEE SOLID-STATE CIRCUITS MAGAZINE



Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Winter 2015

IEEE Solid-State Circuits Magazine - Winter 2015 - Cover1
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