Phase-Mode DC Analysis

The Josephson junction device has unique behavior which complicates simulation with a SPICE-type simulator (see 2.17.1.1). Central is the idea of phase, which is a quantum-mechanical concept and is generally invisible in the non-quantum world. However with superconductivity, and with Josephson junctions in particular, phase becomes not only observable, but a critical parameter describing these devices and the circuits that contain them.

A general superconducting circuit contains loops of inductors and Josephson junctions. When quiescent, the voltage at all nodes of such a circuit is identically zero, and thus it would seem to be impossible to use a SPICE-type simulator to perform DC analysis on this type of circuit. However, by using phase, which is nonzero at each node, instead of voltage, one can perform DC analysis, using ``phase-mode DC''.

* WRspice* after release 4.3.3 offers a DC analysis capability which
uses phase-mode for circuits containing Josephson junctions. Unlike
strictly phase-mode simulators,

Within * WRspice*, every node connected to a Josephson junction,
inductor, or lossless transmission line (treated as an inductor in DC
analysis) has a ``Phase'' flag set. This indicates that the computed
value is phase, not voltage, for these nodes. We have to special-case
the matrix loading functions for inductors, mutual inductors, lossless
transmission lines, and resistors. Other devices are treated
normally.

A Josephson junction can be modeled by the basic formula

I=I_{c}sin(V(i,j))

where *V*(*i*, *j*)
is the ``voltage'' difference between nodes i and j
(across the junction) which is actually the phase. Inductors look
like resistors:

I=V(i,j)/2L

Where *V*(*i*, *j*)
is the phase difference across the inductor. Mutual
inductance adds similar cross terms.

Capacitors are completely ignored as in normal DC analysis. The
treatment of resistors is slightly complicated. The connected nodes
can be ``Ground'', ``Phase'', or ``Voltage'' type. If both nodes are
Voltage or Ground, the resistor is loaded normally. If both nodes are
Phase or Ground, the resistor is not loaded at all. The interresting
case is when one node is Phase, the other Voltage. In this case, we
load the resistor as if the phase node is actually ground (node number
0). In addition we load a voltage-controlled current source template
that injects current into the phase node of value
*V*(*voltage*)/*Resistance*
.

Resistors are the bridge between normal voltage-mode devices and phase nodes. Some circuits may require introduction of resistors to get correct results. For example, assume a Josephson junction logic gate driving a CMOS comparator circuit. If the input MOS gate is connected directly to the junction, the DC operating point will be incorrect, as the comparator will see phase as input. However, if a resistor separates the MOS gate from the junction, the comparator input will be zero, as it should be.

There is (at present) a topological requirement that all phase nodes must be at ground potential. This means that for a network of Josephson junctions and inductors, there must be a ground connection to one of these devices. A nonzero voltage source connected to an inductor, which is connected to a resistor to ground, although a perfectly valid circuit, will fail. One must use the equivalent consisting of the voltage source connected to a resistor, connected to the inductor which is grounded. This satisfies the two topological requirements:

- Rule 1

There must be a resistor between a voltage-mode device and a phase-mode device, no direct connections allowed. - Rule 2

Every phase-node subnet must have a connection to ground, so all phase nodes are at ground potential.

With this bit of information, and the warning that controlled sources can cause unexpected behavior, the DC analysis using thes technique can apply to general circuits containing Josephson junctions.

Of course, for this to work, no Josephson junction can be biased above
its critical current or nonconvergence results. Both DC operating
point and DC sweep are available, as is AC analysis. Noise analysis
is available with the internal Josephson junction model. This new
hybrid technique appears to be an important advance, which should
avoid the long-standing need to use ```uic`'' and ramp sources up
from zero, and makes available DC sweep analysis, and for the first
time in any simulator AC small-signal analysis.