Figure: A simple model circuit for estimation of transients in an
RSFQ biasing line.
To illustrate the basic limits on power consumption in RSFQ circuits
we consider a simple model circuit (Fig. 
). It has all the
main features of a typical DC bias line in RSFQ technology: a biasing
resistor r with inductance l establishes current picked up from
the current distribution line with inductance L between two
neighboring taps. Influence of SFQ switching events in Josephson
junctions on the biasing circuit is modeled by voltage sources
which we assume to be a given set of functions, independent
of the processes in the biasing line. This assumption is valid as long
as junctions are biased well inside the working region ()
and variations of the bias currents are small enough to keep the
transient dynamics of the junctions unchanged. With these
simplifications we get a translationally invariant linear system which
can be solved analytically for any combination of functions ,
for any finite (as well as an infinite) number of junctions.
Voltage drop 
between point A and ground in Fig.
can be found in two ways. First,
here dot over 
stands for time derivative. Second,
which is true for the ``A-B-ground'' contour to the right of point
A. From () and () we get
Repeating the same procedure for the contour to the left of point A
and excluding currents 
and 
with the help of the charge
conservation law
we get our main recursive
equation for the currents :
We are going to solve the system of linear differential equations
() for the simplest case of infinite number of
junctions when boundary conditions in the system are not
important. If necessary, corrections for the finiteness of the total
number of junctions in the biasing line will be introduced. We will
also assume that parameters of the biasing line are such that the
switching time 
of the junction is much smaller than all the
characteristic times of the problem (l/r and L/r). In this case
junction number n=0 switching at the moment of time t=0 can be
modeled by
Here 
is the Dirac's delta-function and 
is
the Kroneker symbol. Simultaneously, solution of the system
() with the right-hand side in the form of ()
gives us the Green's function of the problem and solution for an
arbitrary set of functions can be constructed in the usual
way. (This includes the case when switching time is not
negligibly small.) Linear system () is solved by
making a Fourier transform:
and observing that for in the form () we have
Plugging () and () into () we get
a set of independent differential equations
with the initial condition
In view of () this initial condition simply states
that all currents were 0 before junction n=0 switched
at the moment of time t=0.
Solution of () with initial condition () is given by
Here we introduced the parameter
which gives the decay time of a wave with wave vector k,
and 
is the step function:
Plugging () back into () we get our final
solution of the system () with driving voltage
in the form ():
Transformation to the case of a finite number of junctions N in
Eq. () is done by the usual rule:
Generalization for the case of a 2-dimensional biasing line is also
straightforward. Next, we are going to discuss Eq. () in
the two limiting cases: 
and 
.
