The duality principle
can be used very effectively in electrical engineering
In electrical terms, a lot of electrical properties come in dual:
For example, you take the equation of a capacitor:
U = C*dI/dt
Now you change each variable into its dual: voltage to current, capacity to inductance, derivative of current to derivative of voltage. You get:
I = L*dU/dt
...and voila! you get the equation of an inductance. This principle extends to whole circuits. Take the following one:
\ ! !
+---------+ +----! !----+
! + ! ! !
/ \ .-.
!V! ! !
\ / ! !
! - ._.
This is a voltage source (voltage U_0) connected in series with an open switch
, a capacitor
and a resistor
. Then you shut the switch
at t=0. Then the current equation
for t>0 is:
I = (U_0/R) * e^(-t/RC)
To obtain this solution, you have to solve a differential equation, because the current/voltage-equation of the capacitor includes a derivative of the voltage. In this case, it´s not too complicated, but with larger circuits it can become pretty awkward.
Now imagine that two months later you´re looking for the solution of the following circuit:
! ! ! !
/ \ A + MMM .-.
I ! ! MMM ! !
\ / ! ! MMM ! !
! + MMM ._.
! ! ! !
It´s a current source
(current I_0), in parallel
with a closed switch
that opens at t=0, an inductance
and a resistor
As you take a quick look at the duality relations above, you can see that it´s just the dual of the aforementioned circuit! Serial circuit has become parallel circuit, the voltage source is now a current source and so on. Even the resistor is not a problem. A resistor can be seen as either a resistance or a conductance and is kind of a dual of itself.
So you just take the equation of the dual, change the variables and get
U= (I_0/G) * e^(-t/LG)
...which is the correct solution. No work, no endless solving of differential equations, just changing some letters.
This principle means that of all possible circuits, you just need to solve one half of them. You get the solutions of the duals of those circuits just by applying the duality principle. Well, almost; there are circuits that are dual to itselves, just like a resistor is a dual of itself.
Another application is the substitution of certain components by others. For example, big inductances are hard to make. It´s done by making a coil of a long wire. This leads to big components with a lot of unwanted resistance. On integrated circuits it´s almost impossible to make a good inductance. On the other hand, it´s quite simple to make big capacities, small and with low conductance. So, if you use a "dualization device" that can turn inductances into conductances you can save a lot of precious space and have a better component. This device exists; it´s called gyrator.