Next: 2.3.3.3 Linear Capacitor Up: 2.3.3 Devices Previous: 2.3.3.1 Conductor

### 2.3.3.2 Resistor

The constitutive relations for a linear temperature dependent resistor are

 V = I . R() (2.32) R() = R0 . 1 + . ( - ) (2.33) P = V . I (2.34)

The same considerations as for conductors apply for 0 . The stamp is given as
 yx, y I f n1 1 - I n2 -1 I I -1 1 R R0 . . I V - I . R - I I - V P
As the node voltages and the branch current are independent solution variables, it is not guaranteed that the expression V - I . R equals zero while iterating towards the final solution. In the above stamp, the independent solution variable for the current I can be eliminated making I a dependent variable. The resulting stamp reads
 yx, y f n1 G - G V . . R0 . G2 - I n2 - G G - V . . R0 . G2 I -2 . V . G 2 . V . G I2 . . R0 P
with G = 1/R. Since I is a dependent variable, I and V may be used interchangeably, as long as R 0. The above stamp is, of course, equal to the result obtained by directly considering the conductor G = 1/R which, in this case, depends in a non-linear way on . However, the current I can only be eliminated if R 0 so that the second stamp is somewhat more restrictive. Furthermore, this procedure shows how additional currents can be added or eliminated whenever needed as long as a unique inversion V = g-1(I) of the the branch relation I = g(V) exists which is not the case for R = 0. Of course, I must not be used by other device models and hence be a local quantity of the device. In addition, V and I need not necessarily be defined for the same branch as is the case for current-controlled voltage sources.

Another interesting application can be found when adding the branch current for a conductor as an unknown. Neglecting temperature dependencies, the stamp reads

 yx, y I f n1 1 - I n2 -1 I I - G G 1 V . G - I
For an open circuit G = 0 while R = 0 results in a short circuit. By combining the above stamp with the stamp of the ideal resistor an ideal switch can be implemented whose stamp reads
 yx, y I f n1 1 - I n2 -1 I I - S S 1 - S V . S - I . (1 - S)
with S denoting the state of the switch. S = 1 gives a short circuit while S = 0 results in an open circuit.

Next: 2.3.3.3 Linear Capacitor Up: 2.3.3 Devices Previous: 2.3.3.1 Conductor
Tibor Grasser
1999-05-31