2.3.3.2 Resistor

The constitutive relations for a linear temperature dependent resistor are

$$\vartheta$$ V = (2.32)

$$\vartheta \left(\vphantom{ 1 + \alpha \cdot (\vartheta - \vartheta_{\mathit{ref}}) }\right. \alpha \vartheta \vartheta_{\mathit{ref}}^{} \left.\vphantom{ 1 + \alpha \cdot (\vartheta - \vartheta_{\mathit{ref}}) }\right)$$ = (2.33)

P = V ^. I (2.34)

The same considerations as for conductors apply for $\alpha$$\ne$ 0 . The stamp is given as

yx, y	$\varphi_{1}^{}$	$\varphi_{2}^{}$	I	$\vartheta$	f
n1			1		- I
n2			-1		I
I	-1	1	R	R0 . $\alpha$ . I	V - I . R
$\vartheta$	- 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	$\varphi_{1}^{}$	$\varphi_{2}^{}$	$\vartheta$	f
n1	G	- G	V . $\alpha$ . R0 . G2	- I
n2	- G	G	- V . $\alpha$ . R0 . G2	I
$\vartheta$	-2 . V . G	2 . V . G	I2 . $\alpha$ . R0	P

with G = 1/R. Since I is a dependent variable, I and V may be used interchangeably, as long as R$\ne$ 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 $\vartheta$. However, the current I can only be eliminated if R$\ne$ 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	$\varphi_{1}^{}$	$\varphi_{2}^{}$	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	$\varphi_{1}^{}$	$\varphi_{2}^{}$	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.