Next: 2.3.3.3 Linear Capacitor
Up: 2.3.3 Devices
Previous: 2.3.3.1 Conductor
The constitutive relations for a linear temperature dependent resistor are
V 
= 
I^{ . }R() 
(2.32) 
R() 
= 
R_{0}^{ . }1 + ^{ . }(
 ) 
(2.33) 
P 
= 
V^{ . }I 
(2.34) 
The same considerations as for conductors apply for
0 . The stamp is given as
y_{x, y} 


I 

f 
n_{1} 


1 

 I 
n_{2} 


1 

I 
I 
1 
1 
R 
R_{0}^{ . }^{ . }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
y_{x, y} 



f 
n_{1} 
G 
 G 
V^{ . }^{ . }R_{0}^{ . }G^{2} 
 I 
n_{2} 
 G 
G 
 V^{ . }^{ . }R_{0}^{ . }G^{2} 
I 

2^{ . }V^{ . }G 
2^{ . }V^{ . }G 
I^{2 . }^{ . }R_{0} 
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 nonlinear 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
currentcontrolled 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
y_{x, y} 


I 
f 
n_{1} 


1 
 I 
n_{2} 


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
y_{x, y} 


I 
f 
n_{1} 


1 
 I 
n_{2} 


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
19990531