ELECTRICAL CIRCUITS

Solving electrical circuits is a typical problem leading to solving differential equations and the Modern Taylor Series Method can again be used. We start from Kirchhoff's laws. A simple example follows.

For a serial circuit RLC connected to a voltage source u we have

u_L + u_R + u_C = u

or

$L \frac{di}{dt} + R i + \frac{1}{C} \int i dt = u.$

By substituting

y'= i

we have

$i'= \frac{1}{L}( u - Ri - \frac{1}{C}y ).$

Time functions u_R, u_L, u_C for u=1V, $R=1000\Omega$, L=1H, C=10^-6F are












shown in Fig.7.1.

 

Figure 7.1:

The corresponding source text in TKSL/386 is

var i,y,u,UR,UL,UC;
const  R=1000,L=1,C=1E-6,tmax=0.01,dt=0.0001;
system
u=1;
i'=1/L*(u-R*i-1/C*y)   &0;
y'=i                   &0;
UR=R*i;
UC=1/C*y;
UL=u-R*i-1/C*y;
sysend.

Fig.7.2 ( the part labelled LVAR) shows the current I1=i and voltage V1=u_C of the serial circuit RLC ( $R=48.5\Omega, L_0=0.054H, C=200*10^{-6}F$) as functions












of time (for u=0, V1(0)=u_C(0)=500V) if the time function of inductivity is defined as

L=L₀ + 1000 t²[H].

For completness' sake the part labelled LVAR1 shows time functions I1N=i and V1N=u_c for L=L₀.

 

Figure 7.2: