# Lumped Element Modelling

Consider an inductance L (Figure 10.33(a)),

Equation (10.23) can be rearranged to form:

Applying the trapezoidal rule gives: where

Here, R is constant and I varies with time. The equivalent circuit is shown in Figure 10.33(b).

A similar treatment applies to capacitance (C) (see Figure 10.34(a)). Here,

Integrating and applying the trapezoidal rule gives

Rearranging gives:

Where

giving the equivalent circuit of Figure 10.34(b).

Resistance is represented directly by,

The procedure for solving a network using EMTP is as follows:

- 1. Replace all the lumped elements by models described in equations (10.25), (10.26) and (10.27).
- 2. From initial conditions, determine i(t — At) = i(0) and v(t — At) = v(0).
- 3. Solve for i(t) and
*v(t).*Increment time step by At and calculate new values of i and v, and so on. The analysis is carried out by use of the nodal admittance matrix and Gaussian elimination. If mutual coupling between elements exists then the representation becomes very complex.

**Example 10.4**

The equivalent circuit of a network is shown in Figure 10.35. Determine the network which simulates this network for transients using the EMTP method after the first time step of the transient of 5 ms.

**Solution**

By replacing all the lumped elements by models described in equations (10.25), (10.26) and (10.27), the equivalent circuit shown in Figure 10.36(a) was obtained.

**Figure 10.35 **Figure for Example 10.4

At t = 0, note current sources are zero and equivalent circuit is reduced to Figure 10.36(b).

By applying the mesh method to Figure 10.36(b):

Invert

* i_{1}* — 4300 A,

*— —38 A and*

**i**_{2}*— —200 From Figure 10.36(a) and (b):*

**i**_{3}* In —* 4300 A,

*-38 A and*

**In —***-200 A Ici — ii — i*

**In —****'2**— 4338 A and Ic

*2*

**— i***2*—

*— 238 A*

**is**The equivalent circuit after 5 ms is shown in Figure 10.36(c). The process is then repeated for the next 5 ms step using Figure 10.36(c) as the starting condition.