Electrical circuits involving resistors, capacitors and inductors are considered. The behaviour of such systems is governed by Ohm’s law and Kirchhoff’s laws

**Modeling of electrical system**

Electrical circuits involving resistors, capacitors and inductors are considered. The behaviour of such systems is governed by Ohm’s law and Kirchhoff’s laws.

**Resistor: **Consider a resistance of ‘R’ Ω carrying current ‘I’ Amps as shown in Fig (a), then the voltage drop across it is v = R I

**Inductor: **Consider an inductor ― L’ H carrying current ’i ’ Amps as shown in Fig (a),** **then the voltage drop across it can be written as v = L di/dt

**Capacitor: **Consider a capacitor ’C’ F carrying current ’i ’ Amps as shown in Fig (a),** **then the voltage drop across it can be written as v = (1/C)∫ i dt

o Apply Kirchhoff‗s voltage law or Kirchhoff‗s current law to form the differential equations describing electrical circuits comprising of resistors, capacitors, and inductors.

o Form Transfer Functions from the describing differential equations.

o Then simulate the model.

**Example**

LRC circuit. Applying Kirchhoff‗s voltage law to the system shown. We obtain the following equation;

L(di /dt) + Ri + 1/ C ∫ i(t) dt =ei …………………….. (1)

1/ C ∫ i(t) dt =e0 ……………………………………….. (2)

Equation (1) & (2) give a mathematical model of the circuit. Taking the L.T. of equations (1)&(2), assuming zero initial conditions, we obtain

The dc motors have separately excited fields. They are either armature-controlled with fixed field or field-controlled with fixed armature current. For example, dc motors used in instruments employ a fixed permanent-magnet field, and the controlled signal is applied to the armature terminals.

Consider the armature-controlled dc motor shown in the following figure.

Ra = armature-winding resistance, ohms

La = armature-winding inductance, henrys

ia = armature-winding current, amperes

if = field current, a-pares

ea = applied armature voltage, volt

eb = back emf, volts

θ = angular displacement of the motor shaft, radians

T = torque delivered by the motor, Newton*meter

J = equivalent moment of inertia of the motor and load referred to the motor shaft kg.m2

f = equivalent viscous-friction coefficient of the motor and load referred to the motor shaft. Newton*m/rad/s

T = k1 ia ψ where ψ is the air gap flux, ψ = kf if , k1 is constant

For the constant flux

**Where Kb is a back emf constant -------------- (1)**

The differential equation for the armature circuit

The armature current produces the torque which is applied to the inertia and friction; hence

Assuming that all initial conditions are condition are zero/and taking the L.T. of equations (1),

(2) & (3), we obtain

Kps θ (s) = Eb (s)

(Las+Ra ) Ia(s) + Eb (s) = Ea (s) (Js2 +fs)

θ (s) = T(s) = K Ia(s)

The T.F can be obtained is

Let us consider a mechanical (both translational and rotational) and electrical system as shown in the fig.

From the fig (a)

We get M d2 x / dt2 + D d x / dt + K x = f

From the fig (b)

We get M d2 θ / dt2 + D d θ / dt + K θ = T

From the fig (c)

We get L d2 q / dt2 + R d q / dt + (1/C) q = V(t)

Where q = ∫i dt

They are two methods to get analogous system. These are (i) force- voltage (f-v) analogy and (ii) force-current (f-c) analogy

**Problem**

1. Find the system equation for system shown in the fig. And also determine f-v and f-i analogies

The system can be represented in two forms:

1. Block diagram representation

2. Signal flow graph

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