Файл: Microcontroller based applied digital control (D. Ibrahim, 2006).pdf

38 SYSTEM MODELLING

The potential difference across a capacitor depends on the charge the plates hold, and is given by

The relationship between the current through the capacitor and the voltage across it is given by

The energy stored in a capacitor depends on the capacitance and the voltage across the capacitor and is given by

Electrical circuits are modelled using Kirchhoff’s laws. There are two laws: Kirchhoff’s current law and Kirchhoff’s voltage law. To apply these laws effectively, a sign convention should be employed.

Kirchhoff’s current law The sum of the currents at a node in a circuit is zero, i.e. the total current flowing into any junction in a circuit is equal to the total current leaving the junction.

Figure 2.14 shows the sign convention that can be employed when using Kirchhoff’s current law. We can write

i1 + i2 + i3 = 0

for the circuit in Figure 2.14(a),

−(i1 + i2 + i3) = 0

for the circuit in Figure 2.14(b) and

i1 + i2 − i3 = 0

for the circuit in Figure 2.14(c).

Kirchhoff’s voltage law The sum of voltages around any loop in a circuit is zero, i.e. in a circuit containing a source of electromotive force (e.m.f.), the algebraic sum of the potential drops across each circuit element is equal to the algebraic sum of the applied e.m.f.s.

Figure 2.14 Applying Kirchhoff’s current law

R

vR

vL L

vC

C

Figure 2.15 Applying Kirchhoff’s voltage law

It is important to observe the sign convention when applying Kirchhoff’s voltage law. An example circuit is given in Figure 2.15. For this circuit we can write.

vR + vL + vC = 0.

Some examples of the modelling of electrical circuits are given below.

Example 2.8

Figure 2.16 shows a simple electrical circuit consisting of a resistor, an inductor and a capacitor. A voltage Va is applied to the circuit. Derive an expression for the mathematical model for this system.

Solution

Applying Kirchhoff’s voltage law, we can write

For the capacitor we can write

i = C dvC . dt

R

C

Figure 2.16 Simple electrical circuit

Figure 2.17 Electrical circuit for the Example 2.9

Example 2.9

Figure 2.17 shows an electrical circuit consisting of a capacitor, an inductor and a resistor. The inductor and the capacitor are connected in parallel. A voltage Va is applied to the circuit. Derive a mathematical model for the system.

Solution

Applying Kirchhoff’s current law, we can write

Now, the potential difference across the inductor and also across the capacitor is vC . Similarly, the potential difference across the resistor is Va − vC . Thus,

Figure 2.18 Circuit for Example 2.18

Example 2.10

Figure 2.18 shows an electrical circuit consisting of two inductors, two resistors and a capacitor. A voltage Va is applied to the circuit. Derive an expression for the mathematical model for the circuit.

Solution

The circuit consists of two nodes and two loops. We can apply Kirchhoff’s current law to the nodes. For node 1,

42 SYSTEM MODELLING

Equations (2.75) and (2.76) describe the operation of the circuit. These two equations can be represented in matrix form as

2.3 ELECTROMECHANICAL SYSTEMS

Electromechanical systems such as electric motors and electric pumps are used in most industrial and commercial applications. Figure 2.19 shows a simple d.c. motor circuit. The torque produced by the motor is proportional to the applied current and is given by

where T is the torque produced, kt is the torque constant and i is the motor current. Assuming there is no load connected to the motor, the motor torque can be expressed as

As the motor armature coil is rotating in a magnetic field there will be a back e.m.f. induced in the coil in such a way as to oppose the change producing it. This e.m.f. is proportional to the angular speed of the motor and is given by:

where vb is the back e.m.f., ke is the back e.m.f. constant, and ω is the angular speed of the motor.

Using Kirchhoff’s voltage law, we can write the following equation for the motor circuit:

Figure 2.19 Simple d.c. motor