SKEDSOFT

Control Systems - 1

State-space model of Electrical systems

LCR circuit:

 

Fig: 1 Electrical circuit

 

Consider the electrical circuit shown in Figure 1. The circuit consists of an inductance L (henry), a resistance R (ohm), and a capacitance C (farad). Applying Kirchhoff's voltage law to the system, we obtain the following equations:

The above equations give a mathematical model of the circuit.

A transfer function model of the circuit can also be obtained as follows: Taking the Laplace transforms of the above equation, assuming zero initial conditions, we obtain

If ei is assumed to be the input and eo the output, then the transfer function of this system is found to be

 

Complex impedances:

 

 

Fig: 2

 

 In driving transfer functions for electrical circuits, we find it convenient to write the Laplace-transformed equations directly, without writing the differential equations. Consider the system shown in Figure 2 (a). In this system, Z1 and Z2 represent complex impedances. The complex impedance Z(s) of a two-terminal circuit is the ratio of E(s), the Laplace transform of the voltage across the terminals, to I(s), the Laplace transform of the current through the element, under the assumption that the initial conditions are zero, so that Z(s) = E(s)/I(s). If the two terminal elements is a resistance R, capacitance C, or inductance L, then the complex impedance is given by R, 1/Cs, or Ls, respectively. If complex impedances are connected in series, the total impedance is the sum of the individual complex impedances.