Copyright © 2010 Robert D. Throne 1 ECE-205 Dynamical Systems Course Notes Winter 2011-2012 Bob ThroneCopyright © 2010 Robert D. Throne 2Copyright © 2010 Robert D. Throne 3 1.0 Electrical Systems The types of dynamical systems we will be studying can be modeled in terms of algebraic equations, differential equations, or integral equations. We will begin by looking at familiar mathematical models of ideal resistors, ideal capacitors, and ideal inductors. Then we will begin putting these models together to develop models for RL and RC circuits. Finally, we will review solution techniques for the first order differential equation we derive to model the systems. 1.1 Ideal Resistors The governing equation for a resistor with resistanceR is given by Ohm’s law, () ()v t Ri t= where ()vt is the voltage across the resistor and ()it is the current through the resistor. HereRis measured in Ohms,()vtis measured in volts, and()it is measured in amps. The entire expression must be in volts, so we get the unit expression [volts] = [Ohms][amps] 1.2 Ideal Capacitors The governing equation for a capacitor with capacitance C is given by ()()dv tit Cdt= HereCis measured in farads, and again ()vt is measured in volts and ()itis measured in amps. This expression also helps us with the units. The entire expression must be in terms of current , so looking at the differential relationship we can determine the unit expression [amps] = [farads][volts]/[seconds] We can integrate this equation from an initial time 0t up to the current time t as follows: ()()dv tit Cdt= 1() ()i t dt dv tC= Next, since we want to integrate up to a final time t, we need to use a dummy variable in the integral that is not t. This is an important habit to get into—do not use t as the dummy variable of integration if we expect a function of time as the output! Here we have chosen to use the dummy variableλ. Also we incorporate the fact that at time 0t the voltage is 0()vt, while at time t the voltage is ()vtCopyright © 2010 Robert D. Throne 4 0))((1() ()ovttt vti d dvCλλ λ=∫∫ Carrying out the integration we get 001)( ) () (tti d vt vtCλλ= −∫ which we can rearrange as 001)() ( ( )ttvt vt iCdλλ+=∫ This expression tells us there are two components to the voltage across a capacitor, the initial voltage 0()vtand the part due to any current flowing through the capacitor after that time, 01()ttidCλλ∫ Finally, these expressions help us determine some important characteristics of our ideal capacitor: • If the voltage across the capacitor is constant, then the current through the capacitor must be zero since the current is proportional to the rate of change of the voltage. Hence, a capacitor is an open circuit to dc. • It is not possible to change the voltage across a capacitor in zero time .The voltage across a capacitor must be a continuous function of time, otherwise an infinite amount of current would be required. 1.3 Ideal Inductors The governing equation for an inductor with inductance L is given by ()()di tvt Ldt= HereL is measured in henrys, and again ()vt is measured in volts and()it is measured in amps. This expression also helps us with the units. The entire expression must be in terms of voltage , so looking at the differential relationship we can determine the unit expression [volts] = [henrys][amps]/[seconds] We can integrate this equation from an initial time 0t up to the current time t as follows: ()()di tvt Ldt= 1() ()v t dt di tL=Copyright © 2010 Robert D. Throne 5 Next, since we want to integrate up to a final time t, so we again have chosen to use the dummy variableλ. Also we incorporate the fact that at time 0t the current is,0()it while at time t the current is ()it. 0))((1() ()oittt itv d diLλλ λ=∫∫ Carrying out the integration we get 001)( ) () (ttv d it itLλλ= −∫ which we can rearrange as 001)() ( ( )ttit it vLdλλ+=∫ This expression tells us there are two components to the current through an inductor, the initial current 0( )it and the part due to any voltage across the inductor after that time,01()ttvdLλλ∫. Finally, these expressions help us determine some important characteristics of our ideal inductor: • If the current thought an inductor is constant, then the voltage across the inductor must be zero since the voltage is proportional to the rate of change of the current. Hence, an inductor is a short circuit to dc. • It is not possible to change the current through an inductor in zero time .The current through an inductor must be a continuous function of time, otherwise an infinite amount of voltage would be required.Copyright © 2010 Robert D. Throne 6Copyright © 2010 Robert D. Throne 7 2.0 First Order Circuits A first order circuit is a circuit with one effective energy storage element, either an inductor or a capacitor. (In some circuits it may be possible to combine multiple capacitors or inductors into one equivalent capacitor or inductor.) We begin this section with the derivation of the governing differential equation for various first order circuits. We will then put the first order equation into a standard form that allows us to easily determine physical characteristics of the circuit. Next we show an alternative method for checking some parts of the governing differential equations. We then solve the differential equations for the case of piecewise constant inputs, and finish the section with an alternative method of solving the differential equations using integrating factors. 2.1 Governing Differential Equations for First Order Circuits In this section we derive the governing differential equations that model various RL and RC circuits. We then put the governing first order differential equations into a standard form, which allows us to read off descriptive information about the system very easily. The standard form we will use is ()() ()dy ty t Kx tdtτ+= Here we assume the system input is()xt and the system output is()yt. τ is the system time constant, which indicates how long it will take the system to reach steady state for a step (constant) input. K is the static gain of the system. For a constant input of amplitudeA (() ()x t Au t=, where ()ut is the unit step function), in steady state we have ()0dy tdt= and () ()y t Kx t KA= =. Hence the static gain lets us easily compute the steady state value of the output. For circuits with capacitors the differential equation will in general