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system In electrical circuits, we recognize that any circuit containing a single energy storage element (inductor or capacitor) and a combination of sources and resistors (and possibly switches) is a rst-order system In other domains, we also encounter rst-order systems For example, a mechanical system that has mass and damping (eg, friction), but not elasticity, will be a rst-order system A uid system with uid resistance and uid capacitance ( uid storage) will also be of rst order; an example of a rst-order uid system is a storage tank with a valve In thermal systems, we also encounter rst-order systems quite frequently: The ability to store heat (heat capacity) and to dissipate it leads to a rst-order thermal system; heating and cooling of bodies is, at its simplest level, described by rst-order behavior In the present section we analyze the transient response of rst-order circuits In what follows, we shall explain that the initial condition, the steady-state solution, and the time constant of the rst-order system are the three quantities that uniquely determine its response Natural Response of First-Order Circuits Figure 59 compares an RL circuit with the general form of the series RC circuit, showing the corresponding differential equation From Figure 59, it is clear that equation 512 is in the general form of the equation for any rst-order circuit: dx(t) a1 (512) + a2 x(t) = f (t) dt where f is the forcing function and x(t) represents either vC (t) or iL (t) The constant a = a2 /a1 is the inverse of the parameter , called the time constant of the system: a = 1/ To gain some insight into the solution of this equation, consider rst the natural solution, or natural response, of the equation,1 which is obtained by setting the forcing function equal to zero This solution, in effect, describes the response of the circuit in the absence of a source and is therefore characteristic of all RL and RC circuits, regardless of the nature of the excitation Thus, we are interested in the solution of the equation dxN (t) 1 + xN (t) = 0 dt or 1 dxN (t) (514) = xN (t) dt where the subscript N has been chosen to denote the natural solution One can easily verify by substitution that the general form of the solution of the homogeneous equation for a rst-order circuit must be exponential in nature, that is, that xN (t) = Ke at = Ke t/ (515) (513).

cd can also be entered as CMD or CD. However, this is seldom true of the arguments you pass into these commands. For the most part, arguments are case-sensitive.

+ _ vS (t)

126 131

iL(t)

RL circuit:

To evaluate the constant K, we need to know the initial condition The initial condition is related to the energy stored in the capacitor or inductor, as will be further explained shortly Knowing the value of the capacitor voltage or inductor current at t = 0 allows for the computation of the constant K, as follows: xN (t = 0) = Ke 0 = K = x0

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You can also type command instead of cmd, as each will open a Command window However, the Command option opens up an older version of the Command Console window Although this older version works much like the newer one, not all the newer commands will run from it It is used to running older programs and is provided for backward compatibility If you look closely at Figure 7-2, you will see that the older Command window reports that it is for Microsoft s Windows DOS You can also see that it does not display long folder names, but rather places a ~ and a number in place of the rest of the folder name While there is nothing strictly wrong with this window, you would not use it unless you were running a rather old piece of software one that would not run in the standard Command window.

(516)

5

Thus, the natural solution, which depends on the initial condition of the circuit at t = 0, is given by the expression xN (t) = x0 e t/ (517)

227 164

where, once again, xN (t) represents either the capacitor voltage or the inductor current and x0 is the initial condition (ie, the value of the capacitor voltage or inductor current at t = 0) Energy Storage in Capacitors and Inductors

Before delving into the complete solution of the differential equation describing the response of rst-order circuits, it will be helpful to review some basic results pertaining to the response of energy-storage elements to DC sources This knowledge will later greatly simplify the complete solution of the differential equation describing a circuit Consider, rst, a capacitor, which accumulates charge according to the relationship Q = CV The charge accumulated in the capacitor leads to the storage of energy according to the following equation: WC = 1 2 Cv (t) 2 C (518)

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