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Electrical engineers often classify this response as the sum of a transient response and a steady-state response, rather than a sum of a natural response and a forced response The transient response is the response of the circuit following the switching action before the exponential decay terms have died out; that is, the transient response is the sum of the natural and forced responses during the transient readjustment period we have just described The steady-state response is the response of the circuit after all of the exponential terms have died out Equation 527 could therefore be rewritten as x(t) = xT (t) + xSS where xT (t) = (x0 F )e t/ and in the case of a DC excitation, F , xSS (t) = F = x Note that the transient response is not equal to the natural response, but it includes part of the forced response The representation in equations 530 is particularly convenient, because it allows for solution of the differential equation that results from describing the circuit by inspection The key to solving rst-order circuits subject to DC transients by inspection is in considering two separate circuits: the circuit prior to the switching action, to determine the initial condition, x0 ; and the circuit following the switching action, to determine the time constant of the circuit, , and the steady-state ( nal) condition, x Having determined these three values, you can write the solution directly in the form of equation 529, and you can then evaluate it using the initial condition to determine the constant K To summarize, the transient behavior of a circuit can be characterized in three stages Prior to the switching action, the circuit is in a steady-state condition (the initial condition, determined by x0 ) For a period of time following the switching action, the circuit sees a transient readjustment, which is the sum of the effects of the natural response and of the forced response Finally, after a suitably long time (which depends on the time constant of the system), the natural response decays to zero (ie, the term e t/ 0 as t ) and the new steady-state condition of the circuit is equal to the forced response: as t , x(t) xF (t) You may recall that this is exactly the sequence of events described in the introductory paragraphs of Section 53 Analysis of the circuit differential equation has formalized our understanding of the transient behavior of a circuit Continuity of Capacitor Voltages and Inductor Currents As has already been stated, the primary variables employed in the analysis of circuits containing energy-storage elements are capacitor voltages and inductor currents This choice stems from the fact that the energy-storage process in capacitors and inductors is closely related to these respective variables The amount (530) (529).

of charge stored in a capacitor is directly related to the voltage present across the capacitor, while the energy stored in an inductor is related to the current owing through it A fundamental property of inductor currents and capacitor voltages makes it easy to identify the initial condition and nal value for the differential equation describing a circuit: capacitor voltages and inductor currents cannot change instantaneously An instantaneous change in either of these variables would require an in nite amount of power Since power equals energy per unit time, it follows that a truly instantaneous change in energy (ie, a nite change in energy in zero time) would require in nite power Another approach to illustrating the same principle is as follows Consider the de ning equation for the capacitor: iC (t) = C dvC (t) dt

vC (t)

335 124 16

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and assume that the capacitor voltage, vC (t), can change instantaneously, say, from 0 to V volts, as shown in Figure 514 The value of dvC /dt at t = 0 is simply the slope of the voltage, vC (t), at t = 0 Since the slope is in nite at that point, because of the instantaneous transition, it would require an in nite amount of current for the voltage across a capacitor to change instantaneously But this is equivalent to requiring an in nite amount of power, since power is the product of voltage and current A similar argument holds if we assume a step change in inductor current from, say, 0 to I amperes: an in nite voltage would be required to cause an instantaneous change in inductor current This simple fact is extremely useful in determining the response of a circuit Its immediate consequence is that the value of an inductor current or a capacitor voltage just prior to the closing (or opening) of a switch is equal to the value just after the switch has been closed (or opened) Formally, vC (0+ ) = vC (0 ) iL (0+ ) = iL (0 ) (531) (532)

Known Quantities: Source current, IS ; inductor and resistor values Find: Inductor current at t = 0+ and as t Schematics, Diagrams, Circuits, and Given Data: IS = 10 mA Assumptions: The current source has been connected to the circuit for a very long time Figure 515

Just as in the other technologies, you add the Print command to verify that a particular line was reached or that a particular value was set SQL Print statements are often used to test stored procedures Stored procedures are like methods in that they allow you to store a set of one or more SQL commands on the database server Just like methods in a Windows or web application, the store procedure is called when you want to run the code inside of it Of course, it is necessary to debug and test these just like any other method since they contain processing and validation statements Listing 9-22 is a very simple example that gives you an idea of what this would look like..

5

iL(0+)

iL(t)

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