CHAPTER SIX FIRST ORDER CIRCUITS Chapters 2 to 5 have been devoted exclusively t o circuits made of resistors and independent sources The resistors may contain two or more terminals and may be linear or nonlinear time varying o r time invariant We have shown that these resistive circuits are always governed by algebraic equations In this chapter we introduce two new circuit elements namely twoterminal capacitors and inductors We will see that these elements differ from resistors in a fundamental way They are lossless and therefore energy is not dissipated but merely stored in these elements A circuit is said to be dynamic if it includes some capacitor s or some inductor s or both In general dynamic circuits are governed by differential equations In this initial chapter on dynamic circuits we consider the simplest subclass described by only one first order differential equation hence the name first order circuits They include all circuits containing one 2 terminal capacitor or inductor plus resistors and independent sources The important concepts of initial state equilibrium state and time constant allow us to find the solution of any first order linear time invariant circuit driven by dc sources by inspection Sec 3 1 Students should master this material before plunging into the following sections where the inspection method is extended to include linear switching circuits in Sec 4 and piecewiselinear circuits in Sec 5 Here the important concept of a dynamic route plays a crucial role in the analysis of piecewise linear circuits by inspection l TWO TERMINAL CAPACITORS AND INDUCTORS Many devices cannot be modeled accurately using only resistors In this section we introduce capacitors and inductors which together with resistors FIRST ORDER CIRCUlTS 297 Let N be a two terminal element driven by a voltage source v t respectively current source i t Jand let i r respectively v t j denote the corresponding current respectively voltage response If we plot the locus v L of v i t in the v i plane and obtain afixed curve independent o f t h e excitation waveforms then N can be modeled as a two terminal resistor If 2 u i changes with the excitation waveform then N does not behave jike a resistor and a different model must be chosen In this case we can calculate the associated charge waveform q t using Eq 1 2a or the flux waveform t using Eq f 2 b and see whether the corresponding locus 2 q v of q t t in the q v plane or Y 4 i of 4 r in the i plane is a fixed curve independent of the excitation waveforms Exercises 1 Apply at t 0 a voltage source v t A sin w t in volts across a l F capacitor a Calculate the associated current i t flux r and charge 1 2b Assume O 1 Wb and q t for t 2 0 using Eqs 1 2 and q 0 OC b Sketch the loci 2 v i 2 i and T q v in the v i plane 6 i plane and q U plane respectively for the following parameters A is in volts o is in radians per second c Does it make sense to describe this element by a v i characteristic cb i characteristic g U characteristic Explain 2 Apply at t 0 a current source i t A sin wt in amperes across a l H inductor a Calculate the associated voitage v t charge q t and flux t for t O using Eqs 1 2a and 1 2b Assume q 0 1 C and O 0 W b Sketch the loci 2 v i Y q v and Y 4 i in the v i plane q u plane and 6 i plane respectively for the following parameters A is in amperes w is in radians per second c Does it make sense to describe this element by a U icharacteristic q U characteristic 4 1 characteristic Explain 1 1 q v and 4 i characteristics A two terminal element whose charge q t and voltage u t fall on some fixed curve in the q v plane at any time t is called a time invariant capacitor 3 This curve is called the A two terminal element whose flux t and current i t fall on some fixed curve in the 4 i plane at any time t is called a time invariant inductor This curve is called the This definition is generalized to that of a time varying capacitor in Sec 1 2 This definition is generalized to that of a time varying inductor in Sec 1 2 g v characterlsfic of the capacitor It may be represented by the equation5 4 i characterktic of the inductor It may be represenited by the equation7 fJA i v 0 1 3 If Eq 1 3a can be solved for v as a single valued function of q namely If Eq E 3b can be solved for i as a single valued function of namely the capacitor is said to be chargecontrolled If Eq 1 3 can be solved for q as a single valued function of v namely the inductor is said to be fluxcontrolled I f E q 1 36 can be solved for 4 as a single valued function of i namely the capacitor is said to be voltagecontrolled If the function Ij v is differentiable we can apply the chain rule to express the current entering a time invariant voitage controlled capacitor in a form similar to Eq 1 1 the inductor is said to be currenlcontrolled If the function i is differentiable we can apply the chain rule to express the voltage across a tim invariant current controlled inductor in a form similar to Eq 1 1b y where where is called the small signal capacitance at the operating point v is called the small signal inductance at the operating point i Example l a Linear timeinvariant parallel plate capacitor Figure l l a shows a familiar device made of two flat parallel metal plates sepa Example l b Linear timeinvariant toroidal inductor Figure 1 2 shows a familiar device made of a conducting wire wound around a toroid This equation is also called the constitutive relation of the capacitor 6 We will henceforth use the notation 5 A d v t ldt This equation is also called the constitutive relation of the inductor S We will henceforth use the notation di t ldt fcC X 0 1 36 FIRST ORDER CIRCUITS c 299 C Figure 1 1 Parallel plate capacitor Figure 1 2 Toroidal inductor rated in free space by a distance d When a voltage u t 0 is applied we recali from physics that a charge equa1 to made of a nonmetallic material such as wood When a current i t O is applied we recall from physics that a flux equal to is induced at time t on the upper plate and an equal but opposite charge is induced on the lower plate at time t The constant of proportionality is given approximately by is induced at …
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