12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*

Transcription

1 + v C C R L - v i L FIGURE The parallel second-order RLC circuit shown in Figure 2.14a UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure 12.24, which is copied from Figure 2.14a. To analyze the behavior of this circuit we can again employ the node method, and this analysis closely parallels that of Section As in Figure 12.6, a ground node is already selected in Figure 12.24, and the unknown node voltage v is already labeled. So, we may again proceed immediately to Step 3 of the node method. Here, we write KCL in terms of v for the node at which v is defined. This yields C dvt) dt + 1 R vt) + 1 L t v t)d t = ) The first term in Equation is the capacitor current, the second term is the resistor current, and the third term is the inductor current. Because the circuit contains an inductor, Equation contains a time integral. To remove this integral, we differentiate Equation with respect to time, and also divide by C, to obtain d 2 vt) dt dvt) + 1 vt) = 0, 12.82) RC dt LC which is easier to work with. To complete the node analysis, we complete Steps 4 and 5 by solving Equation for v. Then, we use v to determine the other branch variables of interest, for example, i L and v C. Like Equation 12.4, Equation is an ordinary second-order linear differential equation with constant coefficients. Since the circuit does not have a drive, its homogeneous solution is also the complete solution. Therefore, as with Equation 12.4, we expect its solution to be a superposition of two terms of the form Ae st. The substitution of this candidate term into Equation yields from which it follows that A s RC s + 1 ) e st = ) LC s RC s + 1 = ) LC Equation is the characteristic equation of the circuit. To simplify Equation 12.84, and to put it in a form that is more standard for the characteristic 654a

2 equation in second-order circuits, we write it as where s 2 + 2αs + ω 2 = ) α ) 2RC 1 ω LC ; 12.87) note that Equation is the same as Equation Equation is a quadratic equation having two roots. Those roots are s 1 = α + α 2 ω ) s 2 = α α 2 ω ) Therefore, the solution for v is a linear combination of the two functions e s 1t and e s 2t, and takes the form vt) = A 1 e s 1t + A 2 e s 2t 12.90) where A 1 and A 2 are as yet unknown constants that are equivalent to the two constants of integration encountered when integrating Equation twice to find v. To complete the solution to Equation we must again determine A 1 and A 2 from initial conditions v C and i L specified at t = 0, To do so, note that v C t) = vt) ) Further, from KCL applied to either node, that is, from Equation 12.81, i L t) = 1 vt) Cdv R dt ) Equations and can be solved to determine v and dv/dt in terms of i L and v C. Doing so, and evaluating the result at t = 0, then yields v0) = v C 0) 12.93) dv dt 0) = 1 C i L0) 1 RC v C0) ) 654b

3 Now, to find A 1 and A 2, we evaluate Equation and its derivative at t = 0, and set the results equal to Equations and 12.94, respectively. This results in v0) = A 1 + A 2 = v C 0) 12.95) dv dt 0) = s 1A 1 + s 2 A 2 = 1 C i L0) 1 RC v C0). Equations and can be jointly solved for A 1 and A 2 to obtain 12.96) A 1 = 1 + RCs 2)v C 0) + Ri L 0) RCs 2 s 1 ) = s 1v C 0) i L 0)/C s 1 s 2 ) 12.97) A 2 = 1 + RCs 1)v C 0) + Ri L 0) RCs 1 s 2 ) = s 2v C 0) i L 0)/C s 2 s 1 ) 12.98) where we have used the fact that both s 1 and s 2 satisfy Equation 12.84, and the fact that LCs 1 s 2 = 1, to obtain the second equalities. Finally, Equations and can now be substituted into Equation to obtain vt) = s 1v C 0) i L 0)/C s 1 s 2 ) e s1t + s 2v C 0) i L 0)/C e s2t ) s 2 s 1 ) Further, the substitution of Equation into Equations and yields v C t) = s 1v C 0) i L 0)/C s 1 s 2 ) e s1t + s 2v C 0) i L 0)/C e s 2t s 2 s 1 ) ) ) 1 + RCs1 s1 v C 0) i L 0)/C i L t) = R s 1 s 2 ) ) 1 + RCs2 s2 v C 0) i L 0)/C R s 2 s 1 ) = v C0)/L s 2 i L 0) s 1 s 2 ) e s 2t e s 1t e s1t + v C0)/L s 1 i L 0) e s 2t s 2 s 1 ) ) as the states of the parallel circuit. To obtain the second equality in Equation we have again used the fact that both s 1 and s 2 satisfy 654c

4 Equation 12.84, and the fact that LCs 1 s 2 = 1. This completes the formal node analysis of the circuit shown in Figure We will now close this subsection by examining the dynamic behavior of v C and i L for the same three cases defined in Section Those are the cases of under-damped, critically-damped, and over-damped dynamics. As we shall see, the dynamics of the series circuit are essentially identical to those of the parallel circuit for all three cases, except for the details of the role of R. In the series circuit, small R caused light damping while large R caused heavy damping. This role reverses for the parallel circuit because it is in the limit of large R that Figure reduces to Figure UNDER-DAMPED DYNAMICS The case of under-damped dynamics is characterized by α<ω or, after substitution of Equations and 12.87, by 2R > L/C. As R becomes large, the corresponding resistor approaches an open circuit, and so the circuit shown in Figure approaches the LC circuit shown in Figure Therefore, we should expect the under-damped dynamics to be oscillatory in nature. As we shall see shortly, this is indeed the case. With α<ω, the quantity inside the radicals in Equations and is negative, and so the natural frequencies s 1 and s 2 are complex numbers. To simplify matters, let us again define ω d according to ω d ω 2 o α ) With this definition, s 1 and s 2 from Equations and can be written as s 1 = α + jω d ) s 2 = α jω d ) The real and imaginary parts of s 1 and s 2 are now more apparent. Since s 1 and s 2 are now complex, the exponentials in Equations and are also complex. Thus, i L and v C will exhibit both oscillatory and decaying behavior. To see this, we substitute Equations and into 654d

5 Equations and , and use the Euler relation to obtain ) v C t) = v C 0)e αt αcvc 0) + i L 0) cosω d t) e αt sinω d t) Cω d ) = vc 20) + αcvc 0) + i L 0) 2 e αt Cω d )) cos ω d t + tan 1 αcvc 0) + i L 0) ) Cω d v C 0) i L t) = i L 0)e αt vc 0) + αli L 0) cosω d t) + Lω d ) = il 20) + vc 0) + αli L 0) 2 e αt Lω d sin ω d t + tan 1 Lω d i L 0) v C 0) + αli L 0) ) e αt sinω d t) )) ) Sketches of i L and v C are shown in Figure for the special case of i L 0) = 0. As was the case for the series circuit, the states in the parallel circuit display oscillatory and decaying behavior. It is also the case that Equations and reduce to Equations and 12.22, respectively, as the circuit damping characterized by α vanishes. The difference here is that this occurs as R because it is in this limit that Figure reduces to Figure A comparison of Equations and with Equations and shows that the under-damped dynamics of the parallel and series circuits are quite similar. This is to be expected because their characteristic equations are identical. It is for this reason that α, ω, ω d, and Q have the same interpretations for the two circuits. Our comments concerning stored energy also hold for both circuits. Therefore, we will not repeat the details here. Rather, we will identify three important differences. The first difference, which has been mentioned already, is the reversed role of R. A large R in the series circuit corresponds to a small R in the parallel circuit and vice versa. The second difference is the evaluation of the quality factor Q. While Equation still holds for the parallel circuit, that is, Q ω 2α, ) 654e

6 v C v C 0) e -αt 0 π/ω d 2π/ω d 3π/ω d 4π/ω d 5π/ω d 6π/ω d 7π/ω d t i L FIGURE Waveforms of v C and i L in undriven, parallel RLC circuit for the case of i L 0) = 0. I o 0 π/ω d 2π/ω d 3π/ω d 4π/ω d 5π/ω d 6π/ω d 7π/ω d t Equation does not. Rather, for the parallel circuit shown in Figure 12.24, the substitution of Equations and into Equation yields Q = 1 R L C ) or Q = ω ol R ) 654f

7 The third difference is the role of φ in Figure For the parallel circuit, assuming i L 0) = 0 in Equations and , v C is advanced with respect to i L by quadrature plus the additional angle φ, where φ = tan 1 α/ω d ) OVER-DAMPED DYNAMICS As with the case of the series circuit, the case of over-damped dynamics is characterized by α>ω or, after substitution of Equations and 12.87, by 2R < L/C. In this case, the quantity inside the radicals in Equations and is positive, and so both s 1 and s 2 are real. For this reason, the dynamic behavior of i L and v C, as expressed by Equations and , does not exhibit oscillation. Rather, it involves two real exponential functions that decay at different rates, as the two equations show. The expressions for v C and i L for the case of i L 0) = 0 with over-damping are obtained from Equations and , and are shown here: v C t) = s 1v C 0) s 1 s 2 ) e s 1t + s 2v C 0) s 2 s 1 ) e s 2t ) i L t) = v C0) Ls 1 s 2 ) e s1t + v C0) Ls 2 s 1 ) e s2t ) Since α>ω for over-damped circuits, note that s 1 and s 2 are both real in these two equations. As R becomes small, in particular smaller than 1/2 L/C, it becomes a significant short circuit across the capacitor and inductor. In this way it diverts the oscillating current that the capacitor and inductor share for larger values of R. As a consequence, the energy exchange between the capacitor and inductor is interrupted, and the circuit ceases to oscillate. Instead, its behavior is more like that of an independent capacitor and an independent inductor discharging through the resistor. To see this, let us determine the asymptotic values of s 1 and s 2 as R becomes small and hence as α becomes large. They are s 1 = α + α 2 ω 2 = α ) 2 α ω2 α 2α 2 ω = R L ) 654g

8 s 2 = α α 2 ω 2 = α 1 1 ) 2 2α = 1 α RC ) ω As expected the corresponding time constants approach RC and L/R, the time constants of an independent capacitor-resistor circuit and an independent inductor-resistor circuit. Note that, for over-damped dynamics, α>ω from which it follows that RC is the faster time constant and L/R is the slower time constant CRITICALLY-DAMPED DYNAMICS The case of critically-damped dynamics is characterized by α = ω. In this case, it follows from Equations and that s 1 = s 2 = α and that the characteristic equation, Equation 12.85, has a repeated root. Because of this, e s 1t and e s 2t are no longer independent functions, and so the general solution for v is no longer the superposition of these two functions as given by Equation Rather, it is again the superposition of the repeated exponential function e s 1t = e s 2t = e αt and te αt. From this observation, and Equations and 12.92, it follows that v C and i L will exhibit similar behavior. Perhaps the easiest way to determine v C and i L for the case of critical damping is to evaluate Equations and under the conditions of that case. To do so, observe from Equation that, for critical damping ω = α, and so ω d = 0. Therefore, we can obtain v C and i L for the case of critical-damping by evaluating Equations and in the limit ω d 0. This results in v C t) = v C 0)e αt αcv C0) + i L 0) te αt ) C i L t) = i L 0)e αt + v C0) + αli L 0) te αt ) L From Equations and we see that v C and i L contain both the decaying exponential function e αt and the function te αt, as expected. 654h