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70 chapter 44.3 Resistance Becomes Impedance (Theory)We apply Kirchhoff’s laws to the RLC circuit in Figure 4.1 left by summing voltagedrops as we work our way around the circuit. This gives the differential equationfor the current I(t) in the circuit:dV (t)dt= R dIdt + L d2 Idt 2 + I C , (4.13)where we have taken an extra time derivative to eliminate an integral over thecurrent. The analytic solution follows by assuming that the voltage has the formV (t)=V 0 cos ωt and by guessing that the resulting current I(t)=I 0 e −iωt will alsobe complex, with its real part the physical current. Because (4.13) is linear in I, thelaw of linear superposition holds, and so we can solve for the complex I and thenextract its real and imaginary parts:I(t)= 1 Z V 0e −iωt ,⇒ I(t)= V 0|Z| e−i(ωt+θ) = V 0|Z|√( ) 2 1|Z| = R 2 +ωC − ωL ,( ) 1Z = R + iωC − ωL , (4.14)[cos(ωt + θ) − i sin(ωt + θ)] , (4.15)( )1/ωC − ωLθ = tan −1 .RWe see that the amplitude of the current equals the amplitude of the voltage dividedby the magnitude of the complex impedance, and that the phase of the currentrelative to that of the voltage is given by θ.The solution for the two RLC circuits in parallel (Figure 4.1 right) is analogousto that with ordinary resistors. Two impedances in series have the same currentpassing through them, and so we add voltages. Two impedances in parallel havethe same voltage across them, and so we add currents:Z ser = Z 1 + Z 2 ,1Z par= 1 Z 1+ 1 Z 2. (4.16)4.4 Abstract Data Structures, Objects (CS)What do you see when you look at the abstract object on the right of Figure 4.2? Somereaders may see a face in profile, others may see some parts of human anatomy,and others may see a total absence of artistic ability. This figure is abstract in thesense that it does not try to present a true or realistic picture of the object but ratheruses a symbol to suggest more than meets the eye. Abstract or formal concepts−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 70