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Berkeley ELENG 42 - Lecture 13: RC Circuits

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RC CircuitsExample: Circuit ChargingIntuitive Guess at SolutionSetting up EquationsLinear Differential EquationsLinearity and SuperpositionTime InvarianceHomogeneous SolutionSteady-State Homogenous SolutionForced SolutionTransient WaveformsTransientsSolution ProcedureForced SolutionBack to Example...RC Time ConstantComplete SolutionCurrent Through CapacitorCapacitor DischargeCapacitor as a ``Battery''Step ResponseStep Response of CircuitPulse ResponsePulse Reponse (cont)Complete Pulse ResponseFiltering: Short Pulse ResponseDC Blocking CapacitorDC Block (cont)A DifferentiatorAn IntegratorMore Complicated ExampleMore CapacitorsEE 42/100Lecture 13: RC CircuitsELECTRONICSRev B 2/29/2012 (8:31 PM)Prof. Ali M. NiknejadUniversity of California, BerkeleyCopyrightc 2012 by Ali M. NiknejadA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 1/33 – p.RC Circuits•Many circuits can be modeled as a combination of resistors and capacitors. Agood example is the gates inside of a microprocessor. To determine how fastthese gates “switch” (in other words how fast we can run a computer), we canmodel the entire system by an RC circuit.•The general solution of RC circuits is a powerful tool that we can apply to a greatrange of problems and so it deserves some special attention.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 2/33 – p.Example: Circuit Charging•Suppose a battery is connected to a switch and an RC circuit as shown. Initiallywe assume the capacitor is uncharged (v(0) = 0) and the switched is closed attime t = 0. What happens next?A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 3/33 – p.Intuitive Guess at Solution•We might guess that current would begin to flow into the capacitor, where at timezero at least,i(0) =VsR•But as time goes on, the current will change, because as charge accumulates onthe capacitor plates, its voltage rises and so the current through R reducesi(t) =Vs− v(t)R•Note that if v(t) reaches Vs, then the current would drop to zero, which means thatv(t) would then stop changing as well. This is an interesting state because if weput the system in such a state, then dv/dt = 0 and so the system just stays there.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 4/33 – p.Setting up Equations•While we can surmise many properties of this circuit by using intuition, to find theexact solution requires us to setup KCL/KVL equations just as before, but now wemust solve differential equations.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 5/33 – p.Linear Differential Equations•While solving differential equations is a lot harder, we can surmise many thingsabout the solution without actually solving any equations.•For a circuit with a single capacitor and any number of resistors, if the capacitorsand resistors are all linear, then the resulting equations will be a linear andfirst-order.•The order of the equations is determined by the highest derivative term. Since forcapacitors the current is proportional to dv/dt, we end up with a system offirst-order ordinary differential equations. They’re “ordinary” in that they involvenormal derivatives, instead of “partial” derivatives.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 6/33 – p.Linearity and Superposition•The first important property of linear equations is that superposition applies.Superposition means that if the circuit is excited with many inputs simultaneously,the resulting solution is the same as the summation of all the solutions to theindividual stimuli applies one at a time.•This is very handy, even if there is only one stimulus! That’s because a complicatedstimulus can often be broken down into simpler functions (e.g. a Fourier Series).•For example, the solution for the stimulus for vs= V0sin(ωt) can be determinedby solving the system for vs= V0ejωt.•That’s becausesin(x) =ejx− e−jx2jA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 7/33 – p.Time Invariance•Another very important property is the concept of time invariance. If a circuit staysthe same, then no matter when you apply an input, the output should look thesame.•In other words, if you build an RC circuit today and excite a voltage source, you’dexpect to see the same response if you did the same experiment 10 minutes later(assuming no aging of components), assuming you shift your time reference.•More compactly, if shift the stimulus to another time, v′s= vs(t − T ), then thesolution v′(t) = v(t − T ) (we simply shift the time reference).A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 8/33 – p.Homogeneous Solution•When there are no sources present, then the solution of a differential equations iscalled the “homogeneous” solution. That’s a fancy name for the “zero input”solution.•Why is there a solution for zero input? A system described by a differentialequation is not a simple memory-less input-output relation, so even if there are noexplicit inputs, there can be an output!•The output waveform is a function of the state of the system, which in this case isthe capacitor voltage. If the capacitor voltage is initially charged, then there’s goingto be an output, despite there being no input.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 9/33 – p.Steady-State Homogenous Solution•If there are no inputs and we wait long enough, we can say that eventually thesystem will come to rest. When the system comes to rest, the voltages/currents donot change, and so dv/dt = 0 for every capacitor. From this we can find thesteady-state value of the homogenous solution.•We can find this solution directly if we note that dv/dt = 0 implies the currentthrough all the capacitors is zero, or it’s replaced with an open circuits.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 13 p. 10/33 – pForced Solution•Because of linearity, we see that any time we do apply an input to the system, theresponse will be a super-position of the response to “no input” and the response tothe stimulus, or the forced solution. In other words, “zero input” is also animportant input to consider when dealing with differential equations.•The response of the circuit to the stimulus is a function of the stimulus and thecircuit. The


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Berkeley ELENG 42 - Lecture 13: RC Circuits

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Lecture 3

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