Slide 1Simplification for time behavior of RC CircuitsGeneral RC SolutionSolving the RC CircuitFinding the Time ConstantExampleFinding the Initial ConditionExample: Initial ConditionSlide 9Finding the Final ValueExample: Final ValueSlide 12Example: Time ConstantExample: Final AnswerHard ExampleInitial ConditionsTime ConstantFinal ValueApplication: DRAMApplication: Transmission on a wire (not too long a wire)What environment do pulses face?RC RESPONSESlide 23Review of simple exponentials.9/20/2004 EE 42 fall 2004 lecture 9 1Lecture #9 Example problems with capacitors•Next we will start exploring semiconductor materials (chapter 2).Reading: Malvino chapter 2 (semiconductors)9/20/2004 EE 42 fall 2004 lecture 9 2Simplification for time behavior of RC Circuits Before any input change occurs we have a dc circuit problem (that is we can use dc circuit analysis to relate the output to the input).We call the time period during which the output changes the transientWe can predict a lot about the transient behavior from the pre- and post-transient dc solutionstimevoltageinputtimevoltageoutputLong after the input change occurs things “settle down” …. Nothing is changing …. So again we have a dc circuit problem.9/20/2004 EE 42 fall 2004 lecture 9 3General RC Solution• Every current or voltage in an RC circuit has the following form while the sources are unchanging:• x represents any current or voltage• t0 is the time when the source voltage switches• xf is the final (asymptotic) value of the current or voltageAll we need to do is find these values and plug in to solve for any current or voltage in an RC circuit.)RC/()0tt(efx)0t(xfx)t(x9/20/2004 EE 42 fall 2004 lecture 9 4Solving the RC CircuitWe need the following three ingredients to fill in our equation for any current or voltage:• x(t0+) This is the current or voltage of interest just after the voltage source switches. It is the starting point of our transition, the initial value.• xf This is the value that the current or voltage approaches as t goes to infinity. It is called the final value.• RC This is the time constant. It determines how fast the current or voltage transitions between initial and final value.9/20/2004 EE 42 fall 2004 lecture 9 5Finding the Time ConstantTo find the response of a circuit with voltage sources, resistances and capacitances, we just need to find the starting point, the long term steady state, and the RC constant.It seems easy to find the time constant: it equals RC.But what if there is more than one resistor or capacitor?R is the Thevenin equivalent resistance with respect to the capacitor terminals.Remove the capacitor and find RTH. It might help to turn off the voltage source. Use the circuit after switching.9/20/2004 EE 42 fall 2004 lecture 9 6ExampleFind the current I(t).t = 0IR2 = 5 kVs = 5 V+C = 1 mFR1 = 10 k9/20/2004 EE 42 fall 2004 lecture 9 7Finding the Initial ConditionTo find x(t0+), the current or voltage just after the switch, we use the following essential fact:Capacitor voltage is continuous; it cannot jump when a switch occurs.So we can find the capacitor voltage VC(t0+) by finding VC(t0-), the voltage before switching.We assume the capacitor was in steady-state before switching. The capacitor acts like an open circuit in this case, and it’s not too hard to find the voltage over this open circuit.We can then find x(t0+) using VC(t0+) using KVL or the capacitor I-V relationship. These laws hold for every instant in time.9/20/2004 EE 42 fall 2004 lecture 9 8Example: Initial Condition•We first find capacitor voltage right after the switch, (at t=0+) and use it to find the current I at t=0+.•To do this, we look at the circuit before switching, because the capacitor voltage will remain the same after switching. •Assuming the circuit has been “unswitched” for a long time, the capacitor acts like an open circuit connected to a resistor.•The capacitor voltage before switching (at t=0-) is 0 V.•The capacitor voltage after switching (at t=0+) is 0 V.t = 0IR2 = 5 kVs = 5 V+C = 1 mFR1 = 10 k9/20/2004 EE 42 fall 2004 lecture 9 9Example: Initial Condition•We know the capacitor voltage is 0 V right after the switch.•By KVL, the resistor gets all 5 V that the source puts out.•So by Ohm’s law, I(0+) is 5 V / 10 k = 0.5 mAt = 0IR2 = 5 kVs = 5 V+C = 1 mFR1 = 10 k+0 V_+ 5V -9/20/2004 EE 42 fall 2004 lecture 9 10Finding the Final ValueTo find xf , the asymptotic final value, we assume that the circuit will be in steady-state as t goes to infinity.So we assume that the capacitor is acting like an open circuit. We then find the value of current or voltage we are looking for using this open-circuit assumption.Here, we use the circuit after switching along with the open-circuit assumption.When we found the initial value, we applied the open-circuit assumption to the circuit before switching, and found the capacitor voltage which would be preserved through the switch.9/20/2004 EE 42 fall 2004 lecture 9 11Example: Final Value•After the circuit has been switched for a long time, the capacitor will act like an open circuit.•Then no current can flow—eventually, I goes to zero.•If = 0 At = 0IR2 = 5 kVs = 5 V+C = 1 mFR1 = 10 k9/20/2004 EE 42 fall 2004 lecture 9 12Finding the Time ConstantIt seems easy to find the time constant: it equals RC.But what if there is more than one resistor or capacitor?R is the Thevenin equivalent resistance with respect to the capacitor terminals.Remove the capacitor and find RTH. It might help to turn off the voltage source. Use the circuit after switching.We will discuss how to combine capacitors in series and in parallel in the next lecture.9/20/2004 EE 42 fall 2004 lecture 9 13Example: Time Constant•How long does it take for the current to converge after switching?•Look at the circuit after switching to find the time constant.•The 5 k resistor is not “in” the circuit after the switch.•R = 10 k, C = 1 mF so = 10 st = 0IR2 = 5 kVs = 5 V+C = 1 mFR1 = 10 k9/20/2004 EE 42 fall 2004 lecture 9 14Example: Final Answer•Plugging in, we get:I(t) = If + (I(0+) – If) e-t/RCI(t) = 0 + (0.5 mA – 0) e-t/10 = 0.5 e-t/10 mAt = 0IR2 = 5 kVs = 5 V+C = 1 mFR1 = 10 k9/20/2004 EE 42 fall 2004 lecture 9 15Hard Example•Find V1(t) and V2(t).•Find the energy absorbed by the resistors for t >
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