Slide 1Slide 2First-Order CircuitsReview (Conceptual)Slide 5Natural Response of an RL CircuitSolving for the Current (t 0)Solving for the Voltage (t > 0)Time Constant tSlide 10Capacitors and Stored ChargeCapacitors in circuitsCAPACITORSCharging a Capacitor with a constant currentDischarging a Capacitor through a resistorVoltage vs time for an RC dischargeNatural Response of an RC CircuitSolving for the Voltage (t 0)Solving for the Current (t > 0)Slide 20RC Circuit Model for a Digital Logic CircuitRC Circuits Abound in ComputersRC Circuit ModelSlide 24Analysis of RC CircuitSlide 26InsightSlide 28Time ConstantTransient vs. Steady-StateGeneral RC SolutionSolving the RC CircuitFinding the Initial ConditionFinding the Final ValueFinding the Time ConstantNatural Response SummaryRC Circuit Transient Analysis ExampleSlide 38Slide 39Transient Excitation of First-Order Circuits1. What is transient excitation and why is it important?2. What is a first-order circuit?3. What are “natural response” and “step response”?4. Transients in RL circuits (briefly)5. Transients in RC circuits application to computer circuitsTypes of Circuit ExcitationLinear Time- Invariant CircuitSteady-State ExcitationLinear Time- Invariant CircuitORLinear Time- Invariant CircuitDigitalPulseSourceTransient ExcitationLinear Time- Invariant CircuitSinusoidal (Single-Frequency) ExcitationAC Steady-State(DC Steady-State)First-Order Circuits•A circuit that contains only sources, resistors and an inductor is called an RL circuit.•A circuit that contains only sources, resistors and a capacitor is called an RC circuit.•RL and RC circuits are called first-order circuits because their voltages and currents are described by first-order differential equations.–+vs LR–+vs CRi iReview (Conceptual)•Any first-order circuit can be reduced to a Thévenin (or Norton) equivalent connected to either a single equivalent inductor or capacitor.–In steady state, an inductor behaves like a short circuit–In steady state, a capacitor behaves like an open circuit–+VTh CRThLRNIN•The natural response of an RL or RC circuit is its behavior (i.e., current and voltage) when stored energy in the inductor or capacitor is released to the resistive part of the network (containing no independent sources).•The step response of an RL or RC circuit is its behavior when a voltage or current source step is applied to the circuit, or immediately after a switch state is changed.Natural Response of an RL Circuit•Consider the following circuit, for which the switch is closed for t < 0, and then opened at t = 0:Notation:0– is used to denote the time just prior to switching0+ is used to denote the time immediately after switching•The current flowing in the inductor at t = 0– is IoLRoRIot = 0i +v–Solving for the Current (t 0)•For t > 0, the circuit reduces to•Applying KVL to the LR circuit yields first-order D.E.:•Solution:LRoRIoi +v–tLReiti)/()0()( = I0e-(R/L)tSolving for the Voltage (t > 0)•Note that the voltage changes abruptly (step response):tLRoeIti)/()(LRoRIo+v–I0RvReIiRtvtvtLRo)0( )( 0,for 0)0()/(Time Constant •In the example, we found that•Define the time constant–At t = , the current has reduced to 1/e (~0.37) of its initial value.–At t = 5, the current has reduced to less than 1% of its initial value.RLtLRotLRoReItveIti)/()/()( )( and(sec)Transient response of RC circuitsand application to computer circuitsdriven by binary voltage pulsesCapacitors and Stored Charge•So far, we have assumed that electrons keep on moving around and around a circuit.•Current doesn’t really “flow through” a capacitor. No electrons can go through the insulator.•But, we say that current flows through a capacitor. What we mean is that positive charge collects on one plate and leaves the other.•A capacitor stores charge. Theoretically, if we did a KCL surface around one plate, KCL could fail. But we don’t do that.•When a capacitor stores charge, it has nonzero voltage. In this case, we say the capacitor is “charged”. A capacitor with zero voltage has no charge differential, and we say it is “discharged”.Capacitors in circuits•If you have a circuit with capacitors, you can use KVL and KCL, nodal analysis, etc.•The voltage across the capacitor is related to the current through it by a differential equation instead of Ohm’s law.dtdVCi CAPACITORS CidtdV So capacitance is defined byCi(t)| (V +dtdVCi Charging a Capacitor with a constant current CidtdV(t)Ci| (V(t) + CidtdV(t)t0t0 dtdt Cti CiV(t)t0dttimevoltageDischarging a Capacitor through a resistorRCV(t) Ci(t)dtdV(t)CiV(t) +RThis is an elementary differential equation, whose solution is the exponential://1dtdttee/0)(teVtVSince:iVoltage vs time for an RC dischargeVoltageTime•Consider the following circuit, for which the switch is closed for t < 0, and then opened at t = 0:Notation:0– is used to denote the time just prior to switching0+ is used to denote the time immediately after switching•The voltage on the capacitor at t = 0– is VoNatural Response of an RC CircuitCRoRVot = 0++v–Solving for the Voltage (t 0)•For t > 0, the circuit reduces to•Applying KCL to the RC circuit:•Solution:+v–RCtevtv/)0()(CRoRVo+iSolving for the Current (t > 0)•Note that the current changes abruptly:RCtoeVtv/)(RVieRVRvtitioRCto)0( )( 0,for 0)0(/+v–CRoRVo+iTime Constant •In the example, we found that•Define the time constant–At t = , the voltage has reduced to 1/e (~0.37) of its initial value.–At t = 5, the voltage has reduced to less than 1% of its initial value.RCRCtoRCtoeRVtieVtv//)( )( and(sec)RC Circuit Model for a Digital Logic CircuitThe capacitor is used to model the response of a digital circuit to a new voltage input:The digital circuit is modeled by a resistor in series with a capacitor. The capacitor cannotchange its voltage instantly,as charges can’t jump instantlyto the other plate, they must go through the circuit!VoutRCVinVout+_+_RC Circuits Abound in ComputersEvery node in a circuit has natural capacitance, and it is the charging of these capacitances that limits real circuit performance (speed)We compute with pulses We send beautiful
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