EE42/100 Lecture 9Slide 2Slide 3Slide 4Slide 5Natural Response of an RL CircuitSlide 7Solving for the Current (t 0)Slide 9Solving for the Voltage (t > 0)Time Constant tTransient vs. Steady-State ResponseReview (Conceptual)Natural Response of an RC CircuitSolving for the Voltage (t 0)Solving for the Current (t > 0)Slide 17Natural Response SummaryTransient Response of 1st-Order CircuitsProcedure for Finding Transient ResponseProcedure (cont’d)Example: RL Transient AnalysisSlide 23Example: RC Transient AnalysisSlide 25Application to Digital Integrated Circuits (ICs)Digital SignalsCircuit Model for a Logic GateLogic Level TransitionsSequential SwitchingPulse DistortionExampleSlide 33Slide 34Slide 35Slide 36Slide 37Slide 38A 2nd Order RLC CircuitThe Differential EquationSlide 41The Particular SolutionThe Complementary SolutionCharacteristic EquationSlide 45EE42/100 Lecture 9Topics: More on First-Order Circuits Water model and potential plot for RC circuits A bit on Second-Order CircuitsFirst-Order Circuits•A circuit which contains only sources, resistors and an inductor is called an RL circuit.•A circuit which 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 iThe 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 IoLt = 0RoRIoi +v–Recall: The current flowing in an inductor cannot change instantly, and the voltage across a capacitor, which is proportional to the charge stored in the capacitor, cannot change instantly. For a first-order circuit these are called initial values of current and voltage. A long time after the circuit configuration changes, the currents and voltages achieve their final, or steady-state values.Later when we talk about second-order circuits – ones that consist of resistors and the equivalent of two energy storage elements, like an L and a C or two Cs – we’ll take a look at the initial and final values of thesequantities and their time derivatives.Solving for the Current (t 0)•For t > 0, the circuit reduces to•Applying KVL to the LR circuit:•Solution:LRoRIoi +v–tLReiti)/()0()(What Does e-t/ Look Like?with = 10-4• is the amount of time necessary for an exponential to decay to 36.7% of its initial value.•-1/ is the initial slope of an exponential with an initial value of 1.e-t/Solving for the Voltage (t > 0)•Note that the voltage changes abruptly:tLRoeIti)/()(LRoRIo+v–RIvReIiRtvtvotLRo)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)/()/()( )( andTransient vs. Steady-State Response•The momentary behavior of a circuit (in response to a change in stimulation) is referred to as its transient response.•The behavior of a circuit a long time (many time constants) after the change in voltage or current is called the steady-state response.Review (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 CRThLRThITh•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(with R in ohms and C infarads, is in seconds)Natural Response SummaryRL Circuit•Inductor current cannot change instantaneously•time constant RC Circuit•Capacitor voltage cannot change instantaneously•time constantRL/)0()()0()0(teitiiiRiL+v–RC/)0()()0()0(tevtvvvRCTransient Response of 1st-Order Circuits•We saw that the currents and voltages in RL and RC circuits decay exponentially with time, with a characteristic time constant , when an applied current or voltage is suddenly removed.•In general, when an applied current or voltage suddenly changes, the voltages and currents in an RL or RC circuit will change exponentially with time, from their initial values to their final values, with the characteristic time constant as follows:where x(t) is the circuit variable (voltage or current)xf is the final value of the circuit variablet0 is the time at which the change occursThis is a very useful equation! /)(00 )()(ttffextxxtxProcedure for Finding Transient Response1. Identify the variable of interest•For RL circuits, it is usually the inductor current iL(t)•For RC circuits, it is usually the capacitor voltage vc(t)2. Determine the initial value (at t = t0+) of the variable •Recall that iL(t) and vc(t) are continuous variables: iL(t0+) = iL(t0) and
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