EE42 100 Lecture 9 Topics More on First Order Circuits Water model and potential plot for RC circuits A bit on Second Order Circuits First 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 R i i L vs vs R C 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 t 0 Io Ro i L R v Notation 0 is used to denote the time just prior to switching 0 is used to denote the time immediately after switching The current flowing in the inductor at t 0 is Io 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 these quantities and their time derivatives Solving for the Current t 0 For t 0 the circuit reduces to i Io Ro L R v Applying KVL to the LR circuit Solution i t i 0 e R L t What Does e t Look Like e t 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 Solving for the Voltage t 0 i t I o e R L t Io Ro L R v Note that the voltage changes abruptly v 0 0 for t 0 v t iR I o Re v 0 I o R R L t Time Constant In the example we found that i t I o e R L t and v t I o Re R L t Define the time constant L R 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 Transient 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 Th venin or Norton equivalent connected to either a single equivalent inductor or capacitor RTh RTh L VTh ITh C In steady state an inductor behaves like a short circuit In steady state a capacitor behaves like an open circuit Natural Response of an RC Circuit Consider the following circuit for which the switch is closed for t 0 and then opened at t 0 Vo Ro C t 0 v R Notation 0 is used to denote the time just prior to switching 0 is used to denote the time immediately after switching The voltage on the capacitor at t 0 is Vo Solving for the Voltage t 0 For t 0 the circuit reduces to Vo i Ro C v Applying KCL to the RC circuit Solution v t v 0 e t RC R Solving for the Current t 0 i Vo Ro C v R i 0 0 v t Vo e t RC Note that the current changes abruptly v Vo t RC for t 0 i t e R R Vo i 0 R Time Constant In the example we found that v t Vo e t RC Vo t RC and i t e R Define the time constant RC with R in ohms and C in farads is in seconds 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 Natural Response Summary RL Circuit RC Circuit i L R v R C Inductor current cannot change instantaneously Capacitor voltage cannot change instantaneously i 0 i 0 v 0 v 0 i t i 0 e t v t v 0 e t L time constant R time constant RC Transient 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 x t x f x t0 x f e t t0 where x t is the circuit variable voltage or current xf is the final value of the circuit variable t0 is the time at which the change occurs This is a very useful equation Procedure for Finding Transient Response 1 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 vc t0 vc t0 Assuming that the circuit reached steady state before t0 use the fact that an inductor behaves like a short circuit in steady state or that a capacitor behaves like an open circuit in steady state Procedure cont d 3 Calculate the final value of the variable its value as t Again make use of the fact that an inductor behaves like a short circuit in steady state t or that a capacitor behaves like an open circuit in steady state t 4 Calculate the time constant for the circuit L R for an RL circuit where R is the Th venin equivalent resistance seen by the inductor RC for an RC circuit where R is the Th venin equivalent resistance seen by the capacitor Example RL Transient Analysis Find the current i t and the voltage v t t 0 R 50 i …
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