EE40 Transients First Order Circuits Prof Nathan Cheung Prof 09 22 2009 R di Reading H bl Chapter Hambley Ch t 4 EE40 Fall 2009 Slide 1 Prof Cheung First Order Circuits A circuit which contains only sources resistors and a capacitor is called an RC circuit A circuit which contains only sources resistors and an inductor is called an RL circuit RL and d RC circuits i it are called ll d first order fi t d circuits because their voltages and currents are described by first order first order differential equations equations R i i EE40 Fall 2009 vs L Slide 2 vs R C Prof Cheung First Order Circuits RC and RL Circuits one energy gy storage element Natural Response Step Response Forced Response Time constant Particular Solution and Complementary Solution Applications Propagation Delay DRAM EE40 Fall 2009 Slide 3 Prof Cheung Circuit Response The natural response of an RC or RL 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 RC or RL circuit is its behavior when a voltage or current source step is applied to the circuit circuit or immediately after a switch state is changed The force response of an RC or RL circuit is its behavior when a voltage or current source has a particular time dependence e g sin t exp t EE40 Fall 2009 Slide 4 Prof Cheung Example 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 i used d tto d denote t th the titime iimmediately di t l after ft switching it hi The voltage on the capacitor at t 0 is Vo EE40 Fall 2009 Slide 5 Prof Cheung Solving for the Voltage t 0 For t 0 the circuit reduces to Vo i Ro C v R Applying pp y g KCL to the RC circuit Solution EE40 Fall 2009 v t v 0 e t RC Slide 6 Prof Cheung Sketch of EE40 Fall 2009 vC t v 0 e Slide 7 t RC Prof Cheung Solving for the Current t 0 i Vo Ro C v R v t Voe t RC Note that the current changes abruptly i 0 0 v Vo t RC for t 0 i t e R R Vo i 0 R EE40 Fall 2009 Slide 8 Prof Cheung Solving for Power and Energy Delivered t 0 i Vo Ro C v R v t Vo e t RC v 2 Vo2 2 t RC p e R R t t 2 Vo 2 x RC w p x dx e dx R 0 0 x dummy variable for integration 1 CVo2 1 e 2 t RC 2 EE40 Fall 2009 Slide 9 Prof Cheung Time Constant In the example we found that v t Voe t RC Vo t RC and d i t e R Define the time constant RC At t the voltage has reduced to 1 e 0 0 37 37 of its initial value At t 5 5 the voltage has reduced to less than 1 of its initial value EE40 Fall 2009 Slide 10 Prof Cheung 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 tt 0 0 the th entire ti system t is i att steady state t d t t and d the th inductor i d t is like short circuit g in the inductor at t 0 is Io and v The current flowing across is 0 EE40 Fall 2009 Slide 11 Prof Cheung Solving for the Voltage t 0 i t I oe 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 R L t v 0 I0R EE40 Fall 2009 Slide 12 Prof Cheung Solving for the Current t 0 For t 0 the circuit reduces to i Io L Ro R v Applying KVL to the LR circuit v t i t R At t 0 i I0 di t At arbitraryy t 0 i i t and v t L Solution S l ti EE40 Fall 2009 d dt i t i 0 e R L t Slide 13 R L t I0e R L t Prof Cheung Solving for Power and Energy Delivered t 0 i t I o e R L t Io Ro L R v p i 2 R I o2 Re 2 R L t t t 0 0 w p x dx I o2 Re 2 R L x dx 1 2 2 R L t LI o 1 e 2 EE40 Fall 2009 Slide 14 Prof Cheung Time Constant In the example we found that i t I oe R L t R L t and v t I o Re 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 EE40 Fall 2009 Slide 15 Prof Cheung Natural Response Summary RL Circuit RC Circuit i L v R C R Inductor current cannot change g instantaneously y Capacitor voltage cannot change g instantaneously y i 0 i 0 v 0 v 0 i t i 0 e t v t v 0 e t L time ti constant t t R EE40 Fall 2009 Slide 16 time constant RC Prof Cheung Step Response of 1st Order Circuits IIn general l when h an applied li d currentt or voltage lt suddenly changes the voltages and currents in an RC a Co or RL ccircuit cu will cchange a ge e exponentially po e a y with time from their initial values to their final values with the characteristic time constant x t x f x t0 x f e t t0 where x t is the circuit variable v or i xf is the final value of the circuit variable t0 is the time at which the change occurs EE40 Fall 2009 Slide 17 Prof Cheung Example v t across a capacitor v is continuous v t v t to EE40 Fall 2009 t Slide 18 t to Prof Cheung Procedure for Finding Transient Response 1 Identify the variable of interest For RL circuits it is usuallyy 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 …
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