Transient Excitation of First Order Circuits 1 2 3 4 5 What is transient excitation and why is it important What is a first order circuit What are natural response and step response Transients in RL circuits briefly Transients in RC circuits application to computer circuits Types of Circuit Excitation Linear TimeInvariant Circuit Linear TimeInvariant Circuit Steady State Excitation DC Steady State Linear TimeInvariant Circuit Sinusoidal SingleFrequency Excitation AC Steady State OR Digital Pulse Source Linear TimeInvariant Circuit Transient Excitation 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 R i i L vs vs R C 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 RN L VTh IN C In steady state an inductor behaves like a short circuit In steady state a capacitor behaves like an open circuit 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 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 yields first order D E Solution i t i 0 e R L t I0e R L t 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 step response v 0 0 for t 0 v t iR I o Re v 0 I0R 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 sec 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 response of RC circuits and application to computer circuits driven by binary voltage pulses Capacitors 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 dV i C dt CAPACITORS V i t C capacitance is defined by dV i C dt dV i So dt C Charging a Capacitor with a constant current V t C i dV t i dt C t t dV t i dt dt dt C 0 0 voltage t i i t V t dt C C 0 time Discharging a Capacitor through a resistor V t i C i R dV t i t V t dt C RC This is an elementary differential equation whose solution is the exponential V t V0 e t Since d t 1 t e e dt Voltage vs time for an RC discharge Voltage Time 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 sec Define the time constant RC 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 RC Circuit Model for a Digital Logic Circuit The 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 R Vout Vin The capacitor cannot change its voltage instantly as charges can t jump instantly to the other plate they must go through the circuit C Vout We compute with pulses voltage RC Circuits Abound in Computers We send beautiful pulses in voltage But we receive lousy looking pulses at the output time time Capacitor charging effects are responsible Every node in a circuit has natural capacitance and it is the charging of these capacitances that limits real circuit performance speed RC Circuit Model Every digital circuit has natural resistance and capacitance In real life the resistance and capacitance can be estimated using characteristics of the materials used and the layout of the physical device R Vout The value of R and C for a digital circuit Vout Vin C determine how long it will take the capacitor to change its voltage the gate delay RC Circuit Model With the digital context in mind Vin will usually be a time varying voltage that switches instantaneously between logic 1 voltage and logic 0 voltage R Vin Vout C Vout t 0 We often represent this switching voltage with a switch in the circuit diagram i Vs 5 V Vout Analysis of …
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