Lecture 8, Slide 1EECS40, Fall 2004Prof. White Lecture #8 OUTLINE First-order circuits Reading Chapter 3, begin Chapter 4

Lecture 8, Slide 2EECS40, Fall 2004Prof. White 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. –+–+ vsvs L R –+–+ vsvs C R ii

Lecture 8, Slide 3EECS40, Fall 2004Prof. White 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.

Lecture 8, Slide 4EECS40, Fall 2004Prof. White 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 switching 0 + is used to denote the time immediately after switching The current flowing in the inductor at t = 0 – is I o LRoRo RIoIo t = 0 i+v–+v–

Lecture 8, Slide 5EECS40, Fall 2004Prof. White Solving for the Current (t  0) For t > 0, the circuit reduces to Applying KVL to the LR circuit: Solution: LRoRo RIoIo i+v–+v– = I 0 e -(R/L)t

Lecture 8, Slide 6EECS40, Fall 2004Prof. White Solving for the Voltage (t > 0) Note that the voltage changes abruptly: LRoRo RIoIo +v–+v– I0RI0R v ReIiRtvt v tLR o       )0( )( 0,for 0)0( )/(

Lecture 8, Slide 7EECS40, Fall 2004Prof. White 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. (sec)

Lecture 8, Slide 8EECS40, Fall 2004Prof. White 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.

Lecture 8, Slide 9EECS40, Fall 2004Prof. White 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 –+–+ V Th C R Th L I Th

Lecture 8, Slide 10EECS40, Fall 2004Prof. White Water Model of Constant Voltage Source Charging a Capacitor Through a Resistor

Lecture 8, Slide 11EECS40, Fall 2004Prof. White 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 switching 0 + is used to denote the time immediately after switching The voltage on the capacitor at t = 0 – is V o Natural Response of an RC Circuit C RoRo RVoVo t = 0 ++ +v–+v–

Lecture 8, Slide 12EECS40, Fall 2004Prof. White Solving for the Voltage (t  0) For t > 0, the circuit reduces to Applying KCL to the RC circuit: Solution: +v–+v– C RoRo RVoVo ++ i

Lecture 8, Slide 13EECS40, Fall 2004Prof. White Solving for the Current (t > 0) Note that the current changes abruptly: +v–+v– C RoRo RVoVo ++ i

Lecture 8, Slide 14EECS40, Fall 2004Prof. White Time 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. (sec)

Lecture 8, Slide 15EECS40, Fall 2004Prof. White Natural Response Summary RL Circuit Inductor current cannot change instantaneously time constant RC Circuit Capacitor voltage cannot change instantaneously time constant R i L +v–+v– RC

Lecture 8, Slide 16EECS40, Fall 2004Prof. White QuantityVariableUnitUnit Symbol Typical Values Defining Relations Important Equations Symbol ChargeQcoulombC1aC to 1Cmagnitude of × electron charges q e = x C i = dq/dt CurrentIampereA 1  A to 1kA 1A = 1C/s VoltageVvoltV 1  V to 500kV 1V = 1N-m/C Summary of Electrical Quantities

Lecture 8, Slide 17EECS40, Fall 2004Prof. White PowerPwattW 1  W to 100MW 1W = 1J/sP = dU/dt; P=IV EnergyUjouleJ1fJ to 1TJ1J = 1N-mU = QV ForceFnewtonN 1N = 1kg- m/s 2 Timetseconds ResistanceRohm  1  to 10M  V = IR; P = V 2 /R = I 2 R R CapacitanceCfaradF1fF to 5FQ = CV; i = C(dv/dt);U = (1/2)CV 2 C InductanceLhenryH 1  H to 1H v = L(di/dt); U = (1/2)LI 2 L Summary of Electrical Quantities (concluded)