Week 4a, Slide 1EECS 42, Spring 2005Prof. White Transformer – Two Coupled Inductors N 1 turns N 2 turns + - v1v1 + v2v2 - |v 2 |/|v 1 | = N 2 /N 1

Week 4a, Slide 2EECS 42, Spring 2005Prof. White AC Power System

Week 4a, Slide 3EECS 42, Spring 2005Prof. 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

Week 4a, Slide 4EECS 42, Spring 2005Prof. 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)

Week 4a, Slide 5EECS 42, Spring 2005Prof. White Types of Circuit Excitations and Their Uses 1.Steady excitation (“DC”) – steady DC voltage or current sources. Uses: the DC part of all kinds of communications, transistor, sensor, … circuits 2.Transient excitation -- DC voltage or current sources are suddenly turned on or off. Uses: flashbulb circuit where charge a capacitor and suddenly dis- charge it through flashbulb; put an on and an off switching together and get a digital pulse: += 3.Sinusoidal excitation (“AC”) – Uses: AC power systems; communication systems to find frequency response.

Week 4a, Slide 6EECS 42, Spring 2005Prof. White 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. –+–+ vsvs L R –+–+ vsvs C R ii

Week 4a, Slide 7EECS 42, Spring 2005Prof. 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.

Week 4a, Slide 8EECS 42, Spring 2005Prof. 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–

Week 4a, Slide 9EECS 42, Spring 2005Prof. 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–

Week 4a, Slide 10EECS 42, Spring 2005Prof. White Solving for the Voltage (t > 0) Note that the voltage changes abruptly: LRoRo RIoIo +v–+v–

Week 4a, Slide 11EECS 42, Spring 2005Prof. 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.

Week 4a, Slide 12EECS 42, Spring 2005Prof. 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.

Week 4a, Slide 13EECS 42, Spring 2005Prof. 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

Week 4a, Slide 14EECS 42, Spring 2005Prof. 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–

Week 4a, Slide 15EECS 42, Spring 2005Prof. 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

Week 4a, Slide 16EECS 42, Spring 2005Prof. White Solving for the Current (t > 0) Note that the current changes abruptly: +v–+v– C RoRo RVoVo ++ i

Week 4a, Slide 17EECS 42, Spring 2005Prof. 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.

Week 4a, Slide 18EECS 42, Spring 2005Prof. 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