Electric Circuit Theory

Size: px
Start display at page:

Download "Electric Circuit Theory"

Transcription

1 Electric Circuit Theory Nam Ki Min

2 Chapter 8 Natural and Step Responses of RLC Circuits Nam Ki Min

3 8.1 Introduction to the Natural Response of 3 First-Order Circuits (Chapter 7) Circuits with a single storage element (a capacitor or an inductor). The differential equations describing them are first-order. v C dv C dt + v C V s RC = 0 i(t) di dt + Ri V s L = 0 Second-Order Circuits (Chapter 8) Circuits containing two storage elements Their responses are described by differential equations that contain second derivatives. Typical examples of second-order circuits are RLC circuits. A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. There are two basic types of RLC circuits: parallel connected and series connected.

4 Finding Initial Values Chapter 8 Natural and Step Responses of RLC Circuits 8.1 Introduction to the Natural Response of There are two key points to keep in mind in determining the initial conditions. We must carefully handle the polarity of voltage v C (t) across the capacitor and the direction of the current i L (t) through the inductor. Keep in mind that v C (t) and i L (t) are defined strictly according to the passive sign convention. One should carefully observe how these are defined and apply them accordingly. Keep in mind that the capacitor voltage is always continuous so that v c 0 + = v C (0 ) and the inductor current is always continuous so that i L 0 + = i L (0 ) where t = 0 denotes the time just before a switching event and t = 0 + is the time just after the switching event, assuming that the switching event takes place at t = 0. 4 Thus, in finding initial conditions, we first focus on those variables that cannot change abruptly, capacitor voltage and inductor current.

5 8.1 Introduction to the Natural Response of Obtaining of the Differential Equation for Parallel RLC circuits find many practical applications, notably in communications networks and filter designs. Assume initial inductor current I 0 and initial capacitor voltage V 0, i L 0 = 1 L 0 v t dt = I o 5 v c 0 = V o Since the three elements are in parallel, they have the same voltage v across them. Applying KCL at the top node gives i R + i L + i C = 0 v R + 1 L t v t dt + C dv dt = 0 v R + I o + 1 L t v t dt 0 + C dv dt = 0 (8.1) v R + 1 L 0 v t dt + 1 L t v t dt 0 + C dv dt = 0

6 8.1 Introduction to the Natural Response of 6 Obtaining of the Differential Equation for Taking the derivative with respect to t results in 1 dv R dt + v L + C d2 v dt 2 = 0 (8.2) v R + I o + 1 L t v t dt 0 + C dv dt = 0 (8.1) Dividing by C Arranging the derivatives in the descending order, we get d 2 v dt dv RC dt + v LC = 0 (8.3) This is a second-order differential equation and is the reason for calling the RLC circuits in this chapter second-order circuits.

7 8.1 Introduction to the Natural Response of Solution of the Differential Equation Our experience in the preceding chapter on first-order circuits suggests that the solution is of exponential form. So we let v t = Ae st (8.4) where A and s are unknown constants. d 2 v dt dv RC dt + v LC = 0 (8.3) Substituting Eq. (8.4) into Eq. (8.3) and carrying out the necessary differentiations, we obtain or As 2 e st + As RC est + Aest LC = 0 Ae st s RC s + 1 LC = 0 (8.5) 7 Since e st 0 s RC s + 1 LC = 0 (8.6) We cannot A=0 as a general solution because the voltage is not zero for all time and Ae st is the assumed solution we are trying to find.

8 8.1 Introduction to the Natural Response of 8 Solution of the Differential Equation s RC s + 1 LC = 0 (8.6) d 2 v dt dv RC dt + v LC = 0 (8.3) Eq.(8.6) is known as the characteristic equation of the differential Eq. (8.3), since the roots of the equation determine the mathematical character of v(t). The two roots of Eq. (8.8) are s 1 = 1 2RC + 1 2RC 2 1 LC (8.7) s 2 = 1 2RC 1 2RC 2 1 LC (8.8)

9 8.1 Introduction to the Natural Response of 9 Solution of the Differential Equation The two values of s in Eqs.(8.7) and (8.8) indicate that there are two possible solutions for i, each of which is of the form of the assumed solution in Eq. (8.4); that is, v 1 = A 1 e s 1t v 2 = A 2 e s 2t d 2 v dt dv RC dt + v LC = 0 (8.3) Since Eq. (8.3) is a linear equation, any linear combination of the two distinct solutions v 1 and v 2 is also a solution of Eq. (8.3). A complete or total solution of Eq. (8.3) would therefore require a linear combination of v 1 and v 2. Thus, the natural response of the parallel RLC circuit is v t = v 1 + v 2 = A 1 e s 1t +A 2 e s 2t (8.9) where the constants A 1 and A 2 are determined from the initial values v(0) and dv(0)/dt.

10 Solution of the Differential Equation Chapter 8 Natural and Step Responses of RLC Circuits 8.1 Introduction to the Natural Response of We can show that Eq.(8.9) also is a solution. v t = A 1 e s 1t +A 2 e s 2t dv dt = A 1s 1 e s 1t + A 2 s 2 e s 2t (8.9) d 2 v dt 2 = A 1s 1 2 e s 1t + A 2 s 2 2 e s 2t 10 d 2 v dt dv RC dt + v LC = 0 (8.3) A 1 e s 1t s RC s LC + A 2e s 2t s RC s LC = 0 (8.12) Each parenthetical term is zero because by definition s 1 and s 2 are roots of the characteristic equation. s RC s LC = 0 s RC s LC = 0 s RC s + 1 LC = 0 (8.6)

11 Definition of Frequency Terms Chapter 8 Natural and Step Responses of RLC Circuits 8.1 Introduction to the Natural Response of We have shown that v 1 is a solution, v 2 is a solution, and v 1 +v 2 is a solution. Therefore, the general solution of Eq.(8.3) has the form given in Eq.(8.13). 11 v t = A 1 e s 1t +A 2 e s 2t (8.13), (8.9) The behavior of v(t) depends on the values of s 1 and s 2. s 1 = 1 2RC + 1 2RC 2 1 LC = α + α2 ω o 2 (8.14) s 2 = 1 2RC 1 2RC 2 1 LC = α α2 ω o 2 (8.15) α = 1 2RC ω o = 1 LC (8.16) (8.17) Since the exponents s 1 t and s 2 t must be dimensionless, s 1 and s 2 must have the unit of per second.

12 Definition of Frequency Terms Chapter 8 Natural and Step Responses of RLC Circuits 8.1 Introduction to the Natural Response of Therefore, the units of α and ω o must also be per second or s 1. A unit of this type is called frequency. Let us define new terms: Neper frequency (exponential damping coefficient) α = 1 2RC (8.16) Resonant frequency 12 ω o = 1 LC (8.17) Complex frequency s 1, s 2

13 8.2 The Forms of the Natural Response of 13 Types of Responses(solutions) From Eqs. (8.14),(8.15), we can infer that there are three types of solutions depending on the relative sizes of α and ω o : s 1 = α + α 2 ω o 2 s 2 = α α 2 ω o 2 (8.14) (8.15) α = 1 2RC ω o = 1 LC Overdamped response : If α 2 > ω o 2 Critically damped response If, both roots s 1 and s 2 are negative and real. α 2 2 = ω o, s 1 = s 2 = α L > 4R 2 C L = 4R 2 C α > ω o α = ω o Underdamped response If α 2 < ω o 2, both roots s 1 and s 2 are complex and, in addition, are conjugates of each other. s 1 = α + α 2 ω o 2 = α + ω o 2 α 2 = α + jω d L < 4R 2 C α < ω o s 2 = α α 2 ω o 2 = α ω o 2 α 2 = α jω d

14 8.2 The Forms of the Natural Response of 14 What is the damping? Undamped System The following physical systems are some examples of simple harmonic oscillator. An undamped spring mass system undergoes simple harmonic motion The motion of an undamped pendulum approximates to simple harmonic motion if the angle of oscillation is small

15 Damped System Chapter 8 Natural and Step Responses of RLC Circuits 8.2 The Forms of the Natural Response of Damping is a dissipation of energy from a vibrating structure. The term dissipate is used to mean the transformation of mechanical energy into other form of energy and, therefore, a removal of mechanical energy from the vibrating system. 15 Overdamped Response The system returns (exponentially decays) to equilibrium without oscillating. Underdamped spring mass system Critically Damped Response The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors. x o m x = 0 x m Elastic potential energy A release from rest at a position x 0. Underdamped Response The mass will overshoot the zero point and oscillate about x=0. x o Damped harmonic oscillator

16 8.2 The Forms of the Natural Response of Electrical System The capacitor stores energy in its electric field E and the inductor stores energy in its magnetic field B (green). The charge flows back and forth between the plates of the capacitor, through the inductor. The energy oscillates back and forth between the capacitor and the inductor until (if not replenished from an external circuit) internal resistance makes the oscillations die out. V o 16 Energy oscillations in the LC Circuit and the mass-spring system V o x = 0 x o

17 Electrical System Chapter 8 Natural and Step Responses of RLC Circuits 8.2 The Forms of the Natural Response of We now consider a RLC circuit which contains a resistor, an inductor and a capacitor. Unlike the LC circuit energy will be dissipated through the resistor. 17 Charging system.

18 8.2 The Forms of the Natural Response of 18 The Overdamped Voltage Response If L > 4R 2 C, α > ω o. The roots of the characteristic equation are negative real numbers. s 1 = α + α 2 ω o 2 s 2 = α α 2 ω o 2 α = 1 2RC ω o = 1 LC V o The response is v t = A 1 e s 1t +A 2 e s 2t (8.18) The process for finding the overdamped response, v(t): (1) Find s 1 and s 2, using the values of R, L, and C. (2) Find v(0 + ) and dv(0 + ) dt. (3) Find the values of A 1 and A 2 using v(0 + ) and dv(0 + ) dt. (4) Substitute the values for s 1, s 2, A 1 and A 2 into Eq.(8.18).

19 8.2 The Forms of the Natural Response of 19 The Overdamped Voltage Response The process for finding the overdamped response, v(t): (2) Find v(0 + ) and dv(0 + ) dt. Two initial conditions v 0 + = V o V o i C = C dv dt dv(0 + ) dt = i C(0 + ) C (8.21) = i L i R 0 + C i C + i L + i R = 0 i C (0 + ) + i L (0 + ) + i R (0 + ) = 0 = I o C V o CR i C 0 + = i L 0 + i R 0 + = I o V 0 R (8.22)

20 8.2 The Forms of the Natural Response of 20 The Overdamped Voltage Response The process for finding the overdamped response, v(t): (3) Find the values of A 1 and A 2 using v(0 + ) and dv(0 + ) dt. v t = A 1 e s 1t +A 2 e s 2t (8.18) v 0 + = A 1 + A 2 = V o (8.23) dv(t) dt dv(0 + ) dt = s 1 A 1 e s 1t + s 2 A 2 e s 2t = s 1 A 1 + s 2 A 2 (8.24)

21 The Underdamped Voltage Response Chapter 8 Natural and Step Responses of RLC Circuits 8.2 The Forms of the Natural Response of 21 If L < 4R 2 C, α < ω o. The roots of the characteristic equation are complex, and the response is underdamped. s 1 = α + α 2 ω o 2 = α + ω o 2 α 2 = α + j ω o 2 α 2 = α + jω d (8.25) s 2 = α α 2 ω o 2 = α jω d (8.26) ω d = ω o 2 α 2 α = 1 2RC ω o = 1 LC The response is v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) (8.28)

22 The Underdamped Voltage Response Chapter 8 Natural and Step Responses of RLC Circuits 8.2 The Forms of the Natural Response of Finding The Underdamped Voltage Response v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) (8.28) 22 The natural response is v t = A 1 e α jω d t + A 2 e α+jω d t = e αt (A 1 e jωdt + A 2 e jωdt ) Using Euler s identities, e jθ = cos θ + j sin θ e jθ = cos θ j sin θ v t (8.29) = A 1 e s 1t +A 2 e s 2t s 1 = α + jω d s 1 = α jω d We get v(t) = e αt [A 1 (cos ω d t + j sin ω d t) + A 2 (cos ω d t j sin ω d t)] = e αt [(A 1 + A 2 ) cos ω d t + j(a 1 A 2 ) sin ω d t] v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) (8.28) B 1 = A 1 + A 2 B 2 = j(a 1 A 2 )

23 8.2 The Forms of the Natural Response of 23 The Underdamped Voltage Response v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) (8.28) With the presence of sine and cosine functions, it is clear that the natural response for this case is exponentially damped and oscillatory in nature. The response has a time constant of 1/α and a period of T = 2π/ω d. Figure depicts a typical underdamped response. [Figure assumes for each case that i(0) = 0]. e αt V o

24 The Underdamped Voltage Response Chapter 8 Natural and Step Responses of RLC Circuits 8.2 The Forms of the Natural Response of 24 Find the values of B 1 and B 2 using v(0 + ) and dv(0 + ) dt. v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) (8.28) v 0 + = B = V o (8.30) dv(t) dt = ( αe αt )(B 1 cos ω d t ) + e αt ( ω d B 1 sin ω d t ) +( αe αt )(B 2 sin ω d t ) + e αt (ω d B 2 cos ω d t ) dv(0 + ) dt = αb 1 + ω d B 2 (8.31)

25 8.2 The Forms of the Natural Response of 25 The Critically Damped Voltage Response If L = 4R 2 C, α = ω o. The two roots of the characteristic equation are real and equal. s 1 = α + s 2 = α α 2 ω o 2 = α α 2 ω o 2 = α s 1 = s 2 = α = 1 2RC (8.32) α = 1 2RC ω o = 1 LC The response is v t = (D 1 t + D 2 )e αt (8.34)

26 8.2 The Forms of the Natural Response of 26 The Critically Damped Voltage Response Finding the natural response (Eq.8.34) d 2 v dt dv RC dt + v LC = 0 (1) (8.3) d 2 v dv dt2 + 2α dt + α2 v = 0 (2) α = ω o = 1 2RC = 1 LC d dt dv dt + αv + α dv dt + αv = 0 (3) Eq.(3) is a first-order differential equation with solution f = A 1 e αt (6) If we let f = dv dt + αv then Eq.(3) becomes df dt + αf = 0 (4) (5) Eq.(4) then becomes dv dv dt + αv = D 1e αt or eαt dt + eαt αv = D 1 This can be written as d(e αt v) dt = D 1 (8) (7)

27 8.2 The Forms of the Natural Response of 27 The Critically Damped Voltage Response Finding the natural response (Eq.8.34) d(e αt v) dt = D 1 (8) Integrating both sides yields d(e αt v) = D 1 dt V o or e αt v = D 1 t + D 2 (9) v(t) = (D 1 t + D 2 )e αt (8.34) v(t) Figure is a sketch of v(t) = te αt, which reaches a maximum value of e 1 /α at t = 1/α, one time constant, and then decays all the way to zero. This is a typical critically damped response. The natural response of the critically damped circuit is a sum of two terms: a negative exponential and a negative exponential multiplied by a linear term.

28 The Critically Damped Voltage Response Chapter 8 Natural and Step Responses of RLC Circuits 8.2 The Forms of the Natural Response of Finding the values of D 1 and D 2 using v(0 + ) and dv(0 + ) dt. 28 v(t) = (D 1 t + D 2 )e αt (8.34) v 0 + = 0 + D 2 = V o (8.35) dv(t) dt = D e αt + D 1 t + D 2 ( αe αt ) V o dv(0 + ) dt = D 1 αd 2 (8.36)

29 8.3 The Step Response of 29 Responses of Parallel RLC Circuit Natural Response: Summary The circuit is being excited by the energy initially stored in the capacitor and inductor. The energy is represented by the initial capacitor voltage V o and initial inductor current I o. Thus, at t = 0, v C 0 = 1 C 0 Overdamped response: i t dt = V o i 0 = I o α 2 > ω o 2 L > 4R 2 C Both roots s 1 and s 2 are negative and real. s 1 = α + α 2 ω o 2 s 2 = α α 2 ω o 2 v t = A 1 e s 1t +A 2 e s 2t Underdamped response: α 2 2 < ω o L < 4R 2 C The roots of the characteristic equation are complex. s 1 = α + jω d s 2 = α jω d v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) Critically damped response: α 2 2 = ω o L = 4R 2 C The two roots of the characteristic equation are real and equal. s 1 = s 2 = α v(t) = (D 1 t + D 2 )e αt

30 8.3 The Step Response of 30 Responses of Parallel RLC Circuit Step Response The step response is obtained by the sudden application of a dc source. i(t) I 0 t We want to find i due to a sudden application of a dc current. To develop a general approach to finding the step response of a second order circuit, we focus on finding the current in the inductor branch, i L. This current does not approach zero as t increases. i L = I

31 8.3 The Step Response of 31 Step Response Applying KCL at the top node for t > 0, or i L + i R + i C = I i L + v R + C dv dt = I (8.37) From the definition v = L di L dt dv dt = L d2 i L dt 2 (8.38) (8.39) Substituting Eqs.(8.38) and (8.39) into Eq.(8.37) gives i L + L R di L dt + LC d2 i L dt 2 = I d 2 i L dt di L RC dt + i L LC = I LC (8.40) (8.41)

32 8.3 The Step Response of 32 Step Response d 2 i L dt di L RC dt + i L LC = I LC (8.41) The complete solution to Eq. (8.41) consists of the natural response i n and the forced response i f ; that is,, i L t = i n (t) + i f (t) The natural response is the same as what we had in Section 8.2. v t = A 1 e s 1t +A 2 e s 2t v(t) = e αt (B 1 cos ω d t + B 2 sin ω d t) v(t) = (D 1 t + D 2 )e αt i n (t) = A 1 e s 1t +A 2 e s 2t i n (t) = e αt (B 1 cos ω d t + B 2 sin ω d t) i n (t) = (D 1 t + D 2 )e αt The forced response is the steady state or final value of i. In the circuit in Fig.8.11, the final value of the current through the inductor is the same as the source current I. Thus, i f t = I

33 8.3 The Step Response of 33 Step Response The complete solutions: i L t = i n (t) + i f (t) Overdamped response i n (t) = A 1 e s1t +A 2 e s 2t i f t = I i L t = I + A 1 e s 1t +A 2 e s 2t (8.47) Underdamped response i n (t) = e αt (B 1 cos ω d t + B 2 sin ω d t) i f t = I i L t = I + e αt (B 1 cos ω d t + B 2 sin ω d t) (8.48) Critically damped response i n (t) = (D 1 t + D 2 )e αt i f t = I i L t = I + (D 1 t + D 2 )e αt (8.49)

To find the step response of an RC circuit

To find the step response of an RC circuit To find the step response of an RC circuit v( t) v( ) [ v( t) v( )] e tt The time constant = RC The final capacitor voltage v() The initial capacitor voltage v(t ) To find the step response of an RL circuit

More information

Response of Second-Order Systems

Response of Second-Order Systems Unit 3 Response of SecondOrder Systems In this unit, we consider the natural and step responses of simple series and parallel circuits containing inductors, capacitors and resistors. The equations which

More information

ENGR 2405 Chapter 8. Second Order Circuits

ENGR 2405 Chapter 8. Second Order Circuits ENGR 2405 Chapter 8 Second Order Circuits Overview The previous chapter introduced the concept of first order circuits. This chapter will expand on that with second order circuits: those that need a second

More information

Source-Free RC Circuit

Source-Free RC Circuit First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order

More information

Chapter 4 Transients. Chapter 4 Transients

Chapter 4 Transients. Chapter 4 Transients Chapter 4 Transients Chapter 4 Transients 1. Solve first-order RC or RL circuits. 2. Understand the concepts of transient response and steady-state response. 1 3. Relate the transient response of first-order

More information

First and Second Order Circuits. Claudio Talarico, Gonzaga University Spring 2015

First and Second Order Circuits. Claudio Talarico, Gonzaga University Spring 2015 First and Second Order Circuits Claudio Talarico, Gonzaga University Spring 2015 Capacitors and Inductors intuition: bucket of charge q = Cv i = C dv dt Resist change of voltage DC open circuit Store voltage

More information

The RLC circuits have a wide range of applications, including oscillators and frequency filters

The RLC circuits have a wide range of applications, including oscillators and frequency filters 9. The RL ircuit The RL circuits have a wide range of applications, including oscillators and frequency filters This chapter considers the responses of RL circuits The result is a second-order differential

More information

Introduction to AC Circuits (Capacitors and Inductors)

Introduction to AC Circuits (Capacitors and Inductors) Introduction to AC Circuits (Capacitors and Inductors) Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

More information

8. Introduction and Chapter Objectives

8. Introduction and Chapter Objectives Real Analog - Circuits Chapter 8: Second Order Circuits 8. Introduction and Chapter Objectives Second order systems are, by definition, systems whose input-output relationship is a second order differential

More information

MODULE I. Transient Response:

MODULE I. Transient Response: Transient Response: MODULE I The Transient Response (also known as the Natural Response) is the way the circuit responds to energies stored in storage elements, such as capacitors and inductors. If a capacitor

More information

Physics 116A Notes Fall 2004

Physics 116A Notes Fall 2004 Physics 116A Notes Fall 2004 David E. Pellett Draft v.0.9 Notes Copyright 2004 David E. Pellett unless stated otherwise. References: Text for course: Fundamentals of Electrical Engineering, second edition,

More information

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying

More information

Circuits with Capacitor and Inductor

Circuits with Capacitor and Inductor Circuits with Capacitor and Inductor We have discussed so far circuits only with resistors. While analyzing it, we came across with the set of algebraic equations. Hereafter we will analyze circuits with

More information

P441 Analytical Mechanics - I. RLC Circuits. c Alex R. Dzierba. In this note we discuss electrical oscillating circuits: undamped, damped and driven.

P441 Analytical Mechanics - I. RLC Circuits. c Alex R. Dzierba. In this note we discuss electrical oscillating circuits: undamped, damped and driven. Lecture 10 Monday - September 19, 005 Written or last updated: September 19, 005 P441 Analytical Mechanics - I RLC Circuits c Alex R. Dzierba Introduction In this note we discuss electrical oscillating

More information

EE 242 EXPERIMENT 8: CHARACTERISTIC OF PARALLEL RLC CIRCUIT BY USING PULSE EXCITATION 1

EE 242 EXPERIMENT 8: CHARACTERISTIC OF PARALLEL RLC CIRCUIT BY USING PULSE EXCITATION 1 EE 242 EXPERIMENT 8: CHARACTERISTIC OF PARALLEL RLC CIRCUIT BY USING PULSE EXCITATION 1 PURPOSE: To experimentally study the behavior of a parallel RLC circuit by using pulse excitation and to verify that

More information

Handout 10: Inductance. Self-Inductance and inductors

Handout 10: Inductance. Self-Inductance and inductors 1 Handout 10: Inductance Self-Inductance and inductors In Fig. 1, electric current is present in an isolate circuit, setting up magnetic field that causes a magnetic flux through the circuit itself. This

More information

Electric Circuit Theory

Electric Circuit Theory Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 11 Sinusoidal Steady-State Analysis Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 11.1

More information

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING RUTGERS UNIVERSITY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING RUTGERS UNIVERSITY DEPARTMENT OF EECTRICA AND COMPUTER ENGINEERING RUTGERS UNIVERSITY 330:222 Principles of Electrical Engineering II Spring 2002 Exam 1 February 19, 2002 SOUTION NAME OF STUDENT: Student ID Number (last

More information

EE292: Fundamentals of ECE

EE292: Fundamentals of ECE EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 14 121011 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Steady-State Analysis RC Circuits RL Circuits 3 DC Steady-State

More information

The Harmonic Oscillator

The Harmonic Oscillator The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can

More information

Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits

Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits Module 24: Undriven RLC Circuits 1 Module 24: Outline Undriven RLC Circuits Expt. 8: Part 2:Undriven RLC Circuits 2 Circuits that Oscillate (LRC) 3 Mass on a Spring: Simple Harmonic Motion (Demonstration)

More information

EE 40: Introduction to Microelectronic Circuits Spring 2008: Midterm 2

EE 40: Introduction to Microelectronic Circuits Spring 2008: Midterm 2 EE 4: Introduction to Microelectronic Circuits Spring 8: Midterm Venkat Anantharam 3/9/8 Total Time Allotted : min Total Points:. This is a closed book exam. However, you are allowed to bring two pages

More information

Chapter 10: Sinusoids and Phasors

Chapter 10: Sinusoids and Phasors Chapter 10: Sinusoids and Phasors 1. Motivation 2. Sinusoid Features 3. Phasors 4. Phasor Relationships for Circuit Elements 5. Impedance and Admittance 6. Kirchhoff s Laws in the Frequency Domain 7. Impedance

More information

General Response of Second Order System

General Response of Second Order System General Response of Second Order System Slide 1 Learning Objectives Learn to analyze a general second order system and to obtain the general solution Identify the over-damped, under-damped, and critically

More information

Slide 1 / 26. Inductance by Bryan Pflueger

Slide 1 / 26. Inductance by Bryan Pflueger Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

More information

Handout 11: AC circuit. AC generator

Handout 11: AC circuit. AC generator Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For

More information

Electromagnetic Oscillations and Alternating Current. 1. Electromagnetic oscillations and LC circuit 2. Alternating Current 3.

Electromagnetic Oscillations and Alternating Current. 1. Electromagnetic oscillations and LC circuit 2. Alternating Current 3. Electromagnetic Oscillations and Alternating Current 1. Electromagnetic oscillations and LC circuit 2. Alternating Current 3. RLC circuit in AC 1 RL and RC circuits RL RC Charging Discharging I = emf R

More information

Sinusoidal Steady-State Analysis

Sinusoidal Steady-State Analysis Chapter 4 Sinusoidal Steady-State Analysis In this unit, we consider circuits in which the sources are sinusoidal in nature. The review section of this unit covers most of section 9.1 9.9 of the text.

More information

Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations Applications of Second-Order Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition

More information

C R. Consider from point of view of energy! Consider the RC and LC series circuits shown:

C R. Consider from point of view of energy! Consider the RC and LC series circuits shown: ircuits onsider the R and series circuits shown: ++++ ---- R ++++ ---- Suppose that the circuits are formed at t with the capacitor charged to value. There is a qualitative difference in the time development

More information

RLC Circuits. 1 Introduction. 1.1 Undriven Systems. 1.2 Driven Systems

RLC Circuits. 1 Introduction. 1.1 Undriven Systems. 1.2 Driven Systems RLC Circuits Equipment: Capstone, 850 interface, RLC circuit board, 4 leads (91 cm), 3 voltage sensors, Fluke mulitmeter, and BNC connector on one end and banana plugs on the other Reading: Review AC circuits

More information

8 sin 3 V. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0.

8 sin 3 V. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0. Spring 2015, Exam #5, Problem #1 4t Answer: e tut 8 sin 3 V 1 For the circuit

More information

Inductance, RL Circuits, LC Circuits, RLC Circuits

Inductance, RL Circuits, LC Circuits, RLC Circuits Inductance, R Circuits, C Circuits, RC Circuits Inductance What happens when we close the switch? The current flows What does the current look like as a function of time? Does it look like this? I t Inductance

More information

Problem Set 5 Solutions

Problem Set 5 Solutions University of California, Berkeley Spring 01 EE /0 Prof. A. Niknejad Problem Set 5 Solutions Please note that these are merely suggested solutions. Many of these problems can be approached in different

More information

Physics 4 Spring 1989 Lab 5 - AC Circuits

Physics 4 Spring 1989 Lab 5 - AC Circuits Physics 4 Spring 1989 Lab 5 - AC Circuits Theory Consider the series inductor-resistor-capacitor circuit shown in figure 1. When an alternating voltage is applied to this circuit, the current and voltage

More information

4.2 Homogeneous Linear Equations

4.2 Homogeneous Linear Equations 4.2 Homogeneous Linear Equations Homogeneous Linear Equations with Constant Coefficients Consider the first-order linear differential equation with constant coefficients a 0 and b. If f(t) = 0 then this

More information

Electric Circuits Fall 2015 Solution #5

Electric Circuits Fall 2015 Solution #5 RULES: Please try to work on your own. Discussion is permissible, but identical submissions are unacceptable! Please show all intermeate steps: a correct solution without an explanation will get zero cret.

More information

Electric Circuits. Overview. Hani Mehrpouyan,

Electric Circuits. Overview. Hani Mehrpouyan, Electric Circuits Hani Mehrpouyan, Department of Electrical and Computer Engineering, Lecture 15 (First Order Circuits) Nov 16 th, 2015 Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 2015 1 1 Overview

More information

Physics for Scientists & Engineers 2

Physics for Scientists & Engineers 2 Electromagnetic Oscillations Physics for Scientists & Engineers Spring Semester 005 Lecture 8! We have been working with circuits that have a constant current a current that increases to a constant current

More information

QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)

QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE E-BOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF

More information

ECE2262 Electric Circuit

ECE2262 Electric Circuit ECE2262 Electric Circuit Chapter 7: FIRST AND SECOND-ORDER RL AND RC CIRCUITS Response to First-Order RL and RC Circuits Response to Second-Order RL and RC Circuits 1 2 7.1. Introduction 3 4 In dc steady

More information

Linear Circuits. Concept Map 9/10/ Resistive Background Circuits. 5 Power. 3 4 Reactive Circuits. Frequency Analysis

Linear Circuits. Concept Map 9/10/ Resistive Background Circuits. 5 Power. 3 4 Reactive Circuits. Frequency Analysis Linear Circuits Dr. Bonnie Ferri Professor School of Electrical and Computer Engineering An introduction to linear electric components and a study of circuits containing such devices. School of Electrical

More information

Basic RL and RC Circuits R-L TRANSIENTS: STORAGE CYCLE. Engineering Collage Electrical Engineering Dep. Dr. Ibrahim Aljubouri

Basic RL and RC Circuits R-L TRANSIENTS: STORAGE CYCLE. Engineering Collage Electrical Engineering Dep. Dr. Ibrahim Aljubouri st Class Basic RL and RC Circuits The RL circuit with D.C (steady state) The inductor is short time at Calculate the inductor current for circuits shown below. I L E R A I L E R R 3 R R 3 I L I L R 3 R

More information

Some of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e

Some of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)

More information

Lecture 39. PHYC 161 Fall 2016

Lecture 39. PHYC 161 Fall 2016 Lecture 39 PHYC 161 Fall 016 Announcements DO THE ONLINE COURSE EVALUATIONS - response so far is < 8 % Magnetic field energy A resistor is a device in which energy is irrecoverably dissipated. By contrast,

More information

Forced Oscillation and Resonance

Forced Oscillation and Resonance Chapter Forced Oscillation and Resonance The forced oscillation problem will be crucial to our understanding of wave phenomena Complex exponentials are even more useful for the discussion of damping and

More information

Figure Circuit for Question 1. Figure Circuit for Question 2

Figure Circuit for Question 1. Figure Circuit for Question 2 Exercises 10.7 Exercises Multiple Choice 1. For the circuit of Figure 10.44 the time constant is A. 0.5 ms 71.43 µs 2, 000 s D. 0.2 ms 4 Ω 2 Ω 12 Ω 1 mh 12u 0 () t V Figure 10.44. Circuit for Question

More information

RC, RL, and LCR Circuits

RC, RL, and LCR Circuits RC, RL, and LCR Circuits EK307 Lab Note: This is a two week lab. Most students complete part A in week one and part B in week two. Introduction: Inductors and capacitors are energy storage devices. They

More information

arxiv: v1 [physics.class-ph] 15 Oct 2012

arxiv: v1 [physics.class-ph] 15 Oct 2012 Two-capacitor problem revisited: A mechanical harmonic oscillator model approach Keeyung Lee arxiv:1210.4155v1 [physics.class-ph] 15 Oct 2012 Department of Physics, Inha University, Incheon, 402-751, Korea

More information

CS 436 HCI Technology Basic Electricity/Electronics Review

CS 436 HCI Technology Basic Electricity/Electronics Review CS 436 HCI Technology Basic Electricity/Electronics Review *Copyright 1997-2008, Perry R. Cook, Princeton University August 27, 2008 1 Basic Quantities and Units 1.1 Charge Number of electrons or units

More information

Sinusoids and Phasors

Sinusoids and Phasors CHAPTER 9 Sinusoids and Phasors We now begins the analysis of circuits in which the voltage or current sources are time-varying. In this chapter, we are particularly interested in sinusoidally time-varying

More information

Circuits Advanced Topics by Dr. Colton (Fall 2016)

Circuits Advanced Topics by Dr. Colton (Fall 2016) ircuits Advanced Topics by Dr. olton (Fall 06). Time dependence of general and L problems General and L problems can always be cast into first order ODEs. You can solve these via the particular solution

More information

Analogue Meter s Concept

Analogue Meter s Concept M2-3 Lecture 4 (1/2015) August 27, 2015 1 Analogue Meter s Concept Han Oersted, in 1820, noted his finding without any explanation. B N S Compass Lord Kelvin made more sensitivity to a current. Magnetic

More information

Inductance, RL and RLC Circuits

Inductance, RL and RLC Circuits Inductance, RL and RLC Circuits Inductance Temporarily storage of energy by the magnetic field When the switch is closed, the current does not immediately reach its maximum value. Faraday s law of electromagnetic

More information

Transient Response of a Second-Order System

Transient Response of a Second-Order System Transient Response of a Second-Order System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a well-behaved closed-loop

More information

Chapter 32. Inductance

Chapter 32. Inductance Chapter 32 Inductance Joseph Henry 1797 1878 American physicist First director of the Smithsonian Improved design of electromagnet Constructed one of the first motors Discovered self-inductance Unit of

More information

ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT

ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT Chapter 31: ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT 1 A charged capacitor and an inductor are connected in series At time t = 0 the current is zero, but the capacitor is charged If T is the

More information

Inductance. Slide 2 / 26. Slide 1 / 26. Slide 4 / 26. Slide 3 / 26. Slide 6 / 26. Slide 5 / 26. Mutual Inductance. Mutual Inductance.

Inductance. Slide 2 / 26. Slide 1 / 26. Slide 4 / 26. Slide 3 / 26. Slide 6 / 26. Slide 5 / 26. Mutual Inductance. Mutual Inductance. Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

More information

APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS

APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration

More information

Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current.

Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. Inductance Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. Basis of the electrical circuit element called an

More information

Coupled Electrical Oscillators Physics Advanced Physics Lab - Summer 2018 Don Heiman, Northeastern University, 5/24/2018

Coupled Electrical Oscillators Physics Advanced Physics Lab - Summer 2018 Don Heiman, Northeastern University, 5/24/2018 Coupled Electrical Oscillators Physics 3600 - Advanced Physics Lab - Summer 08 Don Heiman, Northeastern University, 5/4/08 I. INTRODUCTION The objectives of this experiment are: () explore the properties

More information

Fourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series

Fourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series Fourier series Electrical Circuits Lecture - Fourier Series Filtr RLC defibrillator MOTIVATION WHAT WE CAN'T EXPLAIN YET Source voltage rectangular waveform Resistor voltage sinusoidal waveform - Fourier

More information

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory

More information

Chapter 30 INDUCTANCE. Copyright 2012 Pearson Education Inc.

Chapter 30 INDUCTANCE. Copyright 2012 Pearson Education Inc. Chapter 30 INDUCTANCE Goals for Chapter 30 To learn how current in one coil can induce an emf in another unconnected coil To relate the induced emf to the rate of change of the current To calculate the

More information

Solutions to these tests are available online in some places (but not all explanations are good)...

Solutions to these tests are available online in some places (but not all explanations are good)... The Physics GRE Sample test put out by ETS https://www.ets.org/s/gre/pdf/practice_book_physics.pdf OSU physics website has lots of tips, and 4 additional tests http://www.physics.ohiostate.edu/undergrad/ugs_gre.php

More information

Circuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18

Circuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18 Circuit Analysis-III Sinusoids Example #1 ü Find the amplitude, phase, period and frequency of the sinusoid: v (t ) =12cos(50t +10 ) Signal Conversion ü From sine to cosine and vice versa. ü sin (A ± B)

More information

Supplemental Notes on Complex Numbers, Complex Impedance, RLC Circuits, and Resonance

Supplemental Notes on Complex Numbers, Complex Impedance, RLC Circuits, and Resonance Supplemental Notes on Complex Numbers, Complex Impedance, RLC Circuits, and Resonance Complex numbers Complex numbers are expressions of the form z = a + ib, where both a and b are real numbers, and i

More information

EE292: Fundamentals of ECE

EE292: Fundamentals of ECE EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 20 121101 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Chapters 1-3 Circuit Analysis Techniques Chapter 10 Diodes Ideal Model

More information

Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.

Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc. Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To

More information

ECE Circuit Theory. Final Examination. December 5, 2008

ECE Circuit Theory. Final Examination. December 5, 2008 ECE 212 H1F Pg 1 of 12 ECE 212 - Circuit Theory Final Examination December 5, 2008 1. Policy: closed book, calculators allowed. Show all work. 2. Work in the provided space. 3. The exam has 3 problems

More information

Vibrations and Waves MP205, Assignment 4 Solutions

Vibrations and Waves MP205, Assignment 4 Solutions Vibrations and Waves MP205, Assignment Solutions 1. Verify that x = Ae αt cos ωt is a possible solution of the equation and find α and ω in terms of γ and ω 0. [20] dt 2 + γ dx dt + ω2 0x = 0, Given x

More information

EE292: Fundamentals of ECE

EE292: Fundamentals of ECE EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 18 121025 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review RMS Values Complex Numbers Phasors Complex Impedance Circuit Analysis

More information

2.4 Models of Oscillation

2.4 Models of Oscillation 2.4 Models of Oscillation In this section we give three examples of oscillating physical systems that can be modeled by the harmonic oscillator equation. Such models are ubiquitous in physics, but are

More information

LECTURE 8 RC AND RL FIRST-ORDER CIRCUITS (PART 1)

LECTURE 8 RC AND RL FIRST-ORDER CIRCUITS (PART 1) CIRCUITS by Ulaby & Maharbiz LECTURE 8 RC AND RL FIRST-ORDER CIRCUITS (PART 1) 07/18/2013 ECE225 CIRCUIT ANALYSIS All rights reserved. Do not copy or distribute. 2013 National Technology and Science Press

More information

Alternating Current Circuits. Home Work Solutions

Alternating Current Circuits. Home Work Solutions Chapter 21 Alternating Current Circuits. Home Work s 21.1 Problem 21.11 What is the time constant of the circuit in Figure (21.19). 10 Ω 10 Ω 5.0 Ω 2.0µF 2.0µF 2.0µF 3.0µF Figure 21.19: Given: The circuit

More information

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory

More information

PHYSICS 110A : CLASSICAL MECHANICS HW 2 SOLUTIONS. Here is a sketch of the potential with A = 1, R = 1, and S = 1. From the plot we can see

PHYSICS 110A : CLASSICAL MECHANICS HW 2 SOLUTIONS. Here is a sketch of the potential with A = 1, R = 1, and S = 1. From the plot we can see PHYSICS 11A : CLASSICAL MECHANICS HW SOLUTIONS (1) Taylor 5. Here is a sketch of the potential with A = 1, R = 1, and S = 1. From the plot we can see 1.5 1 U(r).5.5 1 4 6 8 1 r Figure 1: Plot for problem

More information

Name (print): Lab (circle): W8 Th8 Th11 Th2 F8. θ (radians) θ (degrees) cos θ sin θ π/ /2 1/2 π/4 45 2/2 2/2 π/3 60 1/2 3/2 π/

Name (print): Lab (circle): W8 Th8 Th11 Th2 F8. θ (radians) θ (degrees) cos θ sin θ π/ /2 1/2 π/4 45 2/2 2/2 π/3 60 1/2 3/2 π/ Name (print): Lab (circle): W8 Th8 Th11 Th2 F8 Trigonometric Identities ( cos(θ) = cos(θ) sin(θ) = sin(θ) sin(θ) = cos θ π ) 2 Cosines and Sines of common angles Euler s Formula θ (radians) θ (degrees)

More information

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring

More information

Lecture 4: R-L-C Circuits and Resonant Circuits

Lecture 4: R-L-C Circuits and Resonant Circuits Lecture 4: R-L-C Circuits and Resonant Circuits RLC series circuit: What's V R? Simplest way to solve for V is to use voltage divider equation in complex notation: V X L X C V R = in R R + X C + X L L

More information

EE221 Circuits II. Chapter 14 Frequency Response

EE221 Circuits II. Chapter 14 Frequency Response EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active

More information

R-L-C Circuits and Resonant Circuits

R-L-C Circuits and Resonant Circuits P517/617 Lec4, P1 R-L-C Circuits and Resonant Circuits Consider the following RLC series circuit What's R? Simplest way to solve for is to use voltage divider equation in complex notation. X L X C in 0

More information

Electric Circuits I. Inductors. Dr. Firas Obeidat

Electric Circuits I. Inductors. Dr. Firas Obeidat Electric Circuits I Inductors Dr. Firas Obeidat 1 Inductors An inductor is a passive element designed to store energy in its magnetic field. They are used in power supplies, transformers, radios, TVs,

More information

Active Figure 32.3 (SLIDESHOW MODE ONLY)

Active Figure 32.3 (SLIDESHOW MODE ONLY) RL Circuit, Analysis An RL circuit contains an inductor and a resistor When the switch is closed (at time t = 0), the current begins to increase At the same time, a back emf is induced in the inductor

More information

4.9 Free Mechanical Vibrations

4.9 Free Mechanical Vibrations 4.9 Free Mechanical Vibrations Spring-Mass Oscillator When the spring is not stretched and the mass m is at rest, the system is at equilibrium. Forces Acting in the System When the mass m is displaced

More information

Phasors: Impedance and Circuit Anlysis. Phasors

Phasors: Impedance and Circuit Anlysis. Phasors Phasors: Impedance and Circuit Anlysis Lecture 6, 0/07/05 OUTLINE Phasor ReCap Capacitor/Inductor Example Arithmetic with Complex Numbers Complex Impedance Circuit Analysis with Complex Impedance Phasor

More information

Name: Lab: M8 M2 W8 Th8 Th11 Th2 F8. cos( θ) = cos(θ) sin( θ) = sin(θ) sin(θ) = cos. θ (radians) θ (degrees) cos θ sin θ π/6 30

Name: Lab: M8 M2 W8 Th8 Th11 Th2 F8. cos( θ) = cos(θ) sin( θ) = sin(θ) sin(θ) = cos. θ (radians) θ (degrees) cos θ sin θ π/6 30 Name: Lab: M8 M2 W8 Th8 Th11 Th2 F8 Trigonometric Identities cos(θ) = cos(θ) sin(θ) = sin(θ) sin(θ) = cos Cosines and Sines of common angles Euler s Formula θ (radians) θ (degrees) cos θ sin θ 0 0 1 0

More information

EECS2200 Electric Circuits. RLC Circuit Natural and Step Responses

EECS2200 Electric Circuits. RLC Circuit Natural and Step Responses 5--4 EECS Electric Circuit Chapter 6 R Circuit Natural and Step Repone Objective Determine the repone form of the circuit Natural repone parallel R circuit Natural repone erie R circuit Step repone of

More information

Mixing Problems. Solution of concentration c 1 grams/liter flows in at a rate of r 1 liters/minute. Figure 1.7.1: A mixing problem.

Mixing Problems. Solution of concentration c 1 grams/liter flows in at a rate of r 1 liters/minute. Figure 1.7.1: A mixing problem. page 57 1.7 Modeling Problems Using First-Order Linear Differential Equations 57 For Problems 33 38, use a differential equation solver to determine the solution to each of the initial-value problems and

More information

Chapter 30 Inductance

Chapter 30 Inductance Chapter 30 Inductance In this chapter we investigate the properties of an inductor in a circuit. There are two kinds of inductance mutual inductance and self-inductance. An inductor is formed by taken

More information

dx n a 1(x) dy

dx n a 1(x) dy HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)

More information

Chapter 15. Oscillatory Motion

Chapter 15. Oscillatory Motion Chapter 15 Oscillatory Motion Part 2 Oscillations and Mechanical Waves Periodic motion is the repeating motion of an object in which it continues to return to a given position after a fixed time interval.

More information

Physics 4B Chapter 31: Electromagnetic Oscillations and Alternating Current

Physics 4B Chapter 31: Electromagnetic Oscillations and Alternating Current Physics 4B Chapter 31: Electromagnetic Oscillations and Alternating Current People of mediocre ability sometimes achieve outstanding success because they don't know when to quit. Most men succeed because

More information

WPI Physics Dept. Intermediate Lab 2651 Free and Damped Electrical Oscillations

WPI Physics Dept. Intermediate Lab 2651 Free and Damped Electrical Oscillations WPI Physics Dept. Intermediate Lab 2651 Free and Damped Electrical Oscillations Qi Wen March 11, 2017 Index: 1. Background Material, p. 2 2. Pre-lab Exercises, p. 4 3. Report Checklist, p. 6 4. Appendix,

More information

Read each question and its parts carefully before starting. Show all your work. Give units with your answers (where appropriate).

Read each question and its parts carefully before starting. Show all your work. Give units with your answers (where appropriate). ECSE 10, Fall 013 NAME: Quiz #1 September 17, 013 ID: Read each question and its parts carefully before starting. Show all your work. Give units with your answers (where appropriate). 1. Consider the circuit

More information

Transient Analysis of Electrical Circuits Using Runge- Kutta Method and its Application

Transient Analysis of Electrical Circuits Using Runge- Kutta Method and its Application International Journal of Scientific and Research Publications, Volume 3, Issue 11, November 2013 1 Transient Analysis of Electrical Circuits Using Runge- Kutta Method and its Application Anuj Suhag School

More information

MATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam

MATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam MATH 51 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam A collection of previous exams could be found at the coordinator s web: http://www.math.psu.edu/tseng/class/m51samples.html

More information

Math 211. Substitute Lecture. November 20, 2000

Math 211. Substitute Lecture. November 20, 2000 1 Math 211 Substitute Lecture November 20, 2000 2 Solutions to y + py + qy =0. Look for exponential solutions y(t) =e λt. Characteristic equation: λ 2 + pλ + q =0. Characteristic polynomial: λ 2 + pλ +

More information

Review of 1 st Order Circuit Analysis

Review of 1 st Order Circuit Analysis ECEN 60 Circuits/Electronics Spring 007-7-07 P. Mathys Review of st Order Circuit Analysis First Order Differential Equation Consider the following circuit with input voltage v S (t) and output voltage

More information

Linear Second-Order Differential Equations LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS

Linear Second-Order Differential Equations LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS 11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS A Spring with Friction: Damped Oscillations The differential equation, which we used to describe the motion of a spring, disregards friction. But there

More information