SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course

Size: px
Start display at page:

Download "SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course"

Transcription

1 SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE 26 APPLICATIONS TO ELECTRICAL CIRCUITS Module Topics 1. Complex numbers and alternating currents 2. Complex impedance 3. Differential equations for RLC circuits 4. Forced oscillations and resonance 5. Complex solutions of differential equations 6. Phasors A: Work Scheme based on JAMES (FIFTH EDITION) 1. The first topic uses results from module 22 on complex numbers. Before starting on this topic you should make sure that you understand the relevant topics. If you need to revise this material look at the module again or study section 3.3 of James. 2. Turn to p.225 and study section 3.6 on alternating currents in electrical circuits. Work through Example The key results of this section are as follows: When an alternating current with frequency ω/2π flows in a circuit it is given by Similarly we have i = Isinωt = Im ( Ie jωt). Icosωt = Re ( Ie jωt) The corresponding voltage depends on the resistance, capacitance and inductance of the circuit. However these devices not only determine the magnitude of the voltage but also change the phase of the voltage relative to the current. We may therefore calculate the voltage by multiplying Ie jωt by a complex number Z and then looking at the imaginary part. Note using a complex number deals with the change of phase since Im(e jφ e jωt ) = Im(e j(ωt+φ) ) = sin(ωt+φ), and Re(e jφ e jωt ) = Re(e j(ωt+φ) ) = cos(ωt+φ). The complex number Z is called the complex impedance of the device. The formula for Z for various devices is given as follows: { R for a resistor Z = j ωc for a capacitor jωl for an inductor For a simple RLC circuit (with these devices in series) R L C Fig. 1. 1

2 we simply take the sum of these terms so that Using this result the actual voltage is given by Z = R+jωL j ωc v = Im ( IZe jωt) = I Z sin(ωt+φ) where Z is the impedance of the circuit and the phase φ is given by the argument of Z. Note: it is often convenient to introduce the reactance S given by So that S = ωl 1 ωc Z = R+jS One example of the use of complex impedance is given in Example Here is another example: Example A: Calculate the complex impedance of the element of the circuit shown below with a resistor of R = 20Ω and a capacitor of C = 212 µf, when an alternating current of frequency 50 Hz flows. Use this to find the impedance and the phase. Fig. 2. The complex impedance is the sum of the individual impedance s. Thus Z = R j ωc Here R = 20Ω, ω = 2π 50rad s 1 and C = 212µF, so that 1/Cω = 15 Z = 20 15j Hence the impedance is Z = (15) 2 +(20) 2 = 25Ω and the phase is φ = tan 1 ( 15/20) = 0.64 rad. ***Do Exercises on p.227*** 3. Turn to p.794 and study section on Simple Electrical Circuits. We again summarise the key results: The relationship between voltage V, current i and resistance R for a pure resistor is given by V = ir Similarly for a pure capacitor of capacitance C with charge q we have Finally for a pure inductor of inductance L we have V = q C V = L di dt 2

3 If we now consider the circuit shown below A B Fig. 3. then when the switch is in contact with B then by Kirchhoff s voltage law the total potential difference around the circuit must be zero so we have L di dt +Ri+ q C = 0 (1) The principle of conservation of charge tells us that the current flowing is equal to the rate of change of charge so that i = dq (2) dt Substituting for i (and its time derivative) in (1) using (2) gives L d2 q dt 2 +Rdq dt + 1 C q = 0 (3) This is a differential equation for the charge q. However it is more convenient to work with the current i and differentiating (3) with respect to time gives L d2 i dt 2 +Rdi dt + 1 C i = 0 (4) Alternatively we can use V = q/c in equation (3) to obtain an equation for the voltage LC d2 V dt 2 +RCdV dt +V = 0 (5) We have therefore shown that a simple analysis of a RLC circuit gives rise to a second order (homogeneous) linear differential equation with constant coefficients. Example B: A series circuit (as in Fig. 3) contains an inductor for which L = 1 H, a resistor for which R = 1 kω, and a capacitor for which C = 6.25 µf. The capacitor holds a charge of C, at time t = 0 a switch is moved from A to B and the capacitor discharges through the circuit. Find q and i as functions of t. The differential equation to be solved is d 2 q dt dq dt q = 0 This is a differential equation with constant coefficients so we look at the auxiliary equation m 2 +1,000m+160,000 = 0 3

4 which has roots m 1 = 200 and m 2 = 800. Hence q = Ae 200t +Be 800t ( ) The initial charge is q 0 = so that A+B = (i) To find the other initial condition we differentiate (*) to obtain an equation for the current The initial current is zero so that we have i = 200Ae 200t 800Be 800t 200A 800B = 0 (ii) Solving (i) and (ii) for A and B we get A = , B = Hence and q = e 200t e 800t i = 0.4e 200t +0.4e 800t ***Do Exercise A: A series circuit consists of an inductor for which L = 0.02 H and a capacitor for which C = F. The capacitor holds a charge of C and at time t = 0 a switch is closed allowing the capacitor to discharge through the circuit. Find the charge and the current in the circuit at time t. ***Do Exercise B: For the problem given above also find the voltage drop across the inductor and across the capacitor and show that the sum of these two voltages is zero in accordance with Kirchhoff s law. 4. Linear differential equations with constant coefficients are covered in module 13 (Differential Equations III) in order to be able to study oscillations in RLC circuits you need to make sure that you understand this material. If you need to revise this topic look again at module 13 or else turn to p.852 and read section 10.9 of J. Note that you will also need the material from section which deals with the inhomogeneous equations with constant coefficients. ***Do Exercise 55 on p.857 and Exercise 63(c),(f) on p.864*** 5. Read section on p.850 which deals with oscillations in electrical circuits. The key result is the following. Consider the RLC circuit shown below V i R L C V o Fig. 4. 4

5 Suppose that a voltage V i (t) is applied across the input terminals then the voltage V o (t) across the output terminals is given by LC d2 V o dt 2 +RCdV o dt +V o = V i (t) (6) which is second order inhomogeneous differential equation for constant coefficients. In many cases the input signal is of the form V i = V cosωt (where V is a constant) so that the equation reduces to LC d2 V o dt 2 +RCdV o dt +V o = V cosωt (7) To solve this we need to find a complementary function and a particular integral. The complementary function is just a solution of the homogeneous equation (5) and in section of J it is shown that in this case (since R and C are positive) this is always given by a decaying function of time. For this reason the complementary function is called the transient solution. In order to find the particular integral we look for a solution of the form P cosωt+qsinωt where P and Q are constants which depend upon ω. This can be written in the alternative form AV C cos(ωt+δ) where A(ω) is a constant that depends upon ω and δ gives the phase relative to the input. This is a sinusoidal signal which oscillates with amplitude AV/C. Note that unlike the transient solution this solution does not decay so that this solution gives the long term behaviour of the solution. For this reason it is called the steady-state response. A direct calculation shows that A = 1 ω(r 2 +S 2 ) 1/2 = 1 ω Z where R is the resistance, S is the reactance and Z is the impedance, and the phase is given by δ = tan 1 ( R S ) Therefore if a sinusoidal voltage is applied to the input terminals the voltage produced is also sinusoidal but with an amplitude (and phase) which depends on the frequency of the input. Such a device is called a filter. Note that the corresponding current is obtained by differentiating the voltage and multiplying by C so that the modulus of the output current is the modulus of the input voltage divided by the impedance. We now give an example of how equation (6) may be used to calculate the voltage in an alternating RLC circuit. Example C: An RLC-circuit as shown in Fig. 4 consists of a resistor with R = 11 Ω, an inductor with L = 0.1 H, and a capacitor with C = 10 2 F. An input voltage V i (t) = 1000cos400t is applied. Find the resultant steady state voltage. Find the amplitude of this voltage and the phase lag compared to the the voltage that produced it. The differential equation for V o (t) is given by 10 3d2 V o dt dV o dt +V o = 1000cos400t To find the steady state solution we look for a particular solution of the form V o = P cos400t+qsin400t 5

6 Differentiating this and substituting into the equation we get (44Q 159P)cos400t (44P +159Q)sin400t = 1000cos400t Hence which has solution 159P +44Q = P +159Q = 0 P = 5.84, Q = 1.62 so that V o = 5.84cos400t+1.62sin400t. It is more useful to write this in terms of modulus and phase as V o = 6.06cos(400t+0.27) So that the amplitude is 6.06 and the voltage is 0.27 in advance of the voltage that produced it (since δ is negative). ***Do Exercise C: An RLC-circuit as shown in Fig. 4 consists of a resistor with R = 100 Ω, an inductor with L = 0.1 H, and a capacitor with C = 10 5 F. An input voltage V i (t) = 100cos500t is applied. Find the resultant steady state voltage and current and the amplitude of this current. 6. Complex Solutions The following sections deal with complex solutions and phasors which are not described in J. As we have seen the equation for the output current is where V i (t) is the input voltage. Two cases of interest are L d2 q dt 2 +Rdq dt + 1 C q = V i(t) (8) and V i (t) = V cos(ωt) V i (t) = V sin(ωt) (a) (b) If we solve equation (8) with a complex source given by V i (t) = V(cos(ωt)+jsin(ωt)) = Ve jωt (c) then we obtain a complex solution q(t). However the real part of this is gives the solution corresponding to source (a), while the imaginary part of this solution gives the solution corresponding to source (b). It turns out that it is easier to solve (8) for the complex source (c) and then take the real part rather than solve for the real source (a). For the case of the complex source we want to solve L d2 q dt 2 +Rdq dt + 1 C q = Vejωt (9) The steady-state solution is given by the particular solution and we find this by looking for a solution of the form q(t) = KVe jωt where K is a complex constant to be determined. Differentiating this gives dq dt = jωkvejωt, and d 2 q dt 2 = ω2 KVe jωt 6

7 Substituting into (9) gives ( ω 2 L+jRω + 1 C )KVejωt = Ve jωt and hence using S = ωl 1, we must have ωc So that jω(r+js)k = 1 K = 1 jωz where Z is the complex impedance. Thus the corresponding complex current is given by i c (t) = dq dt = V Z ejωt To find the actual (real) current we write the complex impedance in polar form as Z = Z e jθ where ) Z = R 2 +S 2, and θ = tan 1 ( S R Then the complex current produced is So that the real current produced is i c (t) = V Z ej(ωt θ) i(t) = Re(i c (t)) = V Z cos(ωt θ) To summarise an input voltage of Re(Ve jωt ) produces an output current of Re( V Z ejωt ). 7. Phasors If we are given an alternating current i(t) = I o cos(ωt + φ) in a circuit which is oscillating with angular frequency ω, then all we need to know to find the current is the quantity I o e jφ and the angular frequency ω. This information is provided by the current phasor since the actual current is given by Ĩ(jω) = I o e jφ i(t) = Re(Ĩejωt ) = Re(I o e j(ωt+φ) ) In the same way the voltage v(t) = V 0 cos(ωt+θ) is determined by the voltage phasor Ṽ(jω) = V 0 e jθ In the example of the RLC-circuit considered in section 6 we see that Ĩ(jω) = Ṽ(jω) Z(jω) 7

8 or Ṽ(jω) = Z(jω)Ĩ(jω) where Ṽ(jω) is the phasor for the input voltage, Ĩ(jω) is the phasor for the output current and Z(jω) is the complex impedance of the circuit. Note by taking the modulus of this expression we immediately see that the amplitude of the input voltage is the impedance times the amplitude of the output current. By looking at the argument we also see that the phase of the output current is argz behind that of the input voltage. Example D: Use the phasor method to calculate the amplitude and phase of the output current for the circuit considered in Example C. Use this to compute the modulus of the output voltage. In this example R = 11, ω = 400 and C = Hence S = ωl 1 ωc = 159/4 and hence the complex impedance is Z = 1 4 (44+159j). Thus Z = R 2 +S 2 = , and argz = tan 1 (159/44) = 1.3rad Hence the amplitude of the current is 1000/ = and the phase is 1.3 rad. Hence the output current is i(t) = 24.25cos(400t 1.3) To find the voltage we integrate and divide by C. Hence the modulus of the output voltage is given by = 6.06 in agreement with the previous answer. Note the phase difference of the voltage is π/2 1.3 = 0.27 again in agreement with the previous calculation (the extra π/2 coming from the fact that a cos becomes a sin on integration). B: Work Scheme based on STROUD (SIXTH EDITION) This module is not covered by S., so work through A: Work scheme based on J, presented above. 8

9 Specimen Test An alternating current i = 2 sin 100t flows through a circuit consisting of a resistor with resistance R = 50 Ω, and an inductor with inductance L = 0.5 H, as shown below R L (i) Calculate the complex impedance Z of the circuit. (ii) Calculate the amplitude V of the voltage. (iii) Show that the voltage may be written in the form v(t) = V sin(100t+φ) where V is the amplitude and φ is the phase which you should calculate. 2. The complex impedance of two circuits in parallel with complex impedances Z 1 and Z 2 is given by 1 Z = 1 Z Z 2 If Z 1 = 1+2j and Z 2 = 2 j calculate Z in the form Z = a+bj and use this to calculate the (real) impedance of the circuit. 3. A simple RLC-circuit shown below contains and inductor with L = 1 H, a resistor for which R = 10 3 Ω, and a capacitor for which C = F. With the switch in position A, the battery maintains a charge of C in the capacitor. At time t = 0 the switch is moved to B and the capacitor discharges through the circuit. A B (i) Find the differential equation for the charge q(t) and find the general solution. (ii) Find the expression for the corresponding general solution for the current i(t). (iii) Hence find the solutions q(t) and i(t) which satisfy the initial conditions for this circuit at t = 0. 9

10 4. The RLC-circuit shown below V i R L C V o has R = 8 Ω, L = 0.5 H, C = 0.1 F and input voltage given by V i = 100sin2t. (i) Find the differential equation satisfied by the output current i o (t). (ii) Find a particular solution for the differential equation and hence find the steady state output current and calculate its amplitude. (iii) Confirm your answer for the amplitude of the output current by using the method of phasors, and calculate the phase of the current relative to the input voltage. 10

Chapter 33. Alternating Current Circuits

Chapter 33. Alternating Current Circuits Chapter 33 Alternating Current Circuits 1 Capacitor Resistor + Q = C V = I R R I + + Inductance d I Vab = L dt AC power source The AC power source provides an alternative voltage, Notation - Lower case

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

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

1 Phasors and Alternating Currents

1 Phasors and Alternating Currents Physics 4 Chapter : Alternating Current 0/5 Phasors and Alternating Currents alternating current: current that varies sinusoidally with time ac source: any device that supplies a sinusoidally varying potential

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

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

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

Chapter 10: Sinusoidal Steady-State Analysis

Chapter 10: Sinusoidal Steady-State Analysis Chapter 10: Sinusoidal Steady-State Analysis 1 Objectives : sinusoidal functions Impedance use phasors to determine the forced response of a circuit subjected to sinusoidal excitation Apply techniques

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

Announcements: Today: more AC circuits

Announcements: Today: more AC circuits Announcements: Today: more AC circuits I 0 I rms Current through a light bulb I 0 I rms I t = I 0 cos ωt I 0 Current through a LED I t = I 0 cos ωt Θ(cos ωt ) Theta function (is zero for a negative argument)

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

Sinusoidal Steady-State Analysis

Sinusoidal Steady-State Analysis Sinusoidal Steady-State Analysis Almost all electrical systems, whether signal or power, operate with alternating currents and voltages. We have seen that when any circuit is disturbed (switched on or

More information

REACTANCE. By: Enzo Paterno Date: 03/2013

REACTANCE. By: Enzo Paterno Date: 03/2013 REACTANCE REACTANCE By: Enzo Paterno Date: 03/2013 5/2007 Enzo Paterno 1 RESISTANCE - R i R (t R A resistor for all practical purposes is unaffected by the frequency of the applied sinusoidal voltage or

More information

Part 4: Electromagnetism. 4.1: Induction. A. Faraday's Law. The magnetic flux through a loop of wire is

Part 4: Electromagnetism. 4.1: Induction. A. Faraday's Law. The magnetic flux through a loop of wire is 1 Part 4: Electromagnetism 4.1: Induction A. Faraday's Law The magnetic flux through a loop of wire is Φ = BA cos θ B A B = magnetic field penetrating loop [T] A = area of loop [m 2 ] = angle between field

More information

CIRCUIT ANALYSIS II. (AC Circuits)

CIRCUIT ANALYSIS II. (AC Circuits) Will Moore MT & MT CIRCUIT ANALYSIS II (AC Circuits) Syllabus Complex impedance, power factor, frequency response of AC networks including Bode diagrams, second-order and resonant circuits, damping and

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

09/29/2009 Reading: Hambley Chapter 5 and Appendix A

09/29/2009 Reading: Hambley Chapter 5 and Appendix A EE40 Lec 10 Complex Numbers and Phasors Prof. Nathan Cheung 09/29/2009 Reading: Hambley Chapter 5 and Appendix A Slide 1 OUTLINE Phasors as notation for Sinusoids Arithmetic with Complex Numbers Complex

More information

Sinusoidal Steady-State Analysis

Sinusoidal Steady-State Analysis Sinusoidal Steady-State Analysis Mauro Forti October 27, 2018 Constitutive Relations in the Frequency Domain Consider a network with independent voltage and current sources at the same angular frequency

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

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

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

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

Chapter 9 Objectives

Chapter 9 Objectives Chapter 9 Engr8 Circuit Analysis Dr Curtis Nelson Chapter 9 Objectives Understand the concept of a phasor; Be able to transform a circuit with a sinusoidal source into the frequency domain using phasor

More information

Inductive & Capacitive Circuits. Subhasish Chandra Assistant Professor Department of Physics Institute of Forensic Science, Nagpur

Inductive & Capacitive Circuits. Subhasish Chandra Assistant Professor Department of Physics Institute of Forensic Science, Nagpur Inductive & Capacitive Circuits Subhasish Chandra Assistant Professor Department of Physics Institute of Forensic Science, Nagpur LR Circuit LR Circuit (Charging) Let us consider a circuit having an inductance

More information

AC Circuits Homework Set

AC Circuits Homework Set Problem 1. In an oscillating LC circuit in which C=4.0 μf, the maximum potential difference across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 ma.

More information

Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2

Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2 Module 25: Driven RLC Circuits 1 Module 25: Outline Resonance & Driven LRC Circuits 2 Driven Oscillations: Resonance 3 Mass on a Spring: Simple Harmonic Motion A Second Look 4 Mass on a Spring (1) (2)

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

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

RLC Series Circuit. We can define effective resistances for capacitors and inductors: 1 = Capacitive reactance:

RLC Series Circuit. We can define effective resistances for capacitors and inductors: 1 = Capacitive reactance: RLC Series Circuit In this exercise you will investigate the effects of changing inductance, capacitance, resistance, and frequency on an RLC series AC circuit. We can define effective resistances for

More information

Sinusoidal Response of RLC Circuits

Sinusoidal Response of RLC Circuits Sinusoidal Response of RLC Circuits Series RL circuit Series RC circuit Series RLC circuit Parallel RL circuit Parallel RC circuit R-L Series Circuit R-L Series Circuit R-L Series Circuit Instantaneous

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

Driven RLC Circuits Challenge Problem Solutions

Driven RLC Circuits Challenge Problem Solutions Driven LC Circuits Challenge Problem Solutions Problem : Using the same circuit as in problem 6, only this time leaving the function generator on and driving below resonance, which in the following pairs

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

RLC Circuit (3) We can then write the differential equation for charge on the capacitor. The solution of this differential equation is

RLC Circuit (3) We can then write the differential equation for charge on the capacitor. The solution of this differential equation is RLC Circuit (3) We can then write the differential equation for charge on the capacitor The solution of this differential equation is (damped harmonic oscillation!), where 25 RLC Circuit (4) If we charge

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

Sinusoidal steady-state analysis

Sinusoidal steady-state analysis Sinusoidal steady-state analysis From our previous efforts with AC circuits, some patterns in the analysis started to appear. 1. In each case, the steady-state voltages or currents created in response

More information

Physics 115. AC: RL vs RC circuits Phase relationships RLC circuits. General Physics II. Session 33

Physics 115. AC: RL vs RC circuits Phase relationships RLC circuits. General Physics II. Session 33 Session 33 Physics 115 General Physics II AC: RL vs RC circuits Phase relationships RLC circuits R. J. Wilkes Email: phy115a@u.washington.edu Home page: http://courses.washington.edu/phy115a/ 6/2/14 1

More information

Prof. Anyes Taffard. Physics 120/220. Voltage Divider Capacitor RC circuits

Prof. Anyes Taffard. Physics 120/220. Voltage Divider Capacitor RC circuits Prof. Anyes Taffard Physics 120/220 Voltage Divider Capacitor RC circuits Voltage Divider The figure is called a voltage divider. It s one of the most useful and important circuit elements we will encounter.

More information

Course Updates. Reminders: 1) Assignment #10 due Today. 2) Quiz # 5 Friday (Chap 29, 30) 3) Start AC Circuits

Course Updates. Reminders: 1) Assignment #10 due Today. 2) Quiz # 5 Friday (Chap 29, 30) 3) Start AC Circuits ourse Updates http://www.phys.hawaii.edu/~varner/phys272-spr10/physics272.html eminders: 1) Assignment #10 due Today 2) Quiz # 5 Friday (hap 29, 30) 3) Start A ircuits Alternating urrents (hap 31) In this

More information

Complex Numbers Review

Complex Numbers Review Complex Numbers view ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 4 George Arfken, Mathematical Methods for Physicists Chapter 6 The real numbers (denoted R) are incomplete

More information

BIOEN 302, Section 3: AC electronics

BIOEN 302, Section 3: AC electronics BIOEN 3, Section 3: AC electronics For this section, you will need to have very present the basics of complex number calculus (see Section for a brief overview) and EE5 s section on phasors.. Representation

More information

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

Lecture 24. Impedance of AC Circuits.

Lecture 24. Impedance of AC Circuits. Lecture 4. Impedance of AC Circuits. Don t forget to complete course evaluations: https://sakai.rutgers.edu/portal/site/sirs Post-test. You are required to attend one of the lectures on Thursday, Dec.

More information

P A R T 2 AC CIRCUITS. Chapter 9 Sinusoids and Phasors. Chapter 10 Sinusoidal Steady-State Analysis. Chapter 11 AC Power Analysis

P A R T 2 AC CIRCUITS. Chapter 9 Sinusoids and Phasors. Chapter 10 Sinusoidal Steady-State Analysis. Chapter 11 AC Power Analysis P A R T 2 AC CIRCUITS Chapter 9 Sinusoids and Phasors Chapter 10 Sinusoidal Steady-State Analysis Chapter 11 AC Power Analysis Chapter 12 Three-Phase Circuits Chapter 13 Magnetically Coupled Circuits Chapter

More information

Oscillations and Electromagnetic Waves. March 30, 2014 Chapter 31 1

Oscillations and Electromagnetic Waves. March 30, 2014 Chapter 31 1 Oscillations and Electromagnetic Waves March 30, 2014 Chapter 31 1 Three Polarizers! Consider the case of unpolarized light with intensity I 0 incident on three polarizers! The first polarizer has a polarizing

More information

EM Oscillations. David J. Starling Penn State Hazleton PHYS 212

EM Oscillations. David J. Starling Penn State Hazleton PHYS 212 I ve got an oscillating fan at my house. The fan goes back and forth. It looks like the fan is saying No. So I like to ask it questions that a fan would say no to. Do you keep my hair in place? Do you

More information

Single Phase Parallel AC Circuits

Single Phase Parallel AC Circuits Single Phase Parallel AC Circuits 1 Single Phase Parallel A.C. Circuits (Much of this material has come from Electrical & Electronic Principles & Technology by John Bird) n parallel a.c. circuits similar

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

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1 hapter 31: RL ircuits PHY049: hapter 31 1 L Oscillations onservation of energy Topics Damped oscillations in RL circuits Energy loss A current RMS quantities Forced oscillations Resistance, reactance,

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

Alternating Current Circuits

Alternating Current Circuits Alternating Current Circuits AC Circuit An AC circuit consists of a combination of circuit elements and an AC generator or source. The output of an AC generator is sinusoidal and varies with time according

More information

Phasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis

Phasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis 1 16.202: PHASORS Consider sinusoidal source i(t) = Acos(ωt + φ) Using Eulers Notation: Acos(ωt + φ) = Re[Ae j(ωt+φ) ] Phasor Representation of i(t): = Ae jφ = A φ f v(t) = Bsin(ωt + ψ) First convert the

More information

Lecture 9 Time Domain vs. Frequency Domain

Lecture 9 Time Domain vs. Frequency Domain . Topics covered Lecture 9 Time Domain vs. Frequency Domain (a) AC power in the time domain (b) AC power in the frequency domain (c) Reactive power (d) Maximum power transfer in AC circuits (e) Frequency

More information

Name:... Section:... Physics 208 Quiz 8. April 11, 2008; due April 18, 2008

Name:... Section:... Physics 208 Quiz 8. April 11, 2008; due April 18, 2008 Name:... Section:... Problem 1 (6 Points) Physics 8 Quiz 8 April 11, 8; due April 18, 8 Consider the AC circuit consisting of an AC voltage in series with a coil of self-inductance,, and a capacitor of

More information

Sinusoidal Steady State Analysis

Sinusoidal Steady State Analysis Sinusoidal Steady State Analysis 9 Assessment Problems AP 9. [a] V = 70/ 40 V [b] 0 sin(000t +20 ) = 0 cos(000t 70 ).. I = 0/ 70 A [c] I =5/36.87 + 0/ 53.3 =4+j3+6 j8 =0 j5 =.8/ 26.57 A [d] sin(20,000πt

More information

SINUSOIDAL STEADY STATE CIRCUIT ANALYSIS

SINUSOIDAL STEADY STATE CIRCUIT ANALYSIS SINUSOIDAL STEADY STATE CIRCUIT ANALYSIS 1. Introduction A sinusoidal current has the following form: where I m is the amplitude value; ω=2 πf is the angular frequency; φ is the phase shift. i (t )=I m.sin

More information

Sinusoidal Steady State Analysis (AC Analysis) Part I

Sinusoidal Steady State Analysis (AC Analysis) Part I Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

More information

12 Chapter Driven RLC Circuits

12 Chapter Driven RLC Circuits hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

More information

Physics 142 AC Circuits Page 1. AC Circuits. I ve had a perfectly lovely evening but this wasn t it. Groucho Marx

Physics 142 AC Circuits Page 1. AC Circuits. I ve had a perfectly lovely evening but this wasn t it. Groucho Marx Physics 142 A ircuits Page 1 A ircuits I ve had a perfectly lovely evening but this wasn t it. Groucho Marx Alternating current: generators and values It is relatively easy to devise a source (a generator

More information

Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits. Nov. 7 & 9, 2011

Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits. Nov. 7 & 9, 2011 Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits Nov. 7 & 9, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications,

More information

6.1 Introduction

6.1 Introduction 6. Introduction A.C Circuits made up of resistors, inductors and capacitors are said to be resonant circuits when the current drawn from the supply is in phase with the impressed sinusoidal voltage. Then.

More information

20. Alternating Currents

20. Alternating Currents University of hode sland DigitalCommons@U PHY 204: Elementary Physics Physics Course Materials 2015 20. lternating Currents Gerhard Müller University of hode sland, gmuller@uri.edu Creative Commons License

More information

AC analysis - many examples

AC analysis - many examples AC analysis - many examples The basic method for AC analysis:. epresent the AC sources as complex numbers: ( ). Convert resistors, capacitors, and inductors into their respective impedances: resistor 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

Chapter 21: RLC Circuits. PHY2054: Chapter 21 1

Chapter 21: RLC Circuits. PHY2054: Chapter 21 1 Chapter 21: RC Circuits PHY2054: Chapter 21 1 Voltage and Current in RC Circuits AC emf source: driving frequency f ε = ε sinωt ω = 2π f m If circuit contains only R + emf source, current is simple ε ε

More information

ALTERNATING CURRENT. with X C = 0.34 A. SET UP: The specified value is the root-mean-square current; I. EXECUTE: (a) V = (0.34 A) = 0.12 A.

ALTERNATING CURRENT. with X C = 0.34 A. SET UP: The specified value is the root-mean-square current; I. EXECUTE: (a) V = (0.34 A) = 0.12 A. ATENATING UENT 3 3 IDENTIFY: i Icosωt and I I/ SET UP: The specified value is the root-mean-square current; I 34 A EXEUTE: (a) I 34 A (b) I I (34 A) 48 A (c) Since the current is positive half of the time

More information

Physics 9 Friday, April 18, 2014

Physics 9 Friday, April 18, 2014 Physics 9 Friday, April 18, 2014 Turn in HW12. I ll put HW13 online tomorrow. For Monday: read all of Ch33 (optics) For Wednesday: skim Ch34 (wave optics) I ll hand out your take-home practice final exam

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

EE100Su08 Lecture #11 (July 21 st 2008)

EE100Su08 Lecture #11 (July 21 st 2008) EE100Su08 Lecture #11 (July 21 st 2008) Bureaucratic Stuff Lecture videos should be up by tonight HW #2: Pick up from office hours today, will leave them in lab. REGRADE DEADLINE: Monday, July 28 th 2008,

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

Yell if you have any questions

Yell if you have any questions Class 31: Outline Hour 1: Concept Review / Overview PRS Questions possible exam questions Hour : Sample Exam Yell if you have any questions P31 1 Exam 3 Topics Faraday s Law Self Inductance Energy Stored

More information

L L, R, C. Kirchhoff s rules applied to AC circuits. C Examples: Resonant circuits: series and parallel LRC. Filters: narrowband,

L L, R, C. Kirchhoff s rules applied to AC circuits. C Examples: Resonant circuits: series and parallel LRC. Filters: narrowband, Today in Physics 1: A circuits Solving circuit problems one frequency at a time. omplex impedance of,,. Kirchhoff s rules applied to A circuits. it V in Examples: esonant circuits: i series and parallel.

More information

AC Circuits III. Physics 2415 Lecture 24. Michael Fowler, UVa

AC Circuits III. Physics 2415 Lecture 24. Michael Fowler, UVa AC Circuits III Physics 415 Lecture 4 Michael Fowler, UVa Today s Topics LC circuits: analogy with mass on spring LCR circuits: damped oscillations LCR circuits with ac source: driven pendulum, resonance.

More information

15-884/484 Electric Power Systems 1: DC and AC Circuits

15-884/484 Electric Power Systems 1: DC and AC Circuits 15-884/484 Electric Power Systems 1: DC and AC Circuits J. Zico Kolter October 8, 2013 1 Hydro Estimated U.S. Energy Use in 2010: ~98.0 Quads Lawrence Livermore National Laboratory Solar 0.11 0.01 8.44

More information

Alternating Currents. The power is transmitted from a power house on high voltage ac because (a) Electric current travels faster at higher volts (b) It is more economical due to less power wastage (c)

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

Alternating Current. Chapter 31. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman

Alternating Current. Chapter 31. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Chapter 31 Alternating Current PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 8_8_2008 Topics for Chapter 31

More information

K.K. Gan L3: R-L-C AC Circuits. amplitude. Volts. period. -Vo

K.K. Gan L3: R-L-C AC Circuits. amplitude. Volts. period. -Vo Lecture 3: R-L-C AC Circuits AC (Alternative Current): Most of the time, we are interested in the voltage at a point in the circuit will concentrate on voltages here rather than currents. We encounter

More information

2.4 Harmonic Oscillator Models

2.4 Harmonic Oscillator Models 2.4 Harmonic Oscillator Models In this section we give three important examples from physics of harmonic oscillator models. Such models are ubiquitous in physics, but are also used in chemistry, biology,

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

Experiment 3: Resonance in LRC Circuits Driven by Alternating Current

Experiment 3: Resonance in LRC Circuits Driven by Alternating Current Experiment 3: Resonance in LRC Circuits Driven by Alternating Current Introduction In last week s laboratory you examined the LRC circuit when constant voltage was applied to it. During this laboratory

More information

Learnabout Electronics - AC Theory

Learnabout Electronics - AC Theory Learnabout Electronics - AC Theory Facts & Formulae for AC Theory www.learnabout-electronics.org Contents AC Wave Values... 2 Capacitance... 2 Charge on a Capacitor... 2 Total Capacitance... 2 Inductance...

More information

LINEAR CIRCUIT ANALYSIS (EED) U.E.T. TAXILA 09

LINEAR CIRCUIT ANALYSIS (EED) U.E.T. TAXILA 09 LINEAR CIRCUIT ANALYSIS (EED) U.E.T. TAXILA 09 ENGR. M. MANSOOR ASHRAF INTRODUCTION Thus far our analysis has been restricted for the most part to dc circuits: those circuits excited by constant or time-invariant

More information

An op amp consisting of a complex arrangement of resistors, transistors, capacitors, and diodes. Here, we ignore the details.

An op amp consisting of a complex arrangement of resistors, transistors, capacitors, and diodes. Here, we ignore the details. CHAPTER 5 Operational Amplifiers In this chapter, we learn how to use a new circuit element called op amp to build circuits that can perform various kinds of mathematical operations. Op amp is a building

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

mywbut.com Lesson 16 Solution of Current in AC Parallel and Seriesparallel

mywbut.com Lesson 16 Solution of Current in AC Parallel and Seriesparallel esson 6 Solution of urrent in Parallel and Seriesparallel ircuits n the last lesson, the following points were described:. How to compute the total impedance/admittance in series/parallel circuits?. How

More information

I. Impedance of an R-L circuit.

I. Impedance of an R-L circuit. I. Impedance of an R-L circuit. [For inductor in an AC Circuit, see Chapter 31, pg. 1024] Consider the R-L circuit shown in Figure: 1. A current i(t) = I cos(ωt) is driven across the circuit using an AC

More information

Harman Outline 1A CENG 5131

Harman Outline 1A CENG 5131 Harman Outline 1A CENG 5131 Numbers Real and Imaginary PDF In Chapter 2, concentrate on 2.2 (MATLAB Numbers), 2.3 (Complex Numbers). A. On R, the distance of any real number from the origin is the magnitude,

More information

Refinements to Incremental Transistor Model

Refinements to Incremental Transistor Model Refinements to Incremental Transistor Model This section presents modifications to the incremental models that account for non-ideal transistor behavior Incremental output port resistance Incremental changes

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

A capacitor is a device that stores electric charge (memory devices). A capacitor is a device that stores energy E = Q2 2C = CV 2

A capacitor is a device that stores electric charge (memory devices). A capacitor is a device that stores energy E = Q2 2C = CV 2 Capacitance: Lecture 2: Resistors and Capacitors Capacitance (C) is defined as the ratio of charge (Q) to voltage (V) on an object: C = Q/V = Coulombs/Volt = Farad Capacitance of an object depends on geometry

More information

AC Circuit Analysis and Measurement Lab Assignment 8

AC Circuit Analysis and Measurement Lab Assignment 8 Electric Circuit Lab Assignments elcirc_lab87.fm - 1 AC Circuit Analysis and Measurement Lab Assignment 8 Introduction When analyzing an electric circuit that contains reactive components, inductors and

More information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CIRCUITS holland (9) This print-out should have questions Multiple-choice questions may continue on the next column or page find all choices before answering AP M 99 MC points The power dissipated

More information

Consider a simple RC circuit. We might like to know how much power is being supplied by the source. We probably need to find the current.

Consider a simple RC circuit. We might like to know how much power is being supplied by the source. We probably need to find the current. AC power Consider a simple RC circuit We might like to know how much power is being supplied by the source We probably need to find the current R 10! R 10! is VS Vmcosωt Vm 10 V f 60 Hz V m 10 V C 150

More information

Schedule. ECEN 301 Discussion #20 Exam 2 Review 1. Lab Due date. Title Chapters HW Due date. Date Day Class No. 10 Nov Mon 20 Exam Review.

Schedule. ECEN 301 Discussion #20 Exam 2 Review 1. Lab Due date. Title Chapters HW Due date. Date Day Class No. 10 Nov Mon 20 Exam Review. Schedule Date Day lass No. 0 Nov Mon 0 Exam Review Nov Tue Title hapters HW Due date Nov Wed Boolean Algebra 3. 3.3 ab Due date AB 7 Exam EXAM 3 Nov Thu 4 Nov Fri Recitation 5 Nov Sat 6 Nov Sun 7 Nov Mon

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

1.3 Sinusoidal Steady State

1.3 Sinusoidal Steady State 1.3 Sinusoidal Steady State Electromagnetics applications can be divided into two broad classes: Time-domain: Excitation is not sinusoidal (pulsed, broadband, etc.) Ultrawideband communications Pulsed

More information

DOING PHYSICS WITH MATLAB

DOING PHYSICS WITH MATLAB DOING PHYSIS WITH MATAB THE FINITE DIFFERENE METHOD FOR THE NUMERIA ANAYSIS OF IRUITS ONTAINING RESISTORS, APAITORS AND INDUTORS MATAB DOWNOAD DIRETORY N01.m Voltage and current for a resistor, capacitor

More information

Frequency Response. Re ve jφ e jωt ( ) where v is the amplitude and φ is the phase of the sinusoidal signal v(t). ve jφ

Frequency Response. Re ve jφ e jωt ( ) where v is the amplitude and φ is the phase of the sinusoidal signal v(t). ve jφ 27 Frequency Response Before starting, review phasor analysis, Bode plots... Key concept: small-signal models for amplifiers are linear and therefore, cosines and sines are solutions of the linear differential

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