First and Second Order Circuits. Claudio Talarico, Gonzaga University Spring 2015
|
|
- Dayna Brooks
- 5 years ago
- Views:
Transcription
1 First and Second Order Circuits Claudio Talarico, Gonzaga University Spring 2015
2 Capacitors and Inductors intuition: bucket of charge q = Cv i = C dv dt Resist change of voltage DC open circuit Store voltage (charge) Energy stored = 0.5 C v(t) 2 i(t) intuition: water hose λ = Li v = L di dt Resist change of current DC short circuit Store current (magnetic flux) Energy stored = 0.5 L i(t) 2 C +" v(t) " i(t) + v(t) EE406 Introduction to IC Design 2
3 Characterization of an LTI system s behavior Techniques commonly used to characterize an LTI system: 1. Observe the response of the system when excited by a step input (time domain response) Assumption: x in (t) is causal (i.e. x(t)=0 for t < 0) x in (t) L.T.I. h(t) x out (t) x out = t 0 x in (τ )h(t τ )dτ x in (t)* h(t) 2. Observe the response of the system when excited by sinusoidal inputs (frequency response) X out (s) = X in (s) H(s) EE406 Introduction to IC Design 3
4 Frequency Response The merit of frequency-domain analysis is that it is easier than time domain analysis: L[x(t)] = 0 e st x(t)dt = X(s) The transfer function of any of the LTI circuits we consider Are rational with m n Are real valued coefficients a j and b i Have poles and zeros that are either real or complex conjugated Furthermore, if the system is stable All denominator coefficients are positive The real part of all poles are negative One sided Laplace Transform (assumption: x(t) is causal or is made causal by multiplying it by u(t)) H(s) = a 0 + a 1 s a m sm 1+ b 1 s b n s n = K (s +ω z1)...(s +ω zm) (s +ω p1 )...(s +ω pn ) = = K (s z 1)...(s z m ) (s p 1 )...(s p n ) with K a m b n root form mathematicians style EE406 Introduction to IC Design 4
5 Frequency Response! s # 1+ " H(s) = a 0! # " ω z1 ω p1 $! &...# % " $! s &...# 1+ % " ω zm ω pn $ & % $ & %! s $! # 1 &... 1 s $ # & " z = a 1 % " z m % 0! 1 s $ # " p 1 % &...! 1 s $ # " p & % n EE style NOTE : p i = ω pi z i = ω zi (poles) (zeros) EE406 Introduction to IC Design 5
6 Magnitude and Phase (1) When an LTI system is exited with a sinusoid the output is a sinusoid of the same frequency. The magnitude of the output is equal to the input magnitude multiplied by the magnitude response ( H(jω in ) ). The phase difference between the output and input sinusoid is equal to the phase response (ϕ=phase[h(jω in )]) x in (t) = A in cos(ωt) = A in e jω t + e jω t F X in ( jω) = F[x in (t)] 2 unit circle IM axis θ 1e jθ = cosθ +jsinθ RE axis H( jω) = H( jω) e jωt 0 X out ( jω) = X in ( jω) H( jω) cosθ = e jθ + e jθ 2 Euler s formula EE406 Introduction to IC Design 6
7 Magnitude and Phase (2) X out ( jω) = X in ( jω) H( jω) = X in ( jω) H( jω) e jωt 0 F -1 Time Shift Property: F[x(t t 0 )] = X(f) e j2πft0 x out (t) = H( jω in ) x in (t + t 0 ) = = A in H( jω in ) cos[ω(t + t 0 )] = = A in H( jω in ) cos(ωt +ωt 0 ) EE406 Introduction to IC Design 7
8 First order circuits A first order transfer function has a first order denominator H(s) = A 0 ω p First order low pass transfer function. This is the most commonly encountered transfer function in electronic circuits H(s) = A 0 ω z ω p General first order transfer function. EE406 Introduction to IC Design 8
9 Step Response of first order circuits (1) Case 1: First order low pass transfer function H(s) = A 0 ω p x in (t) = A in u(t) X in (s) = A in s X out (s) = A in s A 0 ω p = A in A 0! x out (t) = A in A 0 u(t) " # 1 e t/τ % & " 1 s 1 % $ ' # $ s +ω p &' with τ =1/ω p EE406 Introduction to IC Design 9
10 Step Response of first order circuits (2) Case 2: General first order transfer function x in (t) = A in u(t) X in (s) = A in s H(s) = A 0 ω z ω p X out (s) = A ina 0 s ω z ω p! ( " x out (t) = A in A 0 u(t) 1 1 ω p * $ )* # ω z % 'e t/τ & + -,- where τ =1/ω p EE406 Introduction to IC Design 10
11 Step Response of first order circuits (3) Notice x out (t) short term and long term behavior x out (0+) = A in A 0 ω p ω z x out ( ) = A in A 0 The short term and long term behavior can also be verified using the Laplace transform x out (0+) = lim s s X out (s) = lim s s A in s H(s) = A ina 0 ω p ω z x out ( ) = lim s 0 s X out (s) = lim s 0 s A in s H(s) = A ina 0 EE406 Introduction to IC Design 11
12 Equation for step response to any first order circuit x out (t) = x out ( ) [ x out ( ) x out (0+)] e t/τ where τ =1/ω p Steady response Transitory response EE406 Introduction to IC Design 12
13 Example #1 R = 1KΩ, C= 1µF. Input is a 0.5V step at time 0 Source: Carusone, Johns and Mar2n H(s) = 1 RC τ = RC =1µs ω p = 1 τ =1Mrad / s f = ω p 3dB 2π 159KHz V out (t) = 0.5 ( 1 e t/τ )u(t) EE406 Introduction to IC Design 13
14 Example #2 (1) R1 = 2KΩ, R2=10KΩ C1= 5pF. C2=10pF Input is a 2V step at time 0 Source: Carusone, Johns and Mar2n H(s 0) = R 2 R 1 + R 2 A 0 ; H(s ) = C 1 C 1 + C 2 A τ p = (R 1 R 2 ) (C 1 C 2 ) = R R 1 2 (C 1 + C 2 ) R 1 + R 2 " % R $ H(s) = 2 R 1 C ' $ 1 R 1 + R 2 $ R R ' 1 2 (C 1 + C 2 )' # $ R 1 + R 2 &' By inspection τ z = R 1 C 1 EE406 Introduction to IC Design 14
15 V out (t) = Example #2 (2) V out (0+)+ $ % V out ( ) V out (0+) 1 e t/τ p 0 fort 0 ( ) & ' fort > 0 V out ( ) V out (t) = A in R 2 R 1 + R 2 A in C 1 C 1 + C 2 = V out (0+) EE406 Introduction to IC Design 15
16 Example #3 Consider an amplifier having a small signal transfer function approximately given by A(s) = A 0 ω p A 0 = 1 x 10 5 ω p = 1 x 10 3 rad/s Find approx. unity gain BW and phase shift at the unity gain frequency since A 0 >> 1: A(s) A 0 s ω p = A 0 ω p s A( jω) A 0ω p jω A 0 ω p jω u =1 ω u A 0 ω p " Phase[A( jω u )] Phase A 0ω p % $ ' = 90! # jω u & EE406 Introduction to IC Design 16
17 Second-order low pass Transfer Function H(s) = a 0 1+ b 1 s + b 2 s 2 = a 0 Q + s2 2 b 1 1 Q ; b Interesting cases: - Poles are real - one of the poles is dominant - Poles are complex ω 3dB 1 b 1 = τ j b 1 ( ) EE406 Introduction to IC Design 17
18 Poles Location Roots of the denominator of the transfer function: Complex Conjugate poles (overshooting in step response) Q + s2 2 = 0 Poles location with Q> 0.5 jω for Q > 0.5 p 1,2 = 2Q ( 1 j 4Q2 1) = ω R jω I - For Q = (Φ=45 ), the -3dB frequency is (Maximally Flat Magnitude or Butterworth Response) - For Q > the frequency response has peaking Real poles (no overshoot in the step response) for Q 0.5 p 1,2 = 2Q 1 1 4Q2 ( ) Φ = Source: Carusone σ EE406 Introduction to IC Design 18
19 Frequency Response 10 H(s) 5 H(s) = 1 Q + s2 2 Magnitude [db] Q=0.5 Q=1/sqrt(2) Q=1 Q= / 0 EE406 Introduction to IC Design 19
20 Step Response Underdamped Q > 0.5 Source: Carusone Q=0.5 Overdamped Q < 0.5 time Ringing for Q > 0.5 The case Q=0.5 is called maximally damped response (fastest settling without any overshoot) EE406 Introduction to IC Design 20
21 Widely- Spaced Real Poles H(s) = a 0 1+ b 1 s + b 2 s 2 = a 0 Q + s2 2 b 1 1 Q ; b Real poles occurs when Q 0.5: for Q 0.5 p 1,2 = ω ( 0 2Q 1 1 ) 4Q2 Real poles widely-spaced (that is one of the poles is dominant) implies: p 1 2Q 2Q 1 4Q2 << p 2 2Q + 2Q! 1 4Q2 0 << 2 1 4Q 2 0 << 1 4Q 2 0 <<1 4Q 2 Q 2 << 1 4 b 2 b 1 2 << 1 4 EE406 Introduction to IC Design 21
22 Widely-Spaced Real Poles a 0 H(s) = 1+ b 1 s + b 2 s = a 0 2 " 1 s % $ # p 1 & ' " $ 1 s # p 2 % ' & = a 0 1 s p 1 s p 2 + s2 p 1 p 2 a 0 1 s p 1 + s2 p 1 p 2 p 1 1 b 1 p 2 1 p 1 b 2 = b 1 b 2 This means that in order to estimate the -3dB bandwidth of the circuit, all we need to know is b 1! H(s) a 0 1 s p 1 ω -3dB p 1 1 b 1 ZVTC method: b 1 = τ j ω 3dB 1 = 1 b 1 τ j EE406 Introduction to IC Design 22
23 Example: Series RLC circuit (1) v in (t) = V I u(t) R L +" " C i(t) +" v out (t) " V out (s) V in (s) = 1 sc R + sl + 1 sc = 1 RC + s 2 LC 1 Q + s2 2 2 = 1 LC ; Q = 1 RC = L / C R Z 0 R EE406 Introduction to IC Design 23
24 Example - Series RLC: Poles location (2) Q + s2 ω = 0 s2 + s 2 0 Q +ω 2 0 = 0 s 1,2 = Q $ ω & 0 % Q 2 ' ) ( s 1,2 = ω ( 0 2Q 1 1 ) 4Q2 2Q α Two possible cases: For Q ½ real poles: s 1,2 = ω ( 0 2Q 1 1 ) 4Q2 For Q > ½ complex poles α ( 1 1 ) 4Q2 s 1,2 = ω ( 0 2Q 1 j 4Q2 1) α 1 j 4Q2 1 ( ) EE406 Introduction to IC Design 24
25 V out (s) V in (s) = 1 sc R + sl + 1 sc 2 = 1 LC ; Q = 1 RC = Example series RLC = 1 RC + s 2 LC 1 L / C R Z 0 R ; 2Q α Q + s2 2 C = 1 L ω ; L = C 1 Q = = L RC R 1 α = 2Q = R 2L EE406 Introduction to IC Design 25
26 Example series RLC s 2 + s Q +ω 2 0 = 0 s 2 + s 2α +ω 2 0 = 0 s 1,2 = α α 2 2 For Q ½ real poles: s 1,2 = α α 2 2 For Q > ½ complex poles! # " for α $ <<1 the polesarewidelyspaced& % IM s 1,2 = α j 2 α 2 = α jω n ω n = natural (damped) frequency = resonant frequency ω n ω p1 X X ω p2 α= /2Q RE EE406 Introduction to IC Design 26
27 v out (t) Example series RLC: Step response time ζ α / >1 (a) Overdamped Q < 0.5 (α > ) ζ α / =1 (b) Critically damped Q=0.5 (α = ) ζ α / <1 (c) Underdamped Q > 0.5 (α < ) ω n = ringing frequency α = damping factor (rate of decay) = resonance frequency ζ = Damping ratio EE406 Introduction to IC Design 27
28 Example RLC series: Quality Factor Q For a system under sinusoidal excitation at a frequency ω, the most fundamental definition for Q is: energy stored Q = ω average power dissipated At the resonant frequency, the current through the network is simply V in /R. Energy in such a network sloshes back and forth between the inductance and the inductor, with a constant sum. The peak inductor current at resonance is I pk =V pk /R, so the energy stored by the network can be computed as: E stored = 1 2 L I 2 pk dimensionless The average power dissipated in the resistor at resonance is: P avg = 1 2 R I 2 pk E Q = ω stored 0 = 1 P avg LC L R = L / C R = Z 0 R Z 0 = Characteristic impedance of the network At resonance Z C = Z L = L=( C) 1 = L / C Z 0 EE406 Introduction to IC Design 28
29 Example series RLC: Ringing and Q Since Q is a measure of the rate of energy loss, one expect a higher Q to be associated with more persistent ringing than a lower Q. IM R large α large Q small Φ= ω n α= RE Q=0.5 à Φ = 0 Q=0.707 à Φ = 45 Q=1 à Φ = 60 Q=10 à Φ 87 Ringing doesn t last long and its excursion is small The decaying envelope is proportional to: e αt = e t 2Q Rule of thumb: Q is roughly equal to the number of cycles of ringing. Frequency of the ringing oscillations: 1/f n = T n = 2π/ω n EE406 Introduction to IC Design 29
30 Example RLC series by intuition (1) We can predict the behavior of the circuit without solving pages of differential equations or Laplace transforms. All we need to know is the characteristics equation (denominator of H(s)) and the initial conditions v I (t) = V I u(t) +" " i(t) L R +" C v(t) " Assume the values of R,L,C are such that > α (underdamping). Initial conditions: i(0) = I 0 v(0) = +V 0 The voltage across the capacitor cannot jump: v(0+)=v(0)=+v 0 The current through the inductor cannot jump: i(0+)=i(0)= I 0 The output voltage starts at V 0, it ends at v( )=V I times before settling at V I and it rings about Q EE406 Introduction to IC Design 30
31 Example RLC series using intuition (2) The only question left is to decide if v(t) will start off shooting down or up? But, we know that i(0+) is negative à this means the current flows from the capacitor toward the inductor à which means the capacitor must be discharging à the voltage across the capacitor must be dropping v(t) Ringing stops after Q! cycles V I V I? v (0) i(0) 0 is negative so v(t) must drop π! 2! ω n period of ringing t EE406 Introduction to IC Design 31
32 Example RLC series using intuition (3) In practice for the under damped case it useful to compute two parameters Overshoot Settling time " OS = exp π % $ α' # ω n & Normalized overshoot = % overshoot w.r.t. final value t s 1 α ln # ε ω & n % ( ε is the % error that we are willing $ ' to tolerate w.r.t. the ideal final value EE406 Introduction to IC Design 32
33 First order vs. Second order circuits Behavior First order circuits introduce exponential behavior Second order circuits introduce sinusoidal and exponential behavior combined Fortunately we will not need to go on analyzing 3 rd, 4 th, 5 th and so on circuits because they are not going to introduce fundamentally new behavior EE406 Introduction to IC Design 33
Electric Circuit Theory
Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 8 Natural and Step Responses of RLC Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 8.1 Introduction to the Natural Response
More informationTransient 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 informationSinusoidal 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 informationDynamic circuits: Frequency domain analysis
Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution
More informationSome 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 informationInductance, 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 informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationRefinements 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 informationSource-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 informationCommon Drain Stage (Source Follower) Claudio Talarico, Gonzaga University
Common Drain Stage (Source Follower) Claudio Talarico, Gonzaga University Common Drain Stage v gs v i - v o V DD v bs - v o R S Vv IN i v i G C gd C+C gd gb B&D v s vv OUT o + V S I B R L C L v gs - C
More informationPhysics 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 informationR-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 informationAlternating 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 informationHandout 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 informationMODULE 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 informationLecture 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 informationChapter 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 informationEE221 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 informationResponse 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 informationElectromagnetic 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 informationEE 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 informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationREACTANCE. 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 informationModule 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 informationEE221 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 information09/29/2009 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 informationTo 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 informationFrequency 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 informationEM 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 informationHandout 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 informationELECTROMAGNETIC 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 informationPhysics 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 informationSinusoidal 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 information8 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 informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
More informationECE 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 informationChapter 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 informationRLC 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 informationHomework Assignment 11
Homework Assignment Question State and then explain in 2 3 sentences, the advantage of switched capacitor filters compared to continuous-time active filters. (3 points) Continuous time filters use resistors
More informationDynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology.
Dynamic Response Assoc. Prof. Enver Tatlicioglu Department of Electrical & Electronics Engineering Izmir Institute of Technology Chapter 3 Assoc. Prof. Enver Tatlicioglu (EEE@IYTE) EE362 Feedback Control
More informationAdvanced Analog Building Blocks. Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc
Advanced Analog Building Blocks Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc 1 Topics 1. S domain and Laplace Transform Zeros and Poles 2. Basic and Advanced current
More informationPhasors: 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 informationFilters and Tuned Amplifiers
Filters and Tuned Amplifiers Essential building block in many systems, particularly in communication and instrumentation systems Typically implemented in one of three technologies: passive LC filters,
More informationECE 255, Frequency Response
ECE 255, Frequency Response 19 April 2018 1 Introduction In this lecture, we address the frequency response of amplifiers. This was touched upon briefly in our previous lecture in Section 7.5 of the textbook.
More informationEE292: 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 informationEE292: 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 informationMODULE-4 RESONANCE CIRCUITS
Introduction: MODULE-4 RESONANCE CIRCUITS Resonance is a condition in an RLC circuit in which the capacitive and inductive Reactance s are equal in magnitude, there by resulting in purely resistive impedance.
More informationProf. 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(amperes) = (coulombs) (3.1) (seconds) Time varying current. (volts) =
3 Electrical Circuits 3. Basic Concepts Electric charge coulomb of negative change contains 624 0 8 electrons. Current ampere is a steady flow of coulomb of change pass a given point in a conductor in
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces
More informationOPERATIONAL AMPLIFIER APPLICATIONS
OPERATIONAL AMPLIFIER APPLICATIONS 2.1 The Ideal Op Amp (Chapter 2.1) Amplifier Applications 2.2 The Inverting Configuration (Chapter 2.2) 2.3 The Non-inverting Configuration (Chapter 2.3) 2.4 Difference
More informationDriven 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 informationChapter 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 informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course
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.
More informationElectronic Circuits EE359A
Electronic Circuits EE359A Bruce McNair B26 bmcnair@stevens.edu 21-216-5549 Lecture 22 578 Second order LCR resonator-poles V o I 1 1 = = Y 1 1 + sc + sl R s = C 2 s 1 s + + CR LC s = C 2 sω 2 s + + ω
More informationEE 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 informationENGR 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 informationP441 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 informationCircuits 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 informationGATE : , Copyright reserved. Web:www.thegateacademy.com
GATE-2016 Index 1. Question Paper Analysis 2. Question Paper & Answer keys : 080-617 66 222, info@thegateacademy.com Copyright reserved. Web:www.thegateacademy.com ANALYSIS OF GATE 2016 Electrical Engineering
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2003 Experiment 17: RLC Circuit (modified 4/15/2003) OBJECTIVES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8. Spring 3 Experiment 7: R Circuit (modified 4/5/3) OBJECTIVES. To observe electrical oscillations, measure their frequencies, and verify energy
More informationLecture 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 informationECE 202 Fall 2013 Final Exam
ECE 202 Fall 2013 Final Exam December 12, 2013 Circle your division: Division 0101: Furgason (8:30 am) Division 0201: Bermel (9:30 am) Name (Last, First) Purdue ID # There are 18 multiple choice problems
More informationChapter 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 informationC 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 informationPHYSICS 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 informationRC Circuits (32.9) Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1
(32.9) We have only been discussing DC circuits so far. However, using a capacitor we can create an RC circuit. In this example, a capacitor is charged but the switch is open, meaning no current flows.
More information6.1 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 informationIntroduction 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 informationQUESTION 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 informationAC analysis. EE 201 AC analysis 1
AC analysis Now we turn to circuits with sinusoidal sources. Earlier, we had a brief look at sinusoids, but now we will add in capacitors and inductors, making the story much more interesting. What are
More informationElectric 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 informationChapter 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 informationPhysics 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 informationAssessment Schedule 2015 Physics: Demonstrate understanding of electrical systems (91526)
NCEA Level 3 Physics (91526) 2015 page 1 of 6 Assessment Schedule 2015 Physics: Demonstrate understanding of electrical systems (91526) Evidence Q Evidence Achievement Achievement with Merit Achievement
More informationMAS107 Control Theory Exam Solutions 2008
MAS07 CONTROL THEORY. HOVLAND: EXAM SOLUTION 2008 MAS07 Control Theory Exam Solutions 2008 Geir Hovland, Mechatronics Group, Grimstad, Norway June 30, 2008 C. Repeat question B, but plot the phase curve
More informationELECTROMAGNETIC INDUCTION AND FARADAY S LAW
ELECTROMAGNETIC INDUCTION AND FARADAY S LAW Magnetic Flux The emf is actually induced by a change in the quantity called the magnetic flux rather than simply py by a change in the magnetic field Magnetic
More informationSinusoidal 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 informationDesigning Information Devices and Systems II Fall 2018 Elad Alon and Miki Lustig Discussion 5A
EECS 6B Designing Information Devices and Systems II Fall 208 Elad Alon and Miki Lustig Discussion 5A Transfer Function When we write the transfer function of an arbitrary circuit, it always takes the
More informationName (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 informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationSupplemental 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 informationDEPARTMENT 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 informationInductance, 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 informationSlide 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 informationChapter 8: Converter Transfer Functions
Chapter 8. Converter Transfer Functions 8.1. Review of Bode plots 8.1.1. Single pole response 8.1.2. Single zero response 8.1.3. Right half-plane zero 8.1.4. Frequency inversion 8.1.5. Combinations 8.1.6.
More informationEE292: 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 informationPhysics 2B Spring 2010: Final Version A 1 COMMENTS AND REMINDERS:
Physics 2B Spring 2010: Final Version A 1 COMMENTS AND REMINDERS: Closed book. No work needs to be shown for multiple-choice questions. 1. A charge of +4.0 C is placed at the origin. A charge of 3.0 C
More informationExercise s = 1. cos 60 ± j sin 60 = 0.5 ± j 3/2. = s 2 + s + 1. (s + 1)(s 2 + s + 1) T(jω) = (1 + ω2 )(1 ω 2 ) 2 + ω 2 (1 + ω 2 )
Exercise 7 Ex: 7. A 0 log T [db] T 0.99 0.9 0.8 0.7 0.5 0. 0 A 0 0. 3 6 0 Ex: 7. A max 0 log.05 0 log 0.95 0.9 db [ ] A min 0 log 40 db 0.0 Ex: 7.3 s + js j Ts k s + 3 + j s + 3 j s + 4 k s + s + 4 + 3
More informationHomework 7 - Solutions
Homework 7 - Solutions Note: This homework is worth a total of 48 points. 1. Compensators (9 points) For a unity feedback system given below, with G(s) = K s(s + 5)(s + 11) do the following: (c) Find the
More informationActive 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( s) N( s) ( ) The transfer function will take the form. = s = 2. giving ωo = sqrt(1/lc) = 1E7 [rad/s] ω 01 := R 1. α 1 2 L 1.
Problem ) RLC Parallel Circuit R L C E-4 E-0 V a. What is the resonant frequency of the circuit? The transfer function will take the form N ( ) ( s) N( s) H s R s + α s + ω s + s + o L LC giving ωo sqrt(/lc)
More informationAlternating Current (AC) Circuits
Alternating Current (AC) Circuits We have been talking about DC circuits Constant currents and voltages Resistors Linear equations Now we introduce AC circuits Time-varying currents and voltages Resistors,
More informationEE348L Lecture 1. EE348L Lecture 1. Complex Numbers, KCL, KVL, Impedance,Steady State Sinusoidal Analysis. Motivation
EE348L Lecture 1 Complex Numbers, KCL, KVL, Impedance,Steady State Sinusoidal Analysis 1 EE348L Lecture 1 Motivation Example CMOS 10Gb/s amplifier Differential in,differential out, 5 stage dccoupled,broadband
More informationThe 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 informationControl Systems, Lecture04
Control Systems, Lecture04 İbrahim Beklan Küçükdemiral Yıldız Teknik Üniversitesi 2015 1 / 53 Transfer Functions The output response of a system is the sum of two responses: the forced response and the
More informationEE102 Homework 2, 3, and 4 Solutions
EE12 Prof. S. Boyd EE12 Homework 2, 3, and 4 Solutions 7. Some convolution systems. Consider a convolution system, y(t) = + u(t τ)h(τ) dτ, where h is a function called the kernel or impulse response of
More informationElectronic Circuits EE359A
Electronic Circuits EE359A Bruce McNair B26 bmcnair@stevens.edu 21-216-5549 Lecture 22 569 Second order section Ts () = s as + as+ a 2 2 1 ω + s+ ω Q 2 2 ω 1 p, p = ± 1 Q 4 Q 1 2 2 57 Second order section
More information1 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 informationSingle-Time-Constant (STC) Circuits This lecture is given as a background that will be needed to determine the frequency response of the amplifiers.
Single-Time-Constant (STC) Circuits This lecture is given as a background that will be needed to determine the frequency response of the amplifiers. Objectives To analyze and understand STC circuits with
More information