dynamic behavior of electrical networks

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Lecture slides on RC circuits

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  • Dynamic Behavior of Electrical Networks

    Carlos Coimbra April 14, 2014

  • Overview of Assignments Due in Lab. Sections

  • Grade sheet used for your lab reports Item Maximum ScoreTitle and format of report 5Abstract 5Introduction 5Theory 10Experimental Procedures 10Data and Results 15Discussion and Analysis 20Conclusions 5Error analysis (can be in Discussion) 10Figures, Tables and References 5Raw Data Summaries (Appendix) 5Overall impression 5

    Important point: If you do not have enough time to finish the lab, that is OK. We prefer quality over quantity.

  • Week Date Description Instructor

    1 3/31 Class overview, Introduction to circuits Cattolica, Coimbra

    2 4/7 A/D conversion, sampling rates, error analysis, and lab report writing, LabView

    Cattolica, Mailo

    3 4/14 Filters , frequency analysis, LabView Coimbra, Mailo

    4 4/21 Operational amplifiers: one and two stage Cattolica, Mailo

    5 4/28 Measurement of temperature and heat transfer. Lab View

    Coimbra, Mailo Midterm Exam I

    6 5/5 Pressure transducers and accelerometers LabView

    Coimbra, Mailo

    7 5/12

    Measurement of strain and force, beam vibration. LabView

    Coimbra, Mailo

    8 5/19 Introduction to Position Control. LabView Cattolica, Mailo

    9 5/26 Make Up Labs/Review Lab Practical Final No lecture - Holiday 10 6/2 Lab Practical Final Examination Cattolica, Midterm II

    Lecture Schedule 2014

  • What is due this week?

    Lab Report from week 2

    LabVIEW VI from week 2

    Pre-lab for week 3 (summary of what you will be doing in the lab.+ answers to questions at the start of the lab handout)

  • Objectives of Week 3 lab.

    To investigate the response of a first order RC circuit to step and sine wave inputs, and to determine the value of an unknown capacitance, C, from this response.

    To investigate the use of RC circuits in filtering signals (both low pass and high pass filters).

    Application of low pass filter

  • Basic concepts: Kirchoffs Current Law

    The current flowing into a node is equal to the sum of the individual currents leaving the node (Principle of charge conservation)

    I1I2

    I3

    I1 = I2 + I3

    In

    n = 0

  • Kirchoff's Voltage Law

    The voltage drop around any closed circuit is equal to zero

    0= (Vb Va)+(Vc Vb)+ Vd Vc( )+(Va Vd)

    a

    b c

    d

    (Va Vd )

    Vd Vc( )

    (Vc Vb)

    (Vb Va)

  • Voltage divider

    A most useful circuit, allowing for voltage control

    Vout =Vin

    RbottomRbottom +Rtop

    Vin VoutRtop

    = I =Vout 0Rbottom

  • Capacitors

    An electrical device which stores charge

    The magnitude of the charge stored (Q) is directly proportional to the potential difference (V) between the plates.

    Q=CV Units of C = farad = coulombs/volt

    The current flow through a capacitor is:

    I = dQ

    dt=C dV

    dt

  • Capacitors

    10-6 Coulomb of opposite charge on 10-7 farad capacitor V = Q/C = 10 Volts

    Capacitors commonly found in electronic devices typically range from 10-6 to 10-12 farad.

    10-12 farad is a picofarad or pF

    ~ 6.2 x 1012 electrons lab capacitor + + + + + +

    +

    +

    - - - - - - -

    + C = coulombs/volt = farad

  • RC circuit (in series, DC)

    V0- IR - Q/C = 0 (Kirchoffs 2nd)

    V0

    This system would charge with switch at a and discharge with switch at b.

    dQdt

    =V0R

    Q

    RC=

    1RC

    Q CV0( )

    Q =CV0 1 e

    t

    RC

    When t = 0, Q = 0 When t , Q = CVo http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=31

  • 0.86Vo

    2

    RC circuit - charging

    Charge rises exponentially with time, current decreases exponentially with time

    Q =CV0 1 e

    tRC

    V(t) =Vo 1 e

    t

    RC

    I = dQdt

    =V0R

    e

    tRC = I0e

    t

    RC

    0.63Vo

    When t = RC = = characteristic time

    ( = RC)

    http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=31

  • Note that voltage rises to ~ 6.3 V in 1 ms. = RC = 0.001 sec = characteristic time.

    V( ) = 10(1 e1) = 6.32V

    Illustration of the time constant

    etRC = e1

    when t = RC

  • RC circuit- discharging

    http://www.phy.ntnu.edu.tw/java/rc/rc.html

    Units of RC

    I =dQdt

    =VR

    C =QV

    Q =CV

    dQdt

    = C dVdt

    CdVdt

    =VR

    dVV

    =dtRC

    ln VVo

    =

    tRC

    RC = ohm x farad = volt/(coulomb/sec ) x

    coulombs/volt = sec

    When t = RC = V = Vo/e

    V = 0.37Vo V =Vo exp

    tRC

    V = 0.37Vo

  • Charging capacitor V(t) = Vo(1 - e-t/RC) When t = RC

    1 - e-1 = 0.63 Thus when V(t) = 0.63Vo

    t = RC

    Discharging capacitor V(t) = Voe-t/RC When t = RC

    e-t/RC = 0.37 Thus, when V(t) = 0.37Vo

    t = RC

    t

    0.37 Vo VO

    LTA

    GE

    t

    0.63Vo

  • Capacitive circuit phase angle depends on both capacitance and resistance

  • Impedance

    Describes a measure of opposition to an alternating current Z = V/I

    Describes amplitudes and relative phases Complex quantity Impedance can be

    A resistor A capacitor An inductor

    Z = Z ei

  • Application of complex algebra to impedance

    If we take the voltage V and the current I as complex quantities with frequency and phases and

    V = |V|ei(t+) I = |I|ei(t+) The complex impedance is:

    Z = VI=

    VI

    ei

    Z = Z ei

    = -

    Z Z

    Resistor R 0 R

    Capacitor 1/C -/2 1/iC

    Inductor L /2 iL

  • A

    Frequency response

    The most useful input waveform is the sine wave Vinput = Asin(t) where = 2f

    The output voltage has the same frequency for any linear circuit although its amplitude and phase may change Voutput = Bsin(t + )

    Vi Vo

    B

    volta

    ge

  • Consider when = 1/RC A decibel is one tenth as large as a bel For example, if we have10 times larger

    signal, it is 20 dB

    dB 20log10

    A2A1

    V0Vi

    =12

    G = 20log1012

    Gain = -3 dB (at which power is halved) Gain = 0.707

    Decibels

  • Filters

    A filter is a device that impedes the passage of signals whose frequencies fall within a band called the stop band

    It permits frequencies in the pass band through relatively unchanged

    In signal processing, to remove unwanted parts of the signal such as random noise

  • Low pass filter a RC circuit

    Passes low frequencies Measure voltage across capacitor

    VoutcapacitorVin

    =Gain = 1

    (RC)2 +1

    1iC

    R + 1iC

    1iC

    R 1iC

    *

    =1

    1+ iRC

    1

    1 iRC

    Vinput = R + Zc (Zc = 1/iC) Voutput = Zc

    For an RC circuit, (Vo/Vi)2 =

  • When RC > 1

    The output voltage becomes attenuated Eventually 0

    When RC = 1 = 1/RC

    Low pass filter characteristics

    VoVi

    =1

    RC

    VoVi

    =12

    = 1/RC gain

    -3dB frequency

    0.707

    1

    VoVi

    = 1

    VoutcapacitorVin

    =Gain = 1

    (RC)2 +1

  • High pass filter Passes high frequencies

    A RC circuit Measure voltage across resistor

    VoutVin

    =IR

    IR + IiC

    =iRC

    iRC +1Vout resistorVin

    =Gain = RC 2R 2C 2 +1

    As , Gain 1 As 0, Gain 0 "-3 dB frequency" is frequency where Vo/Vi = 1/2 Gain = 1/2 "-3 dB frequency" occurs when = 1/RC

    0

    1

    f-3dB

    VoVi

    12

    Swap resistor with capacitor

  • THIS WEEK IN THE LAB

    NOTE: All the answers to the lab quiz questions are in this lecture and

    in the laboratory handout

  • Part 1: Finding the value of the capacitor & -3 dB frequency of a RC Filter

    No

    - Use 1 kHz, 5 Vp-p square wave - Measure the voltage across capacitor - Set the frequency to 60 Hz (input and output equal)

  • V = Vo[1 e-t/RC] t = = RC when V = 0.63 Vp-p

    V = 0.63 Vp-p

    t =

    Determining C

  • Question 2

    2a: Compare calculated capacitance with value given in class (mine was about 0.1 F), include plots (via Data Capture) of oscilloscope screens showing voltage and time cursor lines used in determining time constant. Plots should be clear, include notes so that they are self-explanatory. 2b: What units (seconds, microseconds, etc.) must time be in if R is in ohms and C is in farads and e-t/RC is dimensionless?

  • Another way to calculate capacitance - from frequency

    Gain =

    VoutputcapacitorVinput

    =1

    1+ (RC)2

    "-3dB frequency" occurs when

    Gain =12

    = 2 f = 1RC

    Calculate C from determining the frequency corresponding to a gain of 1/2

  • Determining the -3 dB frequency

    On the scope, look at Vo and Vi

    Vary the frequency until Vo = 1/2Vi = 0.707Vi

    -3.01 = 20log(1/2) Vo/Vi = 1/2

    Note - input and output frequencies are the same but phase is shifted.

    f = 164.6 Hz= -3dB frequency Then C = 1/(2fR)

    Ch. 1 (Vi) Ch. 2 (Vo)

    4.99 V 3.53 V

  • Frequency response of a low pass filter

    Gain decreases with frequency; note 3 dB point

  • Plot frequency response (in Excel)

  • Question 3

    What is a linear circuit?

    (a) Include plot of oscilloscope screen used

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