# dynamic behavior of electrical networks

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Lecture slides on RC circuitsTRANSCRIPT

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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