LECTURE 3: Dynamic Behavior of Electrical Networks

1
LECTURE 3 Dynamic Behavior of Electrical
Networks

• October 12, 2009

2
OBJECTIVES OF THE WEEK

• To investigate the response of a first order (one
  pair) RC circuit
• To determine the value of unknown capacitance
• To investigate the use of RC circuits in
  filtering signals
• - Low pass
• - High pass
• To investigate the response of a LC circuit

3
REVIEW OF BASIC CIRCUITS CONCEPTSKIRCHHOFFS LAWS

• The algebraic sum of the currents entering and
  leaving
• a junction is zero (KCL)
• The algebraic sum of the changes in potential
  around a
• closed loop is zero (KVL)

I2
I1
I1I2I3
I3
http//www.rfcafe.com/references/electrical/kirchh
offs_law.htm
b
c
0(Vb-Va)(Vc-Vb)(Vd-Vc)(Va-Vd)
a
d
4
REVIEW OF BASIC CIRCUITS CONCEPTSDC CIRCUIT
ANALYSIS

• The circuit elements Ideal resistors, Ideal
  voltage and current sources.
• Ideal voltage source
• Ideal current source
• Ideal resistor is described by Ohms law
• The algebraic signs of voltage and current are
  very important.
• Useful First choose the reference direction for
  current through the resistor. The value of this
  current is the potential at the tail of the arrow
  minus the potential at the head of the arrow,
  divided by R.
• Try memorizing this rule.

5
REVIEW OF BASIC CIRCUITS CONCEPTSDC CIRCUIT
ANALYSIS

• GENERAL METHOD for circuit analysis
• 1. First draw the circuit diagram, identify
  nodes, and assign labels and symbols to all
  quantities.
• 2. Assign directions to currents in each part of
  the circuit. Although the assignment of current
  directions is arbitrary, you must adhere
  rigorously to the assigned directions when
  applying Kirchhoffs laws.
• 3. Apply KCL to any node in the circuit.
• 4. Apply KVL to as many loops in the circuit as
  are needed to solve for the unknowns. Correctly
  identify the change in potential as you cross
  each element in traversing the closed loop.

6
REVIEW OF BASIC CIRCUITS CONCEPTSAC CIRCUIT
ANALYSIS

• Everything we have learned about DC analysis
  remains true in the AC, or time-varying, case.
• The circuit elements - ideal voltage sources,
  current sources, and resistances - behave in the
  same way whether or not the voltages vary in
  time.
• Additional elements that are useful only in AC
  circuits capacitors and inductors.
• Sometimes interaction of AC signals with
  inductors and capacitors can provide striking
  results How the very large voltages - tens kV -
  can appear in the sparkplug in a car where power
  is supplied by a battery of 12 V?
• This amazing voltage transformation is usually
  obtained by using a single inductor.

7
REVIEW OF BASIC CIRCUITS CONCEPTS IDEAL CAPACITOR

• Basic physics of the capacitor lies in the fact
  that the voltage across the capacitor is linearly
  proportional to the charge on its plates.
• QCV gt IdQ/dtCdV/dt
• We have to specify algebraic signs. We will adopt
  the same nomenclature used for resistances.

In dc circuits capacitors behave just like an
open circuit.
Just as with Ohms law, the algebraic signs of
the current and voltage are of greatest
importance.
8
REVIEW OF BASIC CIRCUITS CONCEPTS IDEAL INDUCTOR

• The I-V relationship

If current through an inductor is constant, the
voltage across the inductor will be zero.
9
REVIEW OF BASIC CIRCUITS CONCEPTS ENERGY STORAGE
Resistors dissipate energy whereas capacitors and
inductors store energy
Instantaneous power P(t) V(t)I(t)
Energy stored during interval t0, t is
If the initial voltage was 0,
For inductors
10
RC CIRCUITS
When a capacitor is included in a circuit , the
current varies as a function of time (when
capacitor is charging or discharging) At time tlt0
switch was in position b, at t0 it moves to
a. KVL I CdV/dt
Initial value problem
At t0, V0
Solution general solution of homogeneous diff.
eq-n particular solution of non-homogeneous
eq-n
11
RC CIRCUITS
Solution general solution of homogeneous diff.
eq-n particular solution of non-homogeneous
eq-n.
(homogeneous eq-n)
general solution of homogeneous equation,
is unknown constant.
Particular solution of non-homogeneous equation
gt particular solution
At t0 V0 gt
RC is the circuits time constant
12
RC CIRCUITS
Units

• The time constant tells us how long it takes
  to charge to rise to
• VVb(1-e-1)0.63Vb
• Small values of RC means a fast response
• Large RC values means slower response

13
DISCHARGING A CAPACITOR

• At time tlt0 switch was in position a, at t0 it
  moves to b.

KVL VR - VC 0 VR IR I - C dVC/dt
At t0 V Vo
gt BV0
is the circuits time constant
14
SUMMARY CHARGING AND DISCHARCHING A CAPACITOR
15
Useful Rules

• RULE 1 The voltage across a capacitor cannot
  change instantaneously
• I CdV/dt If the voltage V were to change
  instantaneously, the current through the
  capacitor would be infinite. But infinite current
  is impossible.
• RULE 2 The current through an inductor cannot
  change instantaneously
• VLdI/dt In order to have a sudden change
  of current, an infinite voltage would be required.

16
REVIEW OF BASIC CIRCUITS CONCEPTSAC CIRCUIT
ANALYSIS
Periodic signals x(tnT) x(t), n1,2,
T is the period of x(t)
Sinusoidal signals
- amplitude
- angular frequency (radians/sec)
- phase
- period of a sinusoid
- ordinary frequency (Hz)
Two sinusoids having the same frequency and
amplitude but different phases
Question Which phase is larger?
17
(No Transcript)
18
PHASORS
Phasor is a complex number to represent a
sinusoid.
In a circuit with the sinusoidal source is
known. Currents through and voltages across
circuit elements are sinusoids with different
amplitudes and phases. Phasors contain
information about amplitude and phase of the
sinusoids.
is phasor of v(t)
RULE 1 If a sinusoid is described by formula
the phasor representing the sinusoid is
Example
Find i
19
PHASORS
RULE 2 To obtain the sinusoid corresponding to a
given phasor, multiply the phasor by and
take the real part. Thus the sinusoid
corresponding to the phasor is
Find v(t)
Example
RULE 3 The amplitude of the sinusoid
corresponding to a phasor v is v. The phase of
the sinusoid corresponding to v is arg (v). That
is, if vAjB, the phase angle is tan-1(B/A).

RULE 4 If v is the phasor of the sinusoid v(t),
then the phasor representing the sinusoid
dv(t)/dt is
20
(No Transcript)
21
In case of resistance, the current through it
and voltage across it are connected by Ohms law.
That is, V(t)/I(t)R. Suppose that the phasor v
represents the sinusoidal voltage across the
terminals of a circuit element, and the phasor i
represents the sinusoidal current through the
element. The impedance Z of the circuit element
is the ratio of the phasor of voltage to the
phasor of current
Z v /i
IMPEDANCE
For resistors VIR. Therefore v iR, and
impedance
In general impedance is a complex number ZRj
X R is resistance and X is reactance. R, X are
measured in ohms.
22
IMPEDANCE

• For capacitor

For inductor
(elements in series)
(elements in parallel)
23
IMPEDANCE

• The impedance of a capacitor

24
PHASOR FORM OF KIRCHHOFFS LAWS

• If the sum of several sinusoids is zero, the sum
  of their phasors must be zero.
• KCL, in phasor form

KVL, in phasor form
25
The system transfer function in terms of
variation of output voltage as function of the
input voltage
26
FREQUENCY RESPONSE LOW PASS FILTER
This is voltage gain.
27
FREQUENCY RESPONSE LOW PASS FILTER

• When wRC ltlt 1
  vout/vin 1
• Low frequency signals are passed through
• the circuit with little attenuation
• When wRC gtgt 1
  vout/vin1/wRC
• The output voltage becomes more and more
  attenuated
• When wRC 1 vout/vin1/SQRT(2)

28
BODE PLOT, DECIBELS

• The log-log graph of frequency response is called
  Bode plot.

A set of two straight line segments. The
frequency at which these segments meet is a break
frequency. More precisely
29
BODE PLOT, DECIBELS

• Decibel is one tenth as large as something
  called a bel
• For example, if we have 10 times large
  signal, it is 20 dB

Gain at the transition point-3dB
30
FREQUENCY RESPONSEHIGH PASS FILTER
31
FREQUENCY RESPONSE BAND PASS FILTER

• The system transfer function

Resonant frequency
32
PARALLEL RESONANT CIRCUIT

• Impedance between A and B

Has a singularity at
Resonant frequency of the circuit
33
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 those in the pass band
  through
• relatively unchanged
• In signal processing, it removes unwanted
  parts of the
• signal, such as random noise, or to
  extract useful parts
• of the signal
• EXAMPLES
• - Low pass filter
• - High pass filter

34
Frequency Response of a Series LC circuit -
Basic Concepts

• Capacitors store energy in form of an electric
  field
• - potential (static voltage)
• Inductors store energy in form of a magnetic
  field
• - kinetic motion of electrons or
  current
• Capacitors and inductors
• - are complementary (releasing and
  storing energy)
• - when connected together, produce
  unusual results
• - will exchange energy between them
• - if connected to a battery, capacitor
  will be quickly charged,
• inductor opposing the change in
  current

http//www.ibiblio.org/obp/electricCircuits/AC/AC_
6.html
35
Frequency Response of a Series LC circuit -
Resonant Frequency

• The reactance of a capacitor
  XC1/wC
• The reactance of an inductor XLwL
• When XCXL
• There is no reactive component to Z
• The voltage source behaves as it is short
  circuited

http//www.ibiblio.org/obp/electricCircuits/AC/AC_
6.html
36
THIS WEEK IN THE LAB
37
Question 1
No
38
Question 2
39
First experiment
40
Determining CV Vo1 e-t/RC ? t ? RC
V 0.63 Vo
? t ?
gt
lt
gt
? V 0.63 Vp-p
lt
41
Question 3
What is a linear circuit?
42
Another way to calculate capacitance - from
frequency
Gain Voutputcapacitor / Vinput 1 /
(R2C2?2 1)1/2 -3 dB frequency occurs when
output power is 1/2 of input power or Gain
1/?2 so ? 1/(RC) CAN ALSO CALCULATE C FROM
DETERMINING THE FREQUENCY (f 2? ?)
CORRESPONDING TO A GAIN OF 1/?2
43
Determining -3dB frequency
On scope look at input and output signals. Vary
frequency until output voltage is equal to 1/?2
or 0.707 of the input voltage. -3.01 10 log
(1/2) output power / input power
1/2 -3.01 20 log (1/ ?2) output voltage/
input voltage 1/?2
Note - input and output frequencies are the same
but phase is shifted.
This is the frequency at which
44
Motivational Question 4
45
Low Pass Filter
Q4 is related to this graph
What is the frequency ratio at -3db?
Repeat the process for high pass filter
46
Motivational Question 5
47
Resonant Frequency of a LC Circuit
Motivational Question 6
Ringing frequency
48
RESONENT FREQUENCY
Forced ringing

• Notice
• - increase in output voltage
• - decrease in input voltage

Ringing frequency
49
Motivational Question 7
50
NEXT WEEK

• Operational amplifiers
• Wheatstone Bridge