INC 112 Basic Circuit Analysis

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INC 112 Basic Circuit Analysis

• Week 7
• Introduction to
• AC Current

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Meaning of AC Current
AC Alternating current means electric current
that change up and down
When we refer to AC current, another variable,
time (t) must be in our consideration.
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Alternating Current (AC)
Electricity which has its voltage or current
change with time.
Example We measure voltage difference between 2
points Time 1pm 2pm 3pm 4pm 5pm 6pm DC
5V 5V 5V 5V 5V 5V AC 5V 3V 2V -3V -1V 2V
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Signals

• Signal is an amount of something at different
  time, e.g. electric signal.
• Signals are mentioned is form of
• Graph
• Equation

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1st Form
Graph Voltage (or current) versus time
V (volts)
t (sec)
v(t) sin 2t
2nd Form
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V (volts)
t (sec)
v(t) 5
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Course requirement of the2nd half
Students must know voltage, current, power at any
point in the given circuits at any
time. e.g. What is the current at point
A? What is the voltage between point B and C at
2pm? What is the current at point D at t2ms?
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Periodic Signals
Periodic signals are signal that repeat
itself. Definition Signal f(t) is a periodic
signal is there is T such that f(tT) f(t) ,
for all t T is called the period, where when f
is the frequency of the signal
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Example v(t) sin 2t Period p Frequency
1/p v(tp) sin 2(tp) sin (2t2p) sin
2t (unit radian) Note sine wave signal has a
form of sin ?t where ? is the angular
velocity with unit radian/sec
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Square wave
Sine wave
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Fact Theorem (continue in Fourier series, INC
212 Signals and Systems) Any periodic signal
can be written in form of a summation of sine
waves at different frequency (multiples of the
frequency of the original signal) e.g. square
wave 1 KHz can be decomposed into a sum of sine
waves of reqeuency 1 KHz, 2 KHz, 3 KHz, 4 KHz, 5
KHz,
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Implication of Fourier Theorem
Sine wave is a basis shape of all waveform. We
will focus our study on sine wave.
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Properties of Sine Wave

• Frequency
• Amplitude
• Phase shift

These are 3 properties of sine waves.
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Frequency
volts
sec
Period 6.28, Frequency 0.1592 Hz
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Amplitude
volts
sec
Blue 1 volts Red 0.8 volts
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Phase Shift
Phase Shift 1
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Sine wave in function of time
Form v(t) Asin(?tf)
Phase (radian)
Amplitude
Frequency (rad/sec)
e.g. v(t) 3sin(8ptp/4) volts
Phase p/4 radian or 45 degree
Amplitude 3 volts
Frequency 8p rad/sec or 4 Hz
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Basic Components

• AC Voltage Source, AC Current Source
• Resistor (R)
• Inductor (L)
• Capacitor (C)

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AC Voltage SourceAC Current Source
Voltage Source
Current Source
????
Amplitude 10V Frequency 1Hz Phase shift 45
degree
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What is the voltage at t 1 sec ?
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Resistors
Same as DC circuits Ohms Law is still
usable V IR R is constant, therefore V and I
have the same shape.
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Find i(t)
Note Only amplitude changes, frequency and phase
still remain the same.
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Power in AC circuits
In AC circuits, voltage and current fluctuate.
This makes power at that time (instantaneous
power) also fluctuate. Therefore, the use of
average power (P) is prefer.
Average power can be calculated by integrating
instantaneous power within 1 period and divide it
with the period.
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Assume v(t) in form
Change variable of integration to ?
Then, find instantaneous power
We get
integrate from 0 to 2p
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Compare with power from DC voltage source
DC
AC
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Root Mean Square Value (RMS)
In DC circuits
In AC, we define Vrms and Irms for convenient in
calculating power
Note Vrms and Irms are constant, independent of
time
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3 ways to tell voltage
V (volts)
311V
t (sec)
0
V peak (Vp) 311 V V peak-to-peak (Vp-p)
622V V rms 220V
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Inductors
Inductance has a unit of Henry (H)
Inductors have V-I relationship as follows
This equation compares to Ohms law for inductors.
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Find i(t)
from
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Phase shift -90
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Phasor Diagram of an inductor
Phasor Diagram of a resistor
v
v
i
i
Power (vi cos?)/2 0
Power (vi cos?)/2 vi/2
Note No power consumed in inductors i lags v
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DC Characteristics
When stable, L acts as an electric wire.
When i(t) is constant, v(t) 0
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Capacitors
Capacitance has a unit of farad (f)
Capacitors have V-I relationship as follows
This equation compares to Ohms law for
capacitors.
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Find i(t)
Phase shift 90
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Phasor Diagram of a capacitor
Phasor Diagram of a resistor
i
v
v
i
Power (vi cos?)/2 0
Power (vi cos?)/2 vi/2
Note No power consumed in capacitors i leads v
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DC Characteristics
When stable, C acts as open circuit.
When v(t) is constant, i(t) 0
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Combination of Inductors
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Combination of Capacitors
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Linearity
Inductors and capacitors are linear components
If i(t) goes up 2 times, v(t) will also goes up 2
times according to the above equations
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Transient Responseand Forced response
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Purpose of the second half

• Know voltage or current at any given time
• Know how L/C resist changes in current/voltage.
• Know the concept of transient and forced response

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Characteristic of R, L, C

• Resistor resist current flow
• Inductor resists change of current
• Capacitor resists change of voltage

L and C have dynamic
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Voltage
2V
1V
time
Current
2A
1A
time
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Voltage
2V
1V
time
Current
2A
1A
time
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Unit Step Input and Switches
Voltage
1V
0V
time
This kind of source is frequently used in circuit
analysis. Step input change suddenly from x
volts to y volts Unit-step input change
suddenly from 0 volts to 1 volt at t0
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This kind of input is normal because it come from
on-off switches.
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PSPICE Example

• All R circuit, change R value
• RL circuit, change L
• RC circuit, change C

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Pendulum Example
I am holding a ball with a rope attached, what is
the movement of the ball if I move my hand to
another point?

• Movements
• Oscillation
• Forced position change

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• Transient Response or Natural Response
• (e.g. oscillation, position change temporarily)
• Fade over time
• Resist changes
• Forced Response
• (e.g. position change permanently)
• Follows input
• Independent of time passed

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Natural response at different time
Forced response
Mechanical systems are similar to electrical
system
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connect
i(t)
Changing
Stable
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Transient Response

• RL Circuit
• RC Circuit
• RLC Circuit