Download Lecture 7

INC 112 Basic Circuit Analysis
Week 7
Introduction to
AC Current
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.
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
Signals
Signal is an amount of something at different time,
e.g. electric signal.
Signals are mentioned is form of
1. Graph
2. Equation
1st Form
Graph Voltage (or current) versus time
V (volts)
t (sec)
2nd Form
v(t) = sin 2t
DC voltage
V (volts)
t (sec)
v(t) = 5
Course requirement of the
2nd 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 t=2ms?
Periodic Signals
Periodic signals are signal that repeat itself.
Definition
Signal f(t) is a periodic signal is there is T such that
f(t+T) = f(t) , for all t
T is called the period, where
1
T
f
when f is the frequency of the signal
Example:
v(t) = sin 2t
Period = π
Frequency = 1/π
v(t+π) = sin 2(t+π) = sin (2t+2π) = sin 2t
(unit: radian)
Note: sine wave signal has a form of sin ωt
where ω is the angular velocity with unit radian/sec
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
Sine wave
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
0
0.01
0.02
0.03
0.04
0.05
0.06
Square wave
0.07
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0.2
-0.2
0
-0.4
-0.2
-0.6
-0.8
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.01
-0.4
-5
-4
-3
-2
-1
0
1
2
3
4
5
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, …
8 sin( t  0.3)  3 sin( 2t  0.5)  1sin( 3t  0.2)  0.5 sin( 4t  1.2)  .....
Implication of Fourier Theorem
Sine wave is a basis shape of all waveform.
We will focus our study on sine wave.
Properties of Sine Wave
1. Frequency
2. Amplitude
3. Phase shift
These are 3 properties of sine waves.
Frequency
period
volts
1
0.8
0.6
0.4
1
T
f
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
8
9
10
Period ≈ 6.28, Frequency = 0.1592 Hz
sec
Amplitude
volts
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
Blue 1 volts
Red 0.8 volts
7
8
9
10
sec
Phase Shift
Phase
Shift =
1
Period=6.28
1
0.8
0.6
0.4
0.2
yblue  sin( t )
0
yred  0.8 sin( t  1)
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
8
Red leads blue 57.3 degree (1 radian)
9
10
1

 360  57.3
6.28
Sine wave in function of time
Form:
v(t) = Asin(ωt+φ)
Phase (radian)
Amplitude
Frequency (rad/sec)
e.g.
Amplitude
3 volts
v(t) = 3sin(8πt+π/4) volts
Frequency
8π rad/sec or 4 Hz
Phase
π/4 radian or 45 degree
Basic Components
• AC Voltage Source, AC Current Source
• Resistor (R)
• Inductor (L)
• Capacitor (C)
AC Voltage Source
AC Current Source
Voltage Source
Current Source
+
AC
AC
+
-
AC
AC
-
เช่ น
+
AC
10sin(2πt + π/4)
-
Amplitude = 10V
Frequency = 1Hz
Phase shift = 45 degree
+
AC
10sin(2πt + π/4)
What is the voltage at t =1 sec ?
-
v(1)  10 sin( 2 (1)   / 4)
 10 sin( 2   / 4)
 10 sin(  / 4)  7.07volts
Resistors
Same as DC circuits
Ohm’s Law is still usable
V = IR
R is constant, therefore V and I have the same shape.
i(t)
+
AC
10sin(2πt + π/4)
2Ω
Find i(t)
-
v(t )  i (t ) R
10 sin( 2t   / 4)  i(t )  2
10 sin( 2t   / 4)
i (t ) 
2
i (t )  5 sin( 2t   / 4)
Note: Only amplitude changes, frequency and phase still remain the same.
Power in AC circuits
2
v
P  i2R 
R
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.
Assume v(t) in form
v(t )  A sin( t )
v  A sin 
We get
Then, find instantaneous power
v 2 A2 sin 2 
p

R
R
1
P
2
2

2
0
Change variable of integration to θ
integrate from 0 to 2π
A2
sin 2  d
R
A2
A2
2

sin  d 

2R 0
2
2
1  cos 2
0 2 d
  2
A2  1
sin 2 
A2


 

2R  2
4  0
2R
Compare with power from DC voltage source
DC
AC
i(t)
i(t)
+
-
+
R
A
Asin(ωt+Ф)
AC
-
A2
P
R
A2
P
2R
R
Root Mean Square Value (RMS)
V2
PI R
R
2
In DC circuits
In AC, we define Vrms and Irms for convenient in calculating power
P  I rms
2
V
R  rms
R
2
For sine wave Asin(ωt+φ)
Note: Vrms and Irms are constant,
independent of time
A
rms _ value 
2
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
Inductors
+
v(t)
-
Inductance has a unit of Henry (H)
i(t)
Inductors have V-I relationship as follows
di (t )
v(t )  L
dt
This equation compares to
Ohm’s law for inductors.
i(t)
+
Asin(ωt)
AC
L
Find i(t)
-
from
di (t )
v(t )  L
dt
1
1
i (t )   v(t )dt   A sin tdt
L
L
A
A   cos t 
  sin tdt  

L
L  
A
A


( cos t ) 
(sin t  )
L
L
2
i(t)
i(t)
+
+
Asin(ωt)
AC
L
Asin(ωt)
AC
R
-
-
A

i (t ) 
(sin t  )
L
2
A
i (t )  (sin t )
R
ωL is called impedance (equivalent resistance)
Phase shift -90
Phasor Diagram of an inductor
Phasor Diagram of a resistor
v
v
i
i
Power = (vi cosθ)/2 = 0
Power = (vi cosθ)/2 = vi/2
Note: No power consumed in inductors
i lags v
DC Characteristics
1V
i(t)
i(t)
1Ω
1Ω
L
1V
When stable, L acts as an electric wire.
di (t )
v(t )  L
dt
When i(t) is constant, v(t) = 0
Capacitors
+
v(t)
-
Capacitance has a unit of farad (f)
i(t)
Capacitors have V-I relationship as follows
dv(t )
i (t )  C
dt
This equation compares to
Ohm’s law for capacitors.
i(t)
+
Asin(ωt)
AC
C
Find i(t)
-
dv(t )
d ( A sin t )
i (t )  C
C
dt
dt
 AC (cos t )
A


sin( t  )
2
 1 


 C 
Phase shift +90
Impedance (equivalent resistance)
Phasor Diagram of a capacitor
Phasor Diagram of a resistor
i
v
v
i
Power = (vi cosθ)/2 = 0
Power = (vi cosθ)/2 = vi/2
Note: No power consumed in capacitors
i leads v
DC Characteristics
1V
i(t)
i(t)
1Ω
1Ω
C
1V
When stable, C acts as open circuit.
dv(t )
i (t )  C
dt
When v(t) is constant, i(t) = 0
Combination of Inductors
L2
L1
L1+L2
Ltotal  L1  L2
L1
L1L2/(L1+L2)
L2
1
Ltotal

1 1

L1 L2
Combination of Capacitors
C1
C2
C1C2/(C1+C2)
1
Ctotal

1
1

C1 C2
C1
C1+C2
C2
Ctotal  C1  C2
Linearity
Inductors and capacitors are linear components
di (t )
v(t )  L
dt
dv(t )
i (t )  C
dt
If i(t) goes up 2 times, v(t) will also goes up 2 times
according to the above equations
Transient Response
and Forced response
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
Characteristic of R, L, C
• Resistor resist current flow
• Inductor resists change of current
• Capacitor resists change of voltage
L and C have “dynamic”
1V
I
I
1Ω
1Ω
2V
I = 1A
I = 2A
Voltage source change from 1V to 2V immediately
Does the current change immediately too?
Voltage
AC voltage
2V
1V
time
Current
2A
1A
time
1V
I
I
1Ω
1Ω
I = 1A
L
2V
I = 2A
Voltage source change from 1V to 2V immediately
Does the current change immediately too?
L
Voltage
AC voltage
2V
1V
time
Current
2A
1A
Forced Response
time
Transient Response + Forced Response
Unit Step Input and Switches
Voltage
1V
0V
time
This kind of source is frequently used in circuit analysis.
AC
u(t)
Step input = change suddenly from x volts to y volts
Unit-step input = change suddenly from 0 volts to 1 volt at t=0
This kind of input is normal because it come from on-off switches.
PSPICE Example
• All R circuit, change R value
• RL circuit, change L
• RC circuit, change C
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
1. Oscillation
2. Forced position change
• 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
Forced response
Natural response
at different time
Mechanical systems are similar to electrical system
i(t)
connect
R
L
AC
i(t)
Changing
Stable
Transient Analysis
Phasor Analysis
Transient Response
• RL Circuit
First-order differential equation
• RC Circuit
• RLC Circuit
Second-order differential equation