Transcript accircuits

AC Circuits
AC Sources
•Often, electrical signals look like sines or cosines V  Vmax sin t 
•AC power, Radio/TV signals, Audio
V  Vmax cos t 
•Sine and cosine look nearly identical
•They are related by a phase shift
•Cosine wave is advanced by /2 (90 degrees) compared to sine
•We will always treat source as sine wave
cos t   sin t  12  
Frequency, period, angular frequency related
f 1 T
  2 f
•This symbol denotes an arbitrary AC source
RMS Voltage
•There are two ways to describe the amplitude
•Maximum voltage Vmax is an overstatement
V  Vmax sin t 
•Average voltage is zero
•Root-mean-square (RMS) voltage is probably the best way
•Plot (V)2, find average value, take square root
•We can do something similar with current
Vrms 
V 2 
2
Vmax
sin 2 t 
House current (US)
is at f = 60 Hz and
Vrms = 120 V
Vmax = 170 V
Vrms
I rms

Vmax

2
I max

2
1
2
2
Vmax
Amplitude, Frequency, and Phase Shift
We will describe any sort of wave in terms of three quantities:
•The amplitude A is how big it gets
•To determine it graphically, measure the peak of the wave
•The frequency is how many times it repeats per second
•To determine it graphically, measure the period T
f 1 T
•Frequency f = 1/T and angular frequency  = 2f
•The phase shift  is how much it is shifted earlier/later
compared to basic sine wave
•Let t0 be when it crosses the origin while rising
•The phase shift is t0 (radians)
t0  0.005 s
T  0.02 s
  100  0.005
f  50 Hz
  100 s 1
  12 
A sin t   
A  1.60 V
  2 f
  t0
Our Goal
•Feed AC source through an arbitrary circuit
•Resistors, capacitors, inductors, or combinations of them
•We will always assume the incoming wave has zero phase shift
V  Vmax sin t 
•We want to find current as a function of time
?
Vmax
I  I max sin t   
•For these components, can show angular frequency  is the same
•We still need to find amplitude Imax and phase shift  for current
•Also want instantaneous power P and average powerP consumed
P  I V
•Generally, maximum current will be proportional to maximum voltage
•Call the ratio the impedance, Z
I max  Vmax Z
 
Degrees vs. Radians
•All my calculations will be done in radians
•Degrees are very commonly used as well
•But the formulas look different
I  I max sin t     I max sin  2 ft   
I  I max sin  360 ft   
  t0
  360 ft0
•Probably best to set your calculator on radians and leave it there
Resistors
V  Vmax sin t 
I  I max sin t   
•Can find the current from Ohm’s Law
V Vmax
I

sin t 
R
R
I max  Vmax R
 0
R = 1.4 k
Vmax  170 V
f  60 Hz
I rms  Vrms R
•The current is in phase
with the voltage
Impedance vector:
•A vector showing relationship between voltage and current
•Length, R is the ratio
•Direction is to the right,
1.4 k
representing the phase shift of zero
Voltage
Current
Power in Resistors
•We want to know
•Instantaneous power
•Average Power
2
2
2


V
I
sin

t

RI
sin
 
t 
P  I V
max max
max
2
P  RI max
sin 2 t 
2
P  12 RI max
2
P  RI rms
Vmax  170 V
f  60 Hz
R = 1.4 k
Capacitors
V  Vmax sin t 
I  I max sin t   
C= 2.0 F
•Charge of capacitor is proportional to voltage Q  C V
•Current is derivative of charge
Vmax  170 V
dQ
d V
 CVmax cos t 
I
C
f  60 Hz
dt
dt
 CVmax sin t  12  
•Current leads voltage by /2
•We say there is a –/2 phase shift:
1.3 k
Impedance vector:
•Define the impedance* for a capacitor as:
1
X


C
•Make a vector out of it

Vmax
C
2
I max 
•Length XC
XC
*We will ignore the
•Pointing down for  = –½
1



2
term “reactance”
Power in Capacitors
C= 2.0 F
We want to know
•Instantaneous Power
•Average Power
Vmax  170 V
P  I V  Vmax I max sin t  cos t 
•Power flows into and out of capacitor
•No net power is consumed by capacitor
P 0
Only resistors contribute to the
average power P consumed
f  60 Hz
Capacitors and Resistors Combined
•Capacitors and resistors both limit the current – they both have
impedance
I max  Vmax R
•Resistors: same impedance at all frequencies
I max  Vmax X C
•Capacitors: more impedance at low frequencies
X C  1 C
Inductors
•Voltage is proportional to change in current
dI
E  L
 Vmax sin t 
dt
•Integrate this equation  LI  Vmax cos t 

Vmax
Vmax
sin t  12  
I 
cos t  
L
L
L= 4.0 H
Vmax  170 V
f  60 Hz
Impedance vector:
•Define the impedance for an inductor as:
•Make a vector out of it X   L
L
•Length XL
•Pointing up for  = +½
1.5 k
•Current lags voltage by /2
•We say there is a +/2 phase shift
2

I max  Vmax X L
  12 
Impedance Table
Resistor
Impedance
R
Phase
0
Vector
Direction
right
Capacitor
XC 
1
C
Inductor
X L  L

 12 
1
2
down
up
I max  Vmax R
I max  Vmax X C
I max  Vmax X L
Adding Impedances Graphically
•Suppose we have 2+ items in series
•Resistors, Capacitors, Inductors
•We can get the total impedance and phase shift by
adding the impedances graphically
The impedance and phase shift of two
components in series can be found by adding
the vector sum of the two separate impedances
•Each impedance is represented by a 1.4 k vector pointing to
the right
•The length of the combination is 2.8 k
•The total impedance is denoted Z
•The total arrow is to the right, so phase shift is 0
1.4 k
1.4 k
Z = 2.8 k
1.4 k
1.4 k
60 Hz
170 V
I max
Vmax

Z
I rms
Vrms

Z
Z  2.8 k
 0
Adding Impedances Graphically (2)
Z  X C2  R2  1.42  1.32 k  1.9 k
•The current is then the voltage over the impedance
Vmax
170 V
I max  88.0 mA
I max 

Z
1930 
•The phase can be found from the diagram
  0.759 rad
1330
tan   
1400
I  I max sin t   
1.4 k
1.3 k
We can add different types of components as well
•The resistor is 1.4 k to the right
1.4 k
•The capacitor is 1.3 k down

•The total is 1.9 k down-right
1
XC 
 1330 
C
2.0 F
60 Hz
170 V
RLC AC Circuits
Z  R   X L  XC 
2
I max 
Vmax
Z
2
 X L  XC 

R


  tan 1 
I  I max sin t   
R
f
Vmax
R

XC
•For this rather general circuit, find
•Current
•Impedance
•Average power
•Phase shift
1. Find the angular frequency    2 f
2. Find the impedance of the capacitor and inductor
3. Find the total impedance Z
1
XC 
X L  L
4. Find the phase shift 
C
5. Find the current
6. Find the average power consumed
C
XL
L
Power in RLC AC Circuits
6. Find the average power consumed
L
I  I max sin t   
I max
Vmax

Z
C
Only resistors
contribute to the
average power
P consumed
2
PR  I 2 R  RI max
sin 2 t   
2
2
P  RI max
sin 2 t     12 RI max
I rms
R
f
Vmax
I max

2
2
P  RI rms
•Most power is delivered to resistor when Imax is maximized
•When Impedance is minimized
•The “resistor” might well represent some useful device
•Like a speaker for a stereo
P 
R
2

V
rms
Z2
Frequency and RLC Circuits
L
•Impedance tends to be dominated by
whichever component has largest impedance
•At low frequencies, that’s the capacitor
•At high frequencies, that’s the inductor
•At intermediate, that’s the resistor
•If the circuit includes a capacitor, it blocks low frequencies
•If the circuit includes an inductor, it blocks high frequencies
High pass filter
Low pass filter
C
R
f
Vmax
A Sample Circuit
1000 F
0.1 mH
10 
X C  1 C
  2 f
X L  L
•What frequencies make it through the capacitor?
1
1

f  16 Hz
R  XC R 
C
RC
•What frequencies make it through the inductor?
R  XL
R
f  16 kHz

L
•These inequalities compatible if:
R  L
1
R

RC L
L  R 2C
f
Vmax = 5 V
A Sample Circuit (2)
X C  1 C
X L  L
1000 F
0.1 mH
10 
f
  2 f
•At low frequencies, blocked by capacitor
•At high frequencies, blocked by inductor
•At intermediate, power goes to resistor
•Frequencies from about 16 Hz–16 kHz get through
•Close to perfect for an audio system
What happens if L > R2C ?
Vmax = 5 V
Power
Phase
Shift
The Narrow Band Filter
X C  1 C
X L  L
  2 f
1.54 H 2.1 pF 2.0 
f
Vmax = 1 mV
•At resonant frequency, capacitor and
inductor cancel
•Perfect for picking up WFDD
Types of RLC Circuits
High Pass Filter
•Lets frequencies through if  > 1/RC
Low Pass Filter
•Lets frequencies through if  < R/L
RLC circuit
•If R2 > L/C, it is a combination of Low and
High pass filter
0 
•If R2 < L/C it only lets a narrow
range of frequencies through
•The smaller R2C/L, the narrower it is
1
CL
Comments on Phase Shifts
V  Vmax sin t 
I  I max sin t   
•The phase shift represents how the timing of the
current compares to the timing of the voltage
•When it is positive, the current lags the voltage
•It rises/falls/peaks later
•When it is negative, the current leads the voltage
•It rises/falls/peaks earlier
 X L  XC 
  tan 

R


1
  t0
Transformers
N1 turns
N2 turns
B
•Let two inductors share the same volume
•You can (should) give them an iron
core too
•The EMF’s can be calculated from the flux
E1  
E2  
•The magnetic flux must be changing
•Only works for AC
d  B1
dt
d B2
dt
  N1
d B
dt
dB
  N2
dt
E1 E2

N1 N 2
What Transformers are Good For
120 V V2

500
5000
V2 = ?
N1 =500
V1 = 120 V
E1 E2

N1 N 2
N2 =5000
•Their main purpose in life is to change the voltage
A 120 V AC source is fed into a transformer,
with N1 = 500 turns on the primary coil, and
N2 = 5000 turns on the secondary. What is
the voltage out of the transformer?
•Voltage can increases, does that mean power increases?
•When you increase voltage, you decrease current
•In an ideal transformer, the product is conserved
5000 120 V 
 1200 V
V2 
500
P  I V
I1V1  I 2 V2
Realistic transformers
are 80-95% efficient
Transformers and Power Transmission
Generator
500 V
Transmission
Line 100 kV
House
Current 120 V
•Why transmit at 10 kV, instead of 500 V or 120 V?
•Transmission wires are long – they have a lot of resistance
•By using a step up transformer, we increase the voltage I  V1 I
2
1
V2
and decrease the current
2
•Power lost for a resistor is:
 V1 
2
2
•You then step it down so you
P = I 2rms R  I1rms R 


V
 2
don’t kill the customer
Power Supplies
To devices
120 V
AC
21 V
AC
20 V
ripply
DC
20 V
smooth
DC
•What if we need a different voltage for a specific device?
•Use a transformer
•What if we want direct current?
•A diode is a device that only lets current through one direction
•What if we don’t like the ripples
•Capacitors store charge from cycle to cycle