Post on 06-Jan-2016
description
Class 34Today we will:•learn about inductors and inductance•learn how to add inductors in series and parallel•learn how inductors store energy•learn how magnetic fields store energy•learn about simple LR circuits
Inductors
An inductor is a coil of wire placed in a circuit. Inductors are usually solenoidal or toroidal windings.
V
R
L
An Inductor in a DC Circuit
In a DC circuit an inductor behaves essentially like a long piece of wire – except the solenoid produces a magnetic field and the wire has a resistance.
V
R
L
An Inductor in a DC Circuit
If we bring a magnet near the inductor, an induced EMF is produced that alters the voltages and current in the circuit.
V
R
L
N
An Inductor in an AC Circuit
In an AC circuit, the current is continually changing. Hence the magnetic field in the inductor is continually changing.
R
L
An Inductor in an AC Circuit
If the magnetic field in the inductor changes, an EMF is produced. Since the inductor’s own field causes an induced EMF, this is called “self inductance.”
R
L
The Inductance of a Solenoid
We can easily calculate the EMF of a solenoidal inductor.
dt
diAnV
dt
dinAN
dt
dNV
niABA
niB
B
B
20
0
0
0
The Inductance of a Solenoid
We define inductance by the equation:
dt
diLV
dt
diAn
dt
dNV B 2
0
So, for a solenoid:
AnL 20
Units of Inductance
Inductance is in units of henrys (H).
dt
diLV
ssA
VH 1
/11
Voltages of Circuit Elements
Note that L is similar to R in these equations.
2
2
1
dt
qdL
dt
diLV
qC
V
dt
dqRRiV
Inductance in Series and Parallel
In series:
In parallel:
21
21
21
LLLdt
diL
dt
diL
dt
diL
VVV
21
21
21
21
111
LLL
L
V
L
V
L
Vdt
di
dt
di
dt
di
iii
Energy in an Inductor
Take a circuit with only an EMF and an inductor.
dt
diLVL
Energy in an Inductor
dt
diL
.2
1 2Lidt
ddt
diiLiP
Energy in an Inductor
From this we can find the energy:
dt
dULi
dt
dP 2
2
1
Energy in an Inductor
From this we can find the energy:
2
2
2
12
1
LiU
dt
dULi
dt
dP
Energy in a Capacitor
Recall that the energy stored in a capacitor is:
2
2
1CVU
Energy Density in a Magnetic Field
Since
2
0
20
0
2
1
2
1
Bvol
Uu
volniU
niB 0
Energy Density in an Electric Field
Recall that in an electric field:
202
1E
vol
Uu
LR Circuits
•An LR circuit consists of an inductor, a resistor, and possibly a battery. •Inductors oppose change in circuits. V
R
L
LR Circuits
•In steady state, inductors act like wires, but they significantly affect circuits when switches are opened and closed.
V
R
L
LR Circuits•When the switch is closed in this circuit, the inductor opposes current flow. Initially, it is able to keep any current from flowing.
V
R
L
LR Circuits
•Current then increases until it reaches a maximum value,
V
R
L
)(ti
t
RV /
RVi /
LR Circuits
•Current in an LR circuit is very similar to charge in an RC circuit.
V
R
L
)(ti
t
RV /
LR Circuits
eR
V 11
t
RV /)(ti
LR Circuits
•Think of the total current being a combination of the induced battery’s current and the induced current.•The current from the batter is I=V/R, and it is constant.
LR Circuits
•The induced current is initially I= – V/R.•The induced current drops to zero as time increases.
LR Circuits•If the inductor is large, it is more effective at making the induced current, so it takes a longer time for the induced current to decrease.•If the resistor is large, it is more effective at decreasing the induced current, so it takes a shorter time for the current to decrease.
LR Circuits
•We’ll show next time that the inductive time constant is:
R
L
Class 35Today we will:•learn more about LR circuits•learn the LR time constant•learn about LC circuits and oscillation•learn about phase angles
LR Circuits
•An LR circuit consists of an inductor, a resistor, and possibly a battery. •Inductors oppose change in circuits. V
R
L
LR Circuits
eR
V 11
t
RV /)(ti
The Definition of Inductance
We define inductance by the equation:
dt
diLV
LR Circuits
•We’ll show that the inductive time constant is:
R
L
Kirchoff’s Loop Law
•The voltage around the loop at any given time must be zero.•It’s important to be careful of signs.
V
R
L
Kirchoff’s Loop Law
•As the current increases, the inductor opposes the increase. It produces an EMF in opposition to the battery.
V
R
L
Kirchoff’s Loop Law
V
R
L
0 iRdt
diLV
Kirchoff’s Loop Law
V
R
L
0 iRdt
diLV
0
0
iRdt
diLV
dt
di
Kirchoff’s Loop Law
0
0
iRdt
diLV
dt
di
R
L
VeeR
VL
VeVeR
VLV
eVeR
VLV
eR
Vti
tt
tt
tt
t
0
0
01
1)(
//
//
//
/
Another LR Circuit
•We can also make an LR circuit in which the current decreases.•At t=0, the switch is moved from 1 to 2.
V
R
L
2
1
Another LR Circuit
•The current is initially RVi /
/)( teR
Vti
V
R
L
2
1
Kirchoff’s Loop Law
•We apply Kirchoff’s Laws again.•This time the induced EMF pushes charge forward.
V
R
L
2
1
Kirchoff’s Loop Law
0 iRdt
diL
V
R
L
2
1
Kirchoff’s Loop Law
0 iRdt
diL
0
0
iRdt
diL
dt
di
V
R
L
2
1
Kirchoff’s Loop Law
0 iRdt
diL
R
L
eR
VRe
R
VL
eR
Vti
tt
t
0
)(
//
/
V
R
L
2
1
Kirchoff’s Loop Law
0 iRdt
diL
R
L
eR
VRe
R
VL
eR
Vti
tt
t
0
)(
//
/
)(ti
t
V
R
L
2
1
Kirchoff’s Loop Law
)(ti
t
eR
V 1
RV /
LC Circuits
•An LC circuit consists of an inductor and a capacitor.
L
C
LC Circuits•Initially the capacitor is charged.•No current flows as the inductor initially prevents it.•Energy is in the electric field of the capacitor.
L
C
LC Circuits•The charge reduces.•Current increases.•Energy is shared by the capacitor and the inductor.
L
C
i
LC Circuits•The charge goes to zero.•Current increases to a maximum value.•Energy is in the magnetic field of the inductor.
LC
i
LC Circuits•The inductor keeps current flowing.•The capacitor begins charging the opposite way.•The capacitor and inductor share the energy.
LC
i
LC Circuits
•The capacitor becomes fully charged.•Current stops.•All the energy is in the capacitor.
LC
LC Circuits•The capacitor begins to discharge.•Current flows counterclockwise.•The capacitor and inductor share the energy.
LC
i
LC Circuits
•Without energy loss, current would continue to oscillate forever.
LC
i
Kirchoff’s Loop Law
•First, we’ll be really sloppy about signs.
L
C
i
Kirchoff’s Loop Law
•First, we’ll be really sloppy about signs.•Know this!
L
C
i
Kirchoff’s Loop Law
L
C
iktkt
Lc
BeAetq
tqLCdt
qd
dt
qdL
dt
diL
C
q
VV
)(
)(1
2
2
2
2
Kirchoff’s Loop Law
L
C
iktkt
Lc
BeAetq
tqLCdt
qd
dt
qdL
dt
diL
C
q
VV
)(
)(1
2
2
2
2 But that’s not oscillatory!
Kirchoff’s Loop Law
L
C
i
tBtAtq
tqLCdt
qd
cossin)(
)(1
2
2
Kirchoff’s Loop Law
tCVtq cos)( 0
If the charge is maximum Q=CV0 at t=0, then we have:
tBtAtq cossin)(
Oscillating Frequency
LC
tCVLC
tCV
tCVtq
tqLCdt
qd
1
cos1
cos
cos)(
)(1
002
0
2
2
Being Careful about Signs
•Whenever we have AC circuits, we have to be really careful about signs. Signs tell us directions, and directions can be confusing.
Resistors
•Choose a direction for positive current.
i
Resistors
•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.
i 0iRVR
i
Resistors
•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.
i 0iRVR
i
positive voltage
Resistors
•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.•We call V negative when the voltage pushes current in the positive direction.
i 0iRVR
i
Resistors
•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.•We call V negative when the voltage pushes current in the positive direction.
i 0iRVR
i
negative voltage
Resistors
t
)(ti
)(tVR
•The voltage and current are in phase.•The phase angle is 0°.
Capacitors
•Choose a direction for positive current.
i
Capacitors
•Choose a direction for positive current. •If the left plate is positive, q>0 and V>0.•If the left plate is negative, q<0 and V<0.
i
i voltage positive
voltage negative
Capacitors
•We plot charge and voltage as a function of time.
t
)(tVC
)(tq
Capacitors
i
i q becoming more positive
•If current is flowing right, dq/dt>0.•If current is flowing left, dq/dt<0.
i
i
q becoming more positive
Capacitors
i
i q becoming more positive
•If current is flowing right, dq/dt>0.•If current is flowing left, dq/dt<0.
i
i
q becoming more positive
dt
dqi
Capacitors
•When q (or V) increases, the current is positive.•When q (or V) decreases, the current is negative.
t
)(tq
)(tVC
)(ti
Capacitors
•When q (or V) increases, the current is positive.•When q (or V) decreases, the current is negative.
t
)(tq
)(tVC
)(ti
Capacitors
t
)(ti
•The voltage lags the current.•The phase angle is −90°.
−90°
)(tVC
Inductors
•Choose a direction for positive current.
i
Inductors
•Choose a direction for positive current.•We need to consider current in both directions.
•In each direction, we need to consider current
increasing and decreasing.
i
Inductors
•In each direction, we need to consider current increasing and decreasing.
t)(tiincreasingi
dt
di
i
0
0
0iinduced
Inductors
• The induced EMF opposes the current.• The voltage positive because it is pushing current in the negative direction.
i 0
dt
diLVL
i 0dt
di
positive voltage
Inductors
•In each direction, we need to consider current increasing and decreasing.
t)(ti
decreasingidt
di
i
0
0
0iinduced
Inductors
•The induced EMF aids the current.•The voltage is positive because it is pushing current in the negative direction.
i 0
dt
diLVL
i 0dt
di
positive voltage
Inductors
•In each direction, we need to consider current increasing and decreasing.
t)(tidecreasingi
dt
di
i
0
0
0iinduced
Inductors
•The induced EMF aids the current.•The voltage is negative because it is pushing current in the positive direction.
i 0
dt
diLVL
i 0dt
di
negative voltage
Inductors
•In each direction, we need to consider current increasing and decreasing.
t)(ti
increasingidt
di
i
0
0
0iinduced
Inductors
•The induced EMF opposes the current.•The voltage is negative because it is pushing current in the positive direction.
i 0
dt
diLVL
i 0dt
di
negative voltage
Inductors
•The voltage is negative when di/dt is negative.•The voltage is positive when di/dt is positive.
t
)(ti
)(tVL
Inductors
•The voltage is negative when di/dt is negative.•The voltage is positive when di/dt is positive.
t
)(ti
)(tVL
Inductors
t
)(ti
•The voltage leads the current.•The phase angle is +90°.
)(tVL
+90°
Voltage and Current in an LC Circuit
t
)(ti
+90°
)(tVC)(tVL
−90°
Class 36Today we will:•learn about phasors•define capacitive and inductive reactance•learn about impedance•apply Kirchoff’s laws to AC circuits
Voltage and Current in AC Circuits
ELI the ICE MAN
t
)(ti
+90°
)(tVC)(tVL
−90°
Voltage and Current in AC Circuits
ELI the ICE MAN
t
)(ti
+90°
)(tVC)(tVL
−90°
In an inductor (L) EMF leads I ELI
Voltage and Current in AC Circuits
ELI the ICE MAN
t
)(ti
+90°
)(tVC)(tVL
−90°
In a capacitor (C) I leads EMF ICE
Representing an AC Current
•If a current varies sinusoidally, we usually represent it graphically.
t
)(ti0i
Representing an AC Current
•But we can also represent it with an arrow that goes up and down in time.
Representing an AC Current•An arrow that goes up and down sinusoidally is just the y component of a vector that rotates at a constant angular frequency.•The length of the arrow is and the angle of the vector is .
0it
0i
Phasors•A phasor is a vector that represents a sinusoidal function. Its magnitude is the amplitude of the sine wave (it’s maximum value) and its angle is the phase angle (the argument of the sine function).
0i
sin)( 0iti
sin0i
Phasors
•Phasors are useful because we can add two sine waves with the same frequency by adding their phasors.
ytiti
xtitiiiadding
ytixtii
ytixtiiphasors
tiitiifunctions
ˆsinsin
ˆcoscos:
ˆsinˆcos
ˆsinˆcos:
sin,sin:
2010
201021
20202
10101
202101
•The y component of the sum of the phasors equals the sum of the functions.
Phasors
•Doesn’t this just make things harder???
Phasors
•Doesn’t this just make things harder???
•No – because phasor diagrams are a lot easier than algebra and trig identities.
•Let’s look at some examples…
Resistors and Phasors•First, we wish to find the maximum voltage across a resistor in terms of the maximum current.•Note these correspond to the length of the phasors.•All we need for this is Ohm’s Law: RiVR 00
Resistors and Phasors
•Now we can make a voltage phasor and a current phasor for the resistor.•The voltage is in phase with the current, so the phasors point in the same way.
iRV2Rif
Resistor Phasors in Time
Inductors and Phasors
•As before, we wish to find the maximum voltage across an inductor in terms of the maximum current.
00
0
0
)cos()(
)sin()(
LiVSo
tLidt
diLtVThen
titiLet
L
L
Inductive Reactance
•Comparing this result to Ohm’s Law, we see that has a role that is similar to resistance – it relates voltage to current.•We define this to be the “inductive reactance” and give it a symbol .
LX
XiV
L
LL
00
L
LX
Inductive Reactance
•This makes sense, because an inductor offers more opposition to the current if the frequencey is large and if the inductance is large.
LX
XiV
L
LL
00
Inductors and Phasors
•We again want a voltage phasor and a current phasor.•The voltage leads the current by 90°, so the phasors are:
i
2LXif
LV
Inductors and Phasors
Capacitors and Phasors
•We find the maximum voltage across a capacitor in terms of the maximum current.
00
0
0
1
)cos(11)(
)(
)sin()(
iC
VSo
dt
dqias
tiC
idtCC
tqtVThen
titiLet
C
C
Capacitive Reactance
•Comparing this result to Ohm’s Law, we see that is similar to resistance.•We define this to be the “capacitive reactance” and give it a symbol .
CX
XiV
C
CC
1
00
C/1
CX
Capacitive Reactance
•This makes sense because a capacitor offers more resistance to current flow if it builds up a big charge. If the frequency is large or the capacitance is large, it is hard to build up much charge.
CX
XiV
C
CC
1
00
Capacitors and Phasors
•We again want a voltage phasor and a current phasor.•The voltage lags the current by 90°, so the phasors are:
i
2CXif
CV
Capacitors and Phasors
Circuit Rules for AC Circuits
If two circuit elements are in parallel:
Circuit Rules for AC Circuits
If two circuit elements are in parallel:1) their voltage phasors are the same.
CL VV
Li
Ci
Circuit Rules for AC Circuits
If two circuit elements are in parallel:1) their voltage phasors are the same.2) their current phasors add to give the
total current.
CL VV
Li
Ci
toti
Circuit Rules for AC Circuits
If two circuit elements are in series:
Circuit Rules for AC Circuits
If two circuit elements are in series:1) their current phasors are the same.
LV
CL ii
CV
Circuit Rules for AC Circuits
If two circuit elements are in series:1) their current phasors are the same.2) their voltage phasors add to give the
total voltage.
LV
CL ii
totV
CV
An Example
We know ε0, ω, the resistances, the capacitance, and the inductance.
1R
2R
L
C
An Example
Label the currents.
1R
2R
L
C
2i
3i
1i
An Example
We don’t know the currents, but we know the relationships between current and voltage.
1R
2R
L
C
2i
3i
1i
An Example
1R
2R
L
C
11010 RiVR
2i
3i
1i
An Example
1R
2R
L
C
22020 RiVR
2i
3i
1i
An Example
1R
2R
L
C
LL XiV 200
2i
3i
1i
An Example
1R
2R
L
C
CC XiV 300
2i
3i
1i
An Example
LRLR ZiV 200
We can combine circuit elements and calculatethe total impedance of these elements.
1R
LRZ
C
2i
3i
1i
An Example
1i
We can combine all the circuit elements into asingle impedance.
Z
An Example
1i
We can combine all the circuit elements into asingle impedance.
Z
Zi100
An Example
We’ll start with the inductor.
1R
2R
L
C
2i
3i
1i
LiXiV LL 20200
An Example
Draw the current in an arbitrary direction – along the x-axis will do – with an arbitrary length.
1R
2R
L
C
2i
3i
1i
LiXiV LL 20200
2i
An Example
We know the direction of the voltage phasor from “ELI.”
1R
2R
L
C
2i
3i
1i
LiXiV LL 20200
2i
LV
An Example
We know the magnitude of the voltage phasor is ωL times i20.
1R
2R
L
C
2i
3i
1i
LiXiV LL 20200
2i
LV
An Example
Now we add the resistor.
1R
2R
L
C
2i
3i
1i
2200 RiVR
2i
LV
An Example
The voltage phasor is in the same direction as the current. It’s magnitude comes from:
1R
2R
L
C
2i
3i
1i
2200 RiVR
2i
LV
RV
An Example
We add the voltages of the two series elements.
1R
2R
L
C
2i
3i
1i
2200 RiVR
2i
LV
RV
LRV
An Example
Let’s look at the math.
2i
LV
RV
LRV
An Example
The LR voltage phasor has two components:
LLR
LRLR
XiVRiV
yVxVV
2002200
00
,
ˆˆ
2i
LV
RV
LRV
An Example
The length of the LR voltage phasor is:
22220
20
200
2002200
00
,
ˆˆ
LLRLR
LLR
LRLR
XRiVVV
XiVRiV
yVxVV
2i
LV
RV
LRV
Impedance
Impedance is the combined “effective resistance” of circuit elements. It is given the symbol Z.
LRLR ZiV 200
2i
LV
RV
LRV
An Example
We can define a total impedance for L and R:
LR
LLRLR
LLR
LRLR
Zi
XRiVVV
XiVRiV
yVxVV
20
22220
20
200
2002200
00
,
ˆˆ
2i
LV
RV
LRV
An Example
We can define a total impedance for L and R:
222 LLR XRZ
2i
LV
RV
LRV
An ExampleWe can calculate the phase angle of the LR phasor:
2
1
0
01 tantanR
X
V
V L
R
LLR
2i
LV
RV
LRV
LR
Impedance
Whenever we know a voltage and a current we can find an impedance. Because of differences in phases, impedances are not added as easily as in DC circuits.
An Example
Now let’s continue the problem, without filling in all the mathematical details.
An Example
We replace the inductor and resistor with a single impedance:
1R
LRZ
C
2i
3i
1i
2i
CLR VV
An Example
The LR voltage phasor is the same as the C voltage phasor, since LR and C are in parallel.
2i
CLR VV
1R
LRZ
C
2i
3i
1i
An Example
We can find the direction and magnitude of .
2i
CLR VV
3i
1R
LRZ
C
2i
3i
1i
An Example
The direction of i3 is given by “ICE.”
2i
CLR VV
3i
1R
LRZ
C
2i
3i
1i
An Example
The magnitude of i3 is given by:
2i
C
LR
LRCLRC
X
Zii
ZiXiVV
2030
203000
CLR VV
3i
1R
LRZ
C
2i
3i
1i
An Example
The total current through LRC is .
2i
3i
32 iiiLRC
LRCi
CLR VV
1R
LRZ
C
2i
3i
1i
An Example
By Kirchoff’s junction rule, we have .
2i
3i
LRCii
1
1i
CLR VV
1R
LRZ
C
2i
3i
1i
An Example
We can replace LRC with a single impedance.
2i
3i
1i
CLR VV
1R
LRCZ
1i
An Example
Now we can find the voltage across the resistor
with Ohm’s Law. 11010 RiVR
1i
CLRC VV
RV
1R
LRCZ
1i
An Example
Finally, we can add the two voltages to get the
EMF. 1R
2R
L
C
2i
3i
1i
LRCV
1RV
11010 RiVR
Class 37Today we will: • study the series LRC circuit in detail.• learn the resonance condition• learn what happens at resonance• calculate power in AC circuits
•Assume we know the maximum voltage of the AC power source, .
•We wish to find the current and the voltages across each circuit element.
The Series LRC Circuit
R
L
i
C
0
•The current phasor is the same everywhere in the series LRC circuit.
•The voltage phasors add:
The Series LRC Circuit
0 CLR VVV
R
L
i
C
•The current phasor is the same everywhere in the series LRC circuit.
•The voltage phasors add:
The Series LRC Circuit
R
L
i
C Zi
C
iXiV
LiXiV
RiV
CC
LL
R
00
000
000
00
0 CLR VVV
The Series LRC Circuit
R
L
i
C
RiVR 00 CC XiV 00
i
LL XiV 00
•We can then draw phasors:
The Series LRC Circuit
R
L
i
C
RiVR 00
CL VV
CC XiV 00
Zi00
LL XiV 00
•The total EMF of the resistor, capacitor, and inductor is the vector sum of the three voltages.
i
The Series LRC Circuit
R
L
i
C
RCX
Z
LX
•Since is a common factor in all the voltages, we can divide it out and simplify the math a little.
CL XX
0i
i
The Series LRC Circuit
R
L
i
C
•Let’s redo this diagram step by step. Be sure you know this well!
The Series LRC Circuit
R
L
i
C
i
•Start with the current direction. It’s easiest just to put this on the +x axis.
The Series LRC Circuit
R
L
i
C
LX
•Put along the +y axis.LX
i
The Series LRC Circuit
R
L
i
CCX
LX
•Put along the --y axis.CX
i
The Series LRC Circuit
R
L
i
C
RCX
Z
LX
•Put along the +x axis.R
i
The Series LRC Circuit
R
L
i
C
RCX
LX
•Now we add all the impedances as vectors.
•First, add and .
CL XX
LX CX
i
The Series LRC Circuit
R
L
i
C
RCX
LX
•Then add R to get Z.
CL XX Z
i
The Series LRC Circuit
R
L
i
C
RCX
LX
CL XX Z
i
LRCV
•We know . and,0 00 ZiVLRC
The Series LRC Circuit
R
L
i
C
RCX
LX
CL XX
Z
i
LRCV
•We know . and,0 00 ZiVLRC
The Series LRC Circuit
R
L
i
C
RCX
LX
•We can find and :
CL XX
Z 0i
Zi
RXXZ CL
00
22
Z
i
Varying the Frequency
•As we vary the frequency, the phasor diagram changes.
•At low frequency, the capacitive reactance is large.
•At high frequency, the inductive reactance is large.
Varying the Frequency
•In this animation, we keep the maximum EMF constant and increase the frequency of the AC power supply.
LX
i
CX
Resonance
•Resonance is where the inductive and capacitive reactances are equal.
Resonance
•Resonance is where the inductive and capacitive reactances are equal.
•The resonant frequency is:
LC
CLXX CL
1
1
Resonance
•Resonance is where the inductive and capacitive reactances are equal.
•The resonant frequency is:
•The impedance is minimum and the current is maximum.
RRXXZ CL 22
LC
CLXX CL
1
1
Power in AC Circuits
•At any given instant, the power dissipated by a resistor is
•Over a full cycle of period T, the average power dissipated in the resistor is:
tRiRi
tVtiP
220
2 sin
)()(
00020
202
20
2
1
2
1
2sin
)(
ViViP
T
T
Ridtt
T
Ri
T
dttP
PTt
t
Tt
t
Power in AC Circuits
•The power provided by a voltage source is:
cos2
1
sincoscossinsin
)sin(sin
)()()(
00
00
00
iP
ttti
tti
ttitP
rms Currents and Voltages
•We want to define an “effective AC current.”
rms Currents and Voltages
•We want to define an “effective AC current.”
•The effective current is less than the maximum current.
rms Currents and Voltages
•We want to define an “effective AC current.”
•The effective current is less than the maximum current.
•The average current is zero.
rms Currents and Voltages
•We want to define an “effective AC current.”
•The effective current is less than the maximum current.
•The average current is zero.
•Absolute values are awkward mathematically.
rms Currents and Voltages
•So, we take the square root of the average value of the current squared.
•This is called the rms or “root mean square” current.
•Similarly:
0
0
220
2
1sin
1iti
Ti
T
rms
02
1VVrms
rms Currents and Voltages
•Normally, when we specify AC voltages or currents, we mean the rms values.
•Standard US household voltage is 110-120V rms.
Power in Terms of rms Quantities
•When we rewrite our power equations in terms of rms currents and voltages, they become similar to DC formulas.
Resistor
Power supply:
R
VRiViViP rms
rmsrmsrms
22
002
1
coscos2
100 rmsrmsiiP