Transcript Electric Circuits

Electric Circuits
Now that we have the concept of voltage, we
can use this concept to understand electric
circuits.
Just like we can use pipes to carry water, we
can use wires to carry electricity. The
flow of water through pipes is caused by
pressure differences, and the flow is
measured by volume of water per time.
Electric Circuits
In electricity, the concept of voltage will be
like pressure. Water flows from high
pressure to low pressure (this is consistent with
our previous analogy that Voltage is like height
since DP = rgh for fluids) ; electricity flows
from high voltage to low voltage.
But what flows in electricity? Charges!
How do we measure this flow? By Current:
current = I = Dq / Dt
UNITS: Amp(ere) = Coulomb / second
Voltage Sources:
batteries and power supplies
A battery or power supply supplies voltage. This is
analogous to what a pump does in a water system.
Question: Does a water pump supply water? If
you bought a water pump, and then plugged it in
(without any other connections), would water
come out of the pump?
Question: Does the battery or power supply
actually supply the charges that will flow
through the circuit?
Voltage Sources:
batteries and power supplies
Just like a water pump only pushes water (gives
energy to the water by raising the pressure of the
water), so the voltage source only pushes the
charges (gives energy to the charges by raising
the voltage of the charges).
Just like a pump needs water coming into it in order
to pump water out, so the voltage source needs
charges coming into it (into the negative terminal)
in order to “pump” them out (of the positive
terminal).
Voltage Sources:
batteries and power supplies
Because of the “pumping” nature of voltage sources,
we need to have a complete circuit before
we have a current.
If we have an air gap (or rubber gap) in the circuit,
no current will flow - just like if we have a solid
block (like a cap) in a water circuit, no water will
flow.
If the gap is small, and the voltage is high enough,
the current will cross over the gap - somewhat like
water, if the pressure is high enough, will break
through a plug.
Circuit Elements
In this first part of the course we will consider
two of the common circuit elements:
resistor
capacitor
The resistor is an element that “resists” the
flow of electricity.
The capacitor is an element that stores
charge for use later (like a water tower).
Resistance
Current is somewhat like fluid flow. Recall
that it took a pressure difference to make
the fluid flow due to the viscosity of the
fluid and the size (area and length) of the
pipe. So to in electricity, it takes a voltage
difference to make electric current flow due
to the resistance in the circuit.
Resistance
By experiment we find that if we increase the
voltage, we increase the current: V is
proportional to I. The constant of
proportionality we call the resistance, R:
V = I*R
Ohm’s Law
UNITS: R = V/I so Ohm = Volt / Amp.
The symbol for resistance is W (capital omega).
Resistance
Just as with fluid flow, the amount of resistance
does not depend on the voltage (pressure) or
the current (volume flow). The formula V=IR
relates voltage to current. If you double the
voltage, you will double the current, not
change the resistance.
As was the case in fluid flow, the amount of
resistance depends on the materials and
shapes of the wires.
Resistance
The resistance depends on material and
geometry (shape). For a wire, we have:
R=rL/A
where r is called the resistivity (in Ohm-m)
and measures how hard it is for current to
flow through the material, L is the length of
the wire, and A is the cross-sectional area of
the wire. The second lab experiment deals with
Ohm’s Law and the above equation.
Electrical Power
The electrical potential energy of a charge is:
PE = q*V .
Power is the change in energy with respect to
time:
Power = DPE / Dt .
Putting these two concepts together we have:
Power = D(qV) / Dt = V(Dq) / Dt = I*V.
Electrical Power
Besides this basic equation for power:
P = I*V
remember we also have Ohm’s Law:
V = I*R .
Thus we can write the following equations for
power: P = I2*R = V2/R = I*V .
To see which one gives the most insight, we
need to understand what is being held
constant.
Example
When using batteries, the battery keeps the
voltage constant. Each D cell battery
supplies 1.5 volts, so four D cell batteries in
series (one after the other) will supply a
constant 6 volts.
When used with four D cell batteries, a light
bulb is designed to use 5 Watts of power.
What is the resistance of the light bulb?
Example
We know V = 6 volts, and P = 5 Watts; we’re
looking for R.
I
We have two equations: V
P = I*V and V = I*R
R
P
which together have
4 quantities: P, I, V & R..
We know two of these (P & V), so we should
be able to solve for the other two (I & R).
Example
Using the power equation we can solve for I:
P = I*V, so 5 Watts = I * (6 volts), or
I = 5 Watts / 6 volts = 0.833 amps.
Now we can use Ohm’s Law to solve for R:
V = I*R, so
R = V/I = 6 volts / 0.833 amps = 7.2 W .
Example extended
If we wanted a higher power light bulb,
should we have a bigger resistance or a
smaller resistance for the light bulb?
We have two relations for power that involve
resistance:
P=I*V; V=I*R; eliminating V gives: P = I2*R and
P=I*V; I=V/R; eliminating I gives:
P = V2 / R .
In the first case, Power goes up as R goes up; in the
second case, Power goes down as R goes up.
Which one do we use to answer the above question?
Example extended
Answer: In this case, the voltage is being held
constant due to the nature of the batteries. This
means that the current will change as we change
the resistance. Thus, the P = V2 / R would be the
most straight-forward equation to use. This means
that as R goes down, P goes up. (If we had used the
P = I2*R formula, as R goes up, I would decrease – so it
would not be clear what happened to power.)
The answer: for more power, lower the
resistance. This will allow more current to
flow at the same voltage, and hence allow
more power!
Connecting Resistors Together
Instead of making and storing all sizes of
resistors, we can make and store just certain
values of resistors. When we need a nonstandard size resistor, we can make it by
connecting two or more standard size
resistors together to make an effective
resistor of the value we need.
The symbol for a resistor is written:
Two basic ways
There are two basic ways of connecting two
resistors: series and parallel.
In series, we connect resistors together like
railroad cars:
+
-
+
high V
-
low
R1
R2
Series
If we include a battery as the voltage source,
the series circuit would look like this:
R1
+
Vbat
R2
Note that there is only one way around the
circuit, and you have to go through BOTH
resistors in making the circuit - no choice!
Parallel
In a parallel hook-up, there is a branch point
that allows you to complete the circuit by
going through either one resistor or the
other: you have a choice!
R1
High V
Low V
R2
Parallel Circuit
If we include a battery, the parallel circuit
would look like this:
+
Vbat
+
R1
-
+
R2
-
Niagara Falls
Image copied from the internet:
http://www.niagarafallslive.com/facts_about_niagara_falls.htm
Formula for Series:
To see how resistors combine to give an
effective resistance when in series, we can
look either at
R1
I
V = I*R,
+
V1 V
R2
or at
Vbat
2
R = rL/A .
Formula for Series
Using V = I*R, we see that in series the
current must move through both resistors.
(Think of water flowing down two water falls in series.)
Thus Itotal = I1 = I2 .
Also, the voltage drop across the two resistors
add to give the total voltage drop:
(The total height that the water fell is the addition of the two
heights of the falls.)
Vtotal = (V1 + V2). Thus, Reff = Vtotal / Itotal =
(V1 + V2)/Itotal = V1/I1 + V2/I2 = R1 + R2.
Formula for Series
Using R = rL/A , we see that we have to go
over both lengths, so the lengths should add.
The lengths are in the numerator, and so the
values should add.
This is just like in R = V/I (from V = IR)
where the V’s are in the numerator and so
add!
Formula for Parallel Resistors
The result for the effective resistance for a
parallel connection is different, but we can
start from the same two places:
(Think of water in a river that splits with some water
flowing over one fall and the rest falling over the
other but all the water ending up joining back
together again.) V=I*R, or R = rL/A .
Itotal
+
Vbat
I1
-
R1 I2
R2
Formula for Parallel Resistors
V=I*R, or R = rL/A
For parallel, both resistors are across the same
voltage, so Vtotal = V1 = V2 . The current
can go through either resistor, so:
Itotal = (I1 + I2 ) .
Since the I’s are in the denominator, we have:
R = Vtotal / Itotal = Vtotal / (I1+I2); or
1/Reff =
(I1+I2) / Vtotal = I1 / V1 + I2 / V2
= 1/R1 + 1/R2.
Formula for Parallel Resistors
If we start from R = rL/A , we can see that
parallel resistors are equivalent to one
resistor with more Area. But A is in the
denominator (just like the current I was in
the previous slide), so we need to add the
inverses:
1/Reff = 1/R1 + 1/R2 .
Review:
Resistors:
V = IR
R = rL/A
Power = IV
Series: Reff = R1 + R2
Parallel: 1/Reff = 1/R1 + 1/R2
series gives largest Reff , parallel gives smallest Reff .
Computer Homework
The Computer Homework, on
Resistors: Basic, Vol 3, #5,
gives both an introduction and problems
dealing with resistors.
Examples
Consider two resistors in series:
a 3W and a 6W resistor:
Reff = R1 + R2 = 3W + 6W = 9W
Note that 9W is larger than the largest single resistor
of 6W.
Consider two resistors in parallel:
a 3W and a 6W resistor:
1/Reff = 1/R1 + 1/R2 =
1/3W + 1/6W = (.333/W + .167/W)
= .500/W, or Reff = W/.500 = 2W.
Note that 2W is smaller than the smallest single resistor
of 3W.
3W
6W
3W
6W
Capacitance
A water tower holds water.
A capacitor holds charge.
The pressure at the base of the water tower
depends on the height (and hence the
amount) of the water. The voltage across a
capacitor depends on the amount of charge
held by the capacitor.
Capacitance
We define capacitance as the amount of charge
stored per volt: C = Qstored / DV.
UNITS:
Farad = Coulomb / Volt
Just as the capacity of a water tower depends on
the size and shape, so the capacitance of a
capacitor depends on its size and shape. Just
as a big water tower can contain more water
per foot (or per unit pressure), so a big
capacitor can store more charge per volt.
Capacitance
While we normally define the capacity of a
water tank by the TOTAL AMOUNT of
water it can hold, we define the capacitance
of an electric capacitor as the AMOUNT
OF CHARGE PER VOLT instead.
There is a TOTAL AMOUNT of charge a
capacitor can hold, and this corresponds to a
MAXIMUM VOLTAGE that can be
placed across the capacitor. Each capacitor
DOES HAVE A MAXIMUM VOLTAGE.
Capacitance
• What happens when a water tower is overfilled? It can break due to the pressure of
the water pushing on the walls.
• What happens when an electric capacitor is
“over-filled” or equivalently a higher
voltage is placed across the capacitor than
the listed maximum voltage? It will
“break” by having the charge “escape”.
This escaping charge is like lightning - a
spark that usually destroys the capacitor.
Capacitors
As we stated before, the capacitance of a capacitor depends on its
size and shape. Basically a capacitor consists of two separated
(at least electrically separated) conductors (usually pieces of
metal) so that we can pull charge from one and deposit it on
the other.
We first look at a common type of capacitor, the
parallel plate capacitor where the two conductors are plates
that are aligned parallel to each other; each of area, A;
separated by a distance, d;
and containing a
Top plate
non-conducting
A
material between
Material between plates
d
the plates.
Bottom plate
Parallel Plate Capacitor
For a parallel plate capacitor, we can pull charge
from one plate (leaving a -Q on that plate) and deposit it
on the other plate (leaving a +Q on that plate). Because of
the charge separation, we have a voltage
difference between the plates, DV. The harder
we pull (the more voltage across the two plates), the more
charge we pull: C = Q /DV.
Note that C is NOT CHANGED by either Q or DV;
C relates Q and DV!
The same applied to resistance:
the resistance did not depend
on the current and voltage –
the resistance related the two. d
Top plate
+Q
Material between plates
Bottom plate
-Q
A
DV
V or DV ?
When we deal with height, h, we usually refer
to the change in height, Dh, between the
base and the top. Sometimes we do refer to
the height as measured from some reference
point. It is usually clear from the context
whether h refers to an actual h or a Dh.
With voltage, the same thing applies. We
often just use V to really mean DV. You
should be able to determine from the
context whether we really mean V or DV
when we say V.
Parallel Plate Capacitor
For this parallel plate capacitor, the capacitance is
related to charge and voltage (C = Q/V), but the
actual capacitance depends on the size and shape:
Cparallel plate = K A / (4 p k d)
where K (called dielectric constant) depends on the
material between the plates, A is the area of each
plate, d is the distance between the plates, and k is
Coulomb’s constant (9 x 109 Nt-m2 / Coul2).
Top plate
d
+Q
Material between plates
Bottom plate
-Q
A
DV
Example:
Parallel Plate Capacitor
Consider a parallel plate capacitor made from
two plates each 5 cm x 5 cm separated by 2
mm with vacuum in between. What is the
capacitance of this capacitor?
Further, if a power supply puts 20 volts across
this capacitor, what is the amount of
charged stored by this capacitor?
Example:
Parallel Plate Capacitor
The capacitance depends on K, A, k and d:
Cparallel plate = K A / (4 p k d)
where K = 1 for vacuum, A = 5 cm x 5 cm = 25
cm2 = 25 x 10-4 m2, d = 2 mm = 2 x 10-3 m,
and k = 9 x 109 Nt-m2/Coul2 , so C =
[(1) * (25 x 10-4 m2) ] / [4 * 3.14 * 9 x 109 Nt-m2/Coul2
* 2 x 10-3 m] = 1.10 x 10-11 F = 11 pF .
Other types of capacitors
Note: We can have other shapes for capacitors.
These other shapes will have formulas for them
that differ from the above formula for parallel
plates. These formulas will also show that the
capacitance depends on the materials and shape of
the capacitor.
Example (cont.)
We can see from the previous example that
a Farad is a huge capacitance!
If we have a DV = 20 volts, then to calculate the
charge, Q, we can use: C = Q/V to get:
Q = C*V = 11 x 10-12 F * 20 volts =
2.2 x 10-10 Coul = 0.22 nCoul = 220 pCoul.
Remember that we often drop the D in front of the V
since we often are concerned by the change in voltage
rather than the absolute value of the voltage - just as
we do when we talk about height!
Capacitance
Note that if we doubled the voltage, we
would not do anything to the capacitance.
Instead, we would double the charge
stored on the capacitor.
However, if we try to overfill the capacitor by
placing too much voltage across it, the
positive and negative plates will attract each
other so strongly that they will spark across
the gap and destroy the capacitor. Thus
capacitors have a maximum voltage!
Energy Storage
If a capacitor stores charge and carries
voltage, it also stores the energy it took to
separate the charge. The formula for this is:
Estored = (1/2)QV = (1/2)CV2 ,
where in the second equation we have used
the relation: C = Q/V .
Energy Storage
Note that previously we had:
PE = q*V ,
and now for a capacitor we have:
E = (1/2)*Q*V .
Why the 1/2 factor for a capacitor?
Energy Storage
The reason is that in charging a capacitor, the
first bit of charge is transferred while there
is very little voltage on the capacitor (recall
that the charge separation creates the
voltage!). Only the last bit of charge is
moved across the full voltage. Thus, on
average, the full charge moves across
only half the voltage!
The battery does supply the full Q*V energy, but the other ½
goes into heat in the resistor during the charging.
Connecting Capacitors Together
Instead of making and storing all sizes of
capacitors, we can make and store just
certain values of capacitors. When we need
a non-standard size capacitor, we can make
it by connecting two or more standard size
capacitors together to make an effective
capacitor of the value we need. (Similar to
what we saw with resistors.)
Two basic ways
Just as with resistors, there are two basic ways
of connecting two capacitors: series and
parallel. In series, we connect capacitors
together like railroad cars; using parallel
plate capacitors it would look like this:
+
-
+
high V
-
low V
C1
C2
Series
If we include a battery as the voltage source,
the series circuit would look like this:
C1
+
+
+
Vbat
C2
Note that there is only one way around the
circuit, and you have to jump BOTH
capacitors in making the circuit - no choice!
Parallel
In a parallel hook-up, there is a branch point
that allows you to complete the circuit by
jumping over either one capacitor or the
other: you have a choice!
+
High V
-
C1
+
C2
Low V
-
Parallel Circuit
If we include a battery, the parallel circuit
would look like this:
+
Vbat
+
C1
+
C2
Formula for Series:
To see how capacitors combine to give an
effective capacitance when in series, we
can look either at C = Q/V, or at
Cparallel plate = KA / [4pkd] .
Formula for Series
Using C = Q/V, we see that in series the
charge moved from capacitor 2’s negative
plate must be moved through the battery to
capacitor 1’s positive plate.
C1
+
+
+Q
+
Vbat
-
-Q
(  +Qtotal)
C2
Formula for Series
But the positive charge on the left plate of C1 will
attract a negative charge on the right plate, and the
negative charge on the bottom plate of C2 will
attract a positive charge on the top plate - just
what is needed to give the negative charge on the
right plate of C1. Thus Qtotal = Q1 = Q2 .
(+Q1  )
C1
+
+Q1
-Q 1
+Q2
Vbat
C2
-
-Q2
(  +Qtotal)
Formula for Series
Also, the voltage drop across the two capacitors add
to give the total voltage drop: Vtotal = (V1 + V2).
Thus, Ceff = Qtotal / Vtotal = Qtotal / (V1 + V2), or
(with Qtotal = Q1 = Q2)
[1/Ceff] = (V1 + V2) / Qtotal = V1/Q1 + V2/Q2 =
1/C1 + 1/C2 = 1/Ceffective .
Note: this is the opposite of resistors when
connected in series! Recall that R =V/I where V
is in the numerator; but with capacitors C = Q/V
where V is in the denominator!
Formula for Series
Using Cparallel plate = KA / [4pkd] , we see that
we have to go over both distances, so the
distances should add. But the distances are
in the denominator, and so the inverses
should add. This is just like in C = Q/V
where the V’s add and are in the
denominator!
Formula for Parallel Capacitors
The result for the effective capacitance for a
parallel connection is different, but we can
start from the same two places:
C = Q/V, or Cparallel plate = KA / [4pkd] .
Parallel Circuit
For parallel, both plates are across the same voltage, so
Vtotal = V1 = V2 . The charge can accumulate on
either plate, so: Qtotal = (Q1 + Q2).
Since the Q’s are in the numerator of C = Q/V, we have:
Ceff = C1 + C2.
+
Vbat
+Q1
C1
-Q1
+Q2
C2
+Q1 
 +Qtotal = (Q1+Q2)
 +Q2
-Q2
Formula for Parallel Capacitors
If we use the parallel plate capacitor formula,
Cparallel plate = KA / [4pkd] , we see that the
areas add, and the areas are in the
numerator, just as the Q’s were in the
numerator in the C = Q/V definition.
Review of Formulas
For capacitors in SERIES we have:
1/Ceff = 1/C1 + 1/C2 .
For capacitors in PARALLEL we have:
Ceff = C1 + C2 .
Note that adding in series gives Ceff being
smaller than the smallest, while adding in
parallel gives Ceff being larger than the
largest!
Review:
Capacitors: C = Q/V
PE = ½CV2; C// = KA/[4pkd]
Series: 1/Ceff = 1/C1 + 1/C2
Parallel: Ceff = C1 + C2
series gives smallest Ceff , parallel gives largest Ceff .
Resistors: V = IR
Power = IV; R = rL/A
Series: Reff = R1 + R2
Parallel: 1/Reff = 1/R1 + 1/R2
series gives largest Reff , parallel gives smallest Reff .
Computer Homework
The Computer Homework on
Capacitors: Basic, Vol 3, #7,
gives both an introduction and problems
dealing with capacitors.