Transcript Solving more complex resistor networks

More basic electricity
Non-Ideal meters, Kirchhoff’s
rules, Power, Power supplies
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What makes for ideal
voltmeters and ammeters?
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Ideal Meters

Ideally when a voltmeter is added to a
circuit, it should not alter the voltage or
current of any of the circuit elements.
These
circuits
should be
the same.
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Voltmeter
Devices in parallel have the same
voltage.
 Voltmeters are placed in parallel with
a circuit element, so they will
experience the same voltage as the
element.

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Theoretical calculation
5 V = (1 k + 3.3 k ) I
Without the
voltmeter, the two
 5 V = (4.3 k ) I
resistors are in
series.
 I = 1.16279 mA
 V3.3 = (3.3 k ) (1.16279 mA)
 V3.3 = 3.837 V
 Slight discrepancy?

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Non-Ideal Voltmeter
Ideally the voltmeter should not affect
current in resistor.
 Let us focus on the resistance of the
voltmeter.

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RV should be large
1
Req

=
1
R3.3
+
If Rv  , then
1
1

Req
R3.3

1
Rv
The voltmeter is in parallel with
the 3.3-k resistor and has an
equivalent resistance Req.
We want the circuit with and without the
voltmeter to be as close as possible.
Thus we want Req to be close to 3.3 k.
This is accomplished in Rv is very large.
Voltmeters should have large resistances.
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Ammeter
Devices in series have the same
current.
 Ammeters are placed in series with a
circuit element, so they will experience
the same current as it.

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RA should be small
ammeter is in series with
Req = (RA + R1 + R3.3 ) The
the 1- and 3.3-k resistors.
 If RA  0
For the ammeter to have a minimal effect on
the equivalent resistance, its resistance
 Req  (R1 + R3.3 ) should be small.
 Ammeters should have small
resistances

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Simplifying circuits using series and
parallel equivalent resistances
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Analyzing a combination of
resistors circuit



Look for resistors which are in series (the
current passing through one must pass
through the other) and replace them with the
equivalent resistance (Req = R1 + R2).
Look for resistors which are in parallel (both
the tops and bottoms are connected by wire
and only wire) and replace them with the
equivalent resistance (1/Req = 1/R1 + 1/R2).
Repeat as much as possible.
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Look for series combinations
Req=3k
Req=3.6 k
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Look for parallel combinations
Req = 1.8947 k
Req = 1.1244 k
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Look for series combinations
Req = 6.0191 k
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Look for parallel combinations
Req = 2.1314 k
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Look for series combinations
Req = 5.1314 k
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Equivalent Resistance
I = V/R = (5 V)/(5.1314 k) = 0.9744 mA
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Kirchhoff’s Rules
When series and parallel
combinations aren’t enough
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Some circuits have resistors which
are neither in series nor parallel
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They can still be analyzed, but one
uses Kirchhoff’s rules.
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Not in series
The 1-k resistor is not in series with the 2.2-k since the
some of the current that went through the 1-k might go
through the 3-k instead of the 2.2-k resistor.
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Not in parallel
The 1-k resistor is not in parallel with the 1.5-k since their
bottoms are not connected simply by wire, instead that 3-k
lies in between.
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Kirchhoff’s Node Rule





A node is a point at which wires meet.
“What goes in, must come out.”
Recall currents have directions, some currents will
point into the node, some away from it.
The sum of the current(s) coming into a node must
equal the sum of the current(s) leaving that node.
 I2
I1 
I 1 + I 2 = I3
The node rule is about
currents!
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 I3
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Kirchhoff’s Loop Rule 1
“If you go around in a circle, you get
back to where you started.”
 If you trace through a circuit keeping
track of the voltage level, it must return
to its original value when you complete
the circuit


Sum of voltage gains = Sum of voltage losses
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Batteries (Gain or Loss)
Loop direction
Whether a battery is a gain or a loss
depends on the direction in which you
are tracing through the circuit
Loop direction

Gain
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Loss
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Resistors (Gain or Loss)
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Loop direction
Gain
I
Current direction
Loss
I
Current direction
Whether a resistor is a gain or a loss
depends on whether the trace direction
and the current direction coincide or not.
Loop direction

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Neither Series Nor Parallel
I1.5 
I1 
I3 
I2.2 
I1.7 
Draw loops such that each current
element is included in at least one loop.
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Apply Current (Node) Rule
I1.5 
I1 
*
I3 
*
I1-I3 
I1.5+I3 
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*Node rule applied.
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Three Loops
Voltage Gains = Voltage Losses
 5 = 1 • I1 + 2.2 • (I1 – I3)
 1 • I1 + 3 • I3 = 1.5 • I1.5
 2.2 • (I1 – I3) = 3 • I3 + 1.7 • (I1.5
+ I3)
 Units: Voltages are in V, currents in mA,
resistances in k

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Simplified Equations







5 = 3.2 • I1 - 2.2 • I3
I1 = 1.5 • I1.5 - 3 • I3
0 = -2.2 • I1 + 1.7 • I1.5 + 6.9 • I3
Substitute middle equation into others
5 = 3.2 • (1.5 • I1.5 - 3 • I3) - 2.2 • I3
0 = -2.2 • (1.5 • I1.5 - 3 • I3) + 1.7 • I1.5 + 6.9 • I3
Multiply out parentheses and combine like terms.
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Solving for I3
5 = 4.8 • I1.5 - 11.8 • I3
 0 = - 1.6 I1.5 + 13.5 • I3
 Solve the second equation for I1.5
and substitute that result into the
first
 5 = 4.8 • (8.4375 I3 ) - 11.8 • I3
 5 = 28.7 • I3
 I3  0.174 mA

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Comparison with Simulation
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Other currents
Return to substitution results to find
other currents.
 I1.5 = 8.4375 I3 = 1.468 mA
 I1 = 1.5 • I1.5 - 3 • I3
 I1 = 1.5 • (1.468) - 3 • (0.174)
 I1 = 1.68 mA

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Power

Recall


Voltage = Energy/Charge
Current = Charge/Time
Voltage  Current = Energy/Time
 The rate of energy per time is known as
power.
 It comes in units called watts.

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Power differences for elements in
“Equivalent” circuits
Same for circuit but
different for individual
resistors
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Resistor
dissipates
25 mW
Resistor
dissipates
100 mW
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Power supplies





Supplies power to a computer
Transforms 120 V (wall socket voltage) down to
voltages used inside computer (12 V, 5 V, 3.3 V).
Converts the AC current to DC current (rectifies).
Regulates the voltage to eliminate spikes and surges
typical of the electricity found in average wall socket.
Sometimes needs help in this last part, especially
with large fluctuations.
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Power supply



Power supplies are rated by the number of
watts they provide.
The more powerful the power supply, the
more watts it can provide to components.
For standard desktop PC, 200 watts is enough


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Full Towers need more
The more cards, drives, etc., the more power
needed
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Surge protection
Takes off extra voltage if it gets too
high (a surge).
 Must be able to react quickly and take a
large hit of energy.
 They are rated by the amount of energy
they can handle.


I read that one wants at least 240 Joules
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Voltage regulator
Most PC’s power supplies deliver 5 V,
but most processors need a little less
than 3.5 V.
 A voltage regulator reduces the voltage
going into the microprocessor.
 Voltage regulators generate a lot of
heat, so they are near the heat sink.

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VRM/VID

Voltage Regulator Module: a small module
that installs on a motherboard to regulate the
voltage fed to the microprocessor.


It’s replaceable
Voltage ID (VID) regulators are
programmable; the microprocessor tells the
regulator the correct voltage during powerup.
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UPS

Uninterruptible Power Supply, a power supply
that includes a battery to continue supplying
power during a brown-outs and power
outages


Line conditioning
A typical UPS keeps a computer running for
several minutes after an outage, allowing you
to save and shut down properly

Recall the data in RAM is volatile (needs power)
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UPS (Cont.)

Some UPSs have an automatic
backup/shut-down option in case the
outage occurs when you're not at the
computer.
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SPS



Standby Power System: checks the power
line and switches to battery power if it
detects a problem.
The switch takes time (several milliseconds –
that’s thousands if not millions of clock
cycles) during the switch the computer gets
no power.
A slight improvement on an SPS is the “Lineinteractive UPS” (provides some conditioning)
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On-line

An on-line UPS avoids these switching
power lapses by constantly providing
power from its own inverter, even when
the power line is fine.


Power (AC) Battery (DC) through
inverter (back to AC)
On-line UPSs are better but much more
expensive
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Laser printers and UPS
Don’t put a laser printer on a UPS
 Laser printers can require a lot of
power, especially when starting, they
probably exceed the UPS rating

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