dc circuits · dc circuits unit 4 review from last unit • let’s review some of the major points...
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DC Circuits
Unit 4
Review From Last Unit
• Let’s review some of the major points we covered at the end of last unit.
• Batteries generate a potential difference (voltage) within a circuit.
• As a result, current flows if there is a complete conductive path between the + and - terminals of the battery.
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Review
• Resistors dissipate energy within the circuit and resist the flow of electrons.
• Voltage drops across a resistor and rises across a battery.
• Current is constant throughout the circuit, and flows from high potential to low potential.
Review
• The amount of current that flows through a circuit is determined by the total resistance of the circuit.
• Current is found using Ohm’s Law.
€
V = IR
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Review
• We have also learned a good number of circuit symbols:
Battery
Capacitor
Resistor
Wire
Switch
Ground
EMF and Terminal Voltage
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EMFs
• We already know that a battery is needed to cause a current to flow in a circuit.
• Batteries transform other types of energy (such as chemical) to electrical energy.
EMFs • A device that generates electrical energy is
commonly known as a source of electromotive force (EMF).
• EMF is not a force measured in Newton’s.
• It is a synonym potential difference, or voltage, and is measured in volts.
• EMF is denoted by the symbol
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EMFs
• It is important to remember that a battery generates a constant voltage.
• However, the current that results from that voltage depends on the total resistance of the circuit.
Terminal Voltage
• Ideally, the potential difference between the two terminals of a battery would exactly equal its emf.
• However, because the battery is made of real materials and involves chemical reactions that occur at a fixed rate, this is never actually the case.
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Terminal Voltage
• The materials of the battery always present at least some hindrance to the current.
• Thus, the battery has an internal resistance. This is denoted with a lower case r.
Terminal Voltage
• A real battery is modeled as a perfect emf followed by a resistor, r.
• Points a and b represent the physical terminals of the battery.
• When we measure the voltage between a and b, we are measuring the terminal voltage.
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Slides missing L
Kirchhoff’s Rules
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Kirchhoff’s Rules
• So far, we have been analyzing circuits by finding equivalent resistances and using Ohm’s law.
• While this works for many circuits, some are too complicated for this method to be effective.
Kirchhoff’s Rules
• An alternative way to analyze circuits is by using Kirchoff’s rules.
• These rules were developed by G. R. Kirchhoff in the mid-19th century.
• They are merely a restatement of two well known physical principles: conservation of charge, and conservation of energy.
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Kirchhoff’s Rules
• Kirchhoff’s first rule is known as the junction rule:
• This is a restatement of the principle of conservation of charge.
At any junction point in a circuit, the sum of all the currents entering the junction must equal the sum of all the currents leaving the junction.
Kirchhoff’s Rules
• Kirchhoff’s second rule is known as the loop rule:
• This is a restatement of the principle of conservation of energy.
The sum of the changes in potential (voltage) around any closed loop in a circuit must be zero.
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Example: Loop Rule
Problem Solving
• When using Kirchhoff’s rules, you will write down several equations to describe the circuit.
• To analyze the circuit, you will be solving these equations as a system.
• This means, you will need as many equations as you have unknowns.
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Problem Solving 1. Start by labeling the current through
each branch of the circuit. – Make sure you differentiate each current
with a subscript. – If you do not immediately know the
direction of a current, guess. If you guess wrong, the current will have a minus sign in the solution.
2. Identify the unkowns. You need as many equations as you have unknowns.
Problem Solving
3. Apply the junction rule at one or more junctions to generate equations.
4. Apply the loop rule around one or more loops generate equations.
– When applying the loop rule, go in one direction only around the loop even if you are going against the current.
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Warnings About the Loop Rule
• When applying the loop rule, pay careful attention to minus signs.
• For a resistor, – The voltage is negative (drops) if the
direction of your loop is the same as the current.
– The voltage is positive (rises) if the direction of the loop is opposite the current.
Warnings About the Loop Rule
• For a battery, – The voltage is positive (rises) if the
direction of the loop goes from the negative terminal to the positive terminal.
– The voltage is negative (drops) if the direction of the loop goes from the positive terminal to the negative terminal.
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Problem Solving
• Once you have written your equations, solve them using any of the common techniques for solving a linear system.
• Substitution is probably your best bet.
• Check your answers afterward to ensure they make sense physically.
Example: Find the Currents
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Example: Find the Currents
Problems
• Do problems 26 and 27 on page 548.
• If you finish early, work on problem 28.
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Problem Day
• Do problems 29, 31, and 32 (part a only) on page 549.
• We will whiteboard at the end of class.
Whiteboarding Groups
Group Members Problem 1 Robert, Piper, Anthony 26 2 Aidan, Bailey, Jacob 27 3 Sarah, Connor, John 28 4 Angi, Armen, Krystiana 29 5 Kaleb, Rachel, Brie, Abbey 31 6 Miggy, Ellen, Jeremiah 32
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Homework
• Read section 19-8.
• Also, read the handout on breadboards.
Introduction to Lab Equipment
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Plan for Today
• Today, we will practice working with the new lab equipment, which includes – Breadboards – Multimeters – Resistors
• You should be taking notes on the use of these devices, as we will be using them to analyze more complex circuits tomorrow.
Breadboards
• A breadboard is a device that allows you to design and test a circuit without having to solder any wires together.
• It consists of a grid of holes designed to fit the leads of electronic components.
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Breadboards
• On the inside of the breadboard, the holes are wired together in a specific way. – Each column on the outside of
the board is connected. – Each row of five holes on the
interior of the board is connected.
– No rows are connected across the central column.
Breadboards
• When you place the lead of any circuit element into a row, it is automatically wired to anything else placed in that row.
• This allows you to make circuits without using lots of excess wires.
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Breadboards
• The breadboards we will be using consist of three of the pictured breadboards placed together.
• The three breadboards are not connected internally.
Breadboards
• Our breadboards are also powered by a wall outlet and are capable of acting as three separate voltage sources.
• One source is fixed at 5 V.
• The other two are adjustable and have a range of 0 V to 15 V.
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Breadboards
• The black terminal is used for ground. You can think of it as the negative terminal of the battery.
• In order to form a circuit, you need to have complete conducting path from one of the red terminals to the black terminal.
Breadboard Rules
• Do not turn on the power to your breadboard until you have finished building your circuit.
• At no point should both leads of a circuit element be placed in the same row. That is the same as shorting the element.
• Generally, we use the + column for the input power from the battery, and the - column for ground.
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The Multimeter
• In order to analyze the circuit, we will be using a digital multimeter.
• The multimeter can contains both an ammeter and a voltmeter.
• It can also measure resistance, capacitance, and temperature.
Measuring Voltage
• When measuring voltage, the black probe should be placed in the lower right socket. This is considered to be ground (0 V) by the meter.
• The red probe should be placed in the upper right socket (red arrow).
• Turn the dial to the DC V position (green arrow).
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Measuring Voltage
• When measuring voltage, the voltmeter should be set up in parallel with the circuit element you are measuring.
• In practice, this just means you touch the two probes to either side of the element you are interested in.
• The red probe should be placed where the current is coming in to the element, black should be where the current is going out.
Measuring Current
• When measuring current, the black probe should be placed in the lower right socket.
• The red probe should be placed in the lower left socket (red arrow).
• Turn the dial to either the mA or the µA position (green arrow) depending on how much current you have.
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Measuring Current
• When measuring current, the ammeter should be set up in series with the circuit element you are measuring.
• In practice, this means you will have to break the circuit at a point and add in the ammeter.
• The red probe should be placed where the current is coming in to the break, black should be where the current is going out.
WARNING • Ammeters are designed with very low internal
resistance.
• If you set it up in parallel with the circuit, all the current will flow through the ammeter and it will get fried. This happens in a fraction of a second.
• If you find yourself hooking up the ammeter without unplugging some part of your circuit, you are not doing it correctly.
• For today, you MUST set up the ammeter with the power to the board off. I will check it, and then you may turn the current on.
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Resistor Color Code
• Resistors are marked with a number of colored bands to indicate the resistance. (see page 499 of the book)
Practice Circuit
• We will use the following series circuit to practice using the equipment.
• Use the resistor color code to determine the values of R1 and R2.
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Practice Circuit
• We will let R1 be the 1.5 kΩ resistor, and R2 will be the 470 Ω resistor. We will also set V at 9 volts.
• In your group, use what we have learned about circuits to predict: – The current in the circuit. – The voltage drop between points a and b. – The voltage drop between points b and c.
Practice Circuit
• Now it’s time to set up this circuit on the board.
• Turn the power on, and then use one of the knobs to set the voltage of one of the power terminals to 9 V. Then, turn the power back off before continuing.
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Practice Circuit
• Connect the power terminal you are using to a + column on the board with a wire.
• Next, connect a - column on the same section of board to the ground terminal with the other wire.
• Lastly, connect the two resistors in series to complete the circuit. Raise your hand when you think your group is done.
Practice Circuit
• Once your group’s setup has been checked, you may turn on the power to the board.
• Use the multimeter to measure the voltage drop from a to b. Then measure from b to c.
• Check these results against your predictions.
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Practice Circuit
• Turn the power to the board off.
• Setup the multimeter to measure the current through the circuit. Then connect it in series with the circuit. Raise your hand when done.
• Once your group’s setup has been checked, use the ammeter to measure the current through the circuit. Check your results against your predictions.
You Try
• Do the previous steps (predict the voltage and current, then measure) again, but this time set up the resistors in parallel.
• Use the wiring of the breadboard to help you. You should not need any extra wires.
• Draw a circuit diagram to help you.
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Homework
• Read section 19-5.
• Do problem 35 on page 549.
Capacitors
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Capacitors
• Like resistors, capacitors can be wired into a circuit in either series or parallel.
• In each case, the network can be modeled as a single capacitor with an equivalent capacitance.
• Today, we will determine how to calculate the equivalent capacitance for the series and parallel cases.
Capacitors in Parallel
• We will consider the parallel case first.
• Notice that the potential across all three capacitors is the same since they are wired in parallel.
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Capacitors in Parallel
• When the battery is connected, charge is driven onto each capacitor.
• The amount of charge driven onto each capacitor is given by
€
Q1 = C1VQ2 = C2VQ3 = C3V
Capacitors in Parallel
• We also know that the total charge that must leave the battery is
• Based on this
€
Q =Q1 +Q2 +Q3
€
Q = C1V +C2V +C3V
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Capacitors in Parallel
• However, we could also have thought of all that charge as being driven onto the places of one “equivalent” capacitor.
• So we can say
€
Q = CeqV
€
Q = C1V +C2V +C3VCeqV = C1V +C2V +C3V
Capacitors in Parallel
• Factoring out V
• The terms in parentheses are the equivalent capacitance.
€
CeqV = C1 +C2 +C3( )V
€
Ceq = C1 +C2 +C3
for capacitors in parallel.
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Capacitors in Parallel
• So, wiring capacitors in parallel increases the net capacitance.
• This should make sense, as this is essentially the same as increasing the overall area of the plates.
Capacitors in Series
• We can also connect capacitors in series.
• When the battery is connected, a certain amount of charge, +Q, flows onto the left place of C1, and -Q flows onto C3.
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Capacitors in Series
• Because the regions A and B were initially neutral, equal charges are induced on the other plates of the capacitors.
• Just like last time, we want to find the equivalent capacitance of this network.
Capacitors in Series
• We know that the voltage across all the capacitors is the sum of the voltages across the individual capacitors.
• We also know
€
V =V1 +V2 +V3
€
Q = CeqV ⇒ V =QCeq
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Capacitors in Series
• So, we can plug in for V
• Dividing both sides by Q, we can find Ceq.
€
QCeq
=QC1
+QC2
+QC3
QCeq
=Q 1C1
+1C2
+1C3
"
# $
%
& '
Capacitors in Series
€
1Ceq
=1C1
+1C2
+1C3
+ ...
for capacitors in series.
WARNING: Just like for resistors in parallel, you must add the fractions first, then take the reciprocal to find the equivalent capacitance.
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Example: Equivalent Capacitance
What is the equivalent capacitance of the network pictured below?
C1 = 400 pF
C2 = 100 pF
C3 = 200 pF
Example: Equivalent Capacitance
For the network shown, find the voltage drop across each capacitor. Assume the battery is 9 V.
C1 = 400 pF
C2 = 100 pF
C3 = 200 pF
Confirm that your answers add up to 9 V
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You Try: Voltage Drops Using our results from a few minutes ago, determine the voltage drop across each capacitor. V = 9 volts.
Hint: The two capacitors in parallel can be reduced to a single capacitor in series with C1.
C1 = 400 pF
C2 = 100 pF
C3 = 200 pF
Homework
• Do problems 36 - 38 on page 549.
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Review of Kirchhoff’s Rules
Kirchhoff’s Rules
Junction Rule (Conservation of Charge)
The sum of all the currents entering a junction point must equal the sum off all the currents leaving the junction.
Loop Rule (Conservation of Energy)
The sum of the voltage rises and voltage drops around any closed loop in a circuit must be zero.
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Example
• Turn to page 530 in your textbooks.
Homework
• Redo problems 27, 29, 30, and 31 on pages 548-549 in the book.
• You may use matrices to solve these problems.
• If you do, you must write out the equations in matrix form as part of your solution.
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RC Circuits
RC Circuits
• We have already learned that a capacitor becomes charged when it is connected to the terminals of a battery.
• However, this process does not happen instantaneously. Like any other motion, time is required for the charge to flow onto the capacitor.
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RC Circuits
• Likewise, when a capacitor is discharged, it takes time for charge to leave the plates.
• It turns out that the charge does not flow at a constant rate when the capacitor is being charged or discharged.
RC Circuits
• An RC circuit is a circuit with a resistor (or lightbulb) and a capacitor wired in series.
• The circuit is designed to take advantage of the time-dependent amount of charge on the capacitor.
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RC Circuits
• When the switch is closed in the circuit, the battery creates a potential difference that causes electrons to flow onto the top plate of the capacitor.
• This means there is a current through the resistor.
RC Circuits
• As charge accumulates on the capacitor, the potential difference across it rises.
• As this happens, the current is reduced (the voltage difference across the resistor decreases).
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RC Circuits
• This process continues until the voltage across the capacitor equals the voltage of the battery.
• Once this happens, no more current flows, and the capacitor is charged.
RC Circuits
• From our analysis, it is clear that the voltage across the capacitor increases with time.
• The increase is described by an exponential curve.
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RC Circuits
€
VC =V 1− e−tRC( )Voltage
across the capacitor
Voltage of the battery.
Resistance Capacitance
Time
RC Circuits
• The product of R and C, which appears in the exponent, is called the time constant, τ, of the circuit.
• The time constant is a measure of how much time it takes for the voltage across the capacitor to reach 63% of the maximum value.
• The time constant has units of seconds.
€
VC =V 1− e−tRC( )
€
τ = RC
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Discharging a Capacitor
• An RC circuit can also be constructed with a charged capacitor and a resistor.
• Since the capacitor is charged, there is a voltage difference between the two plates.
Discharging a Capacitor
• When the switch is closed, there is a conductive path between the two plates of the capacitor.
• Charge flows from one plate of the capacitor to the other, resulting in a current.
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Discharging a Capacitor
• The amount of current that flows through the resistor depends on the voltage across the capacitor.
• But that voltage is decreasing as charge flows off the plates.
Discharging a Capacitor
• Like before, the voltage changes exponentially.
• However, this time it is exponential decay, rather than exponential growth.
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Discharging a Capacitor
€
VC =V0e− tRC
Voltage across the capacitor
Voltage initially across the capacitor.
Resistance Capacitance
Time
Discharging a Capacitor
• As before, the product of R and C is the time constant, τ, of the circuit.
• But now, the time constant is a measure of how much time it takes for the voltage across the capacitor to decrease 63% from the maximum value.
€
τ = RC€
VC =V0e− tRC
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Example: Discharging a Capacitor In this circuit, C = 35 µF and R = 120 Ω.
a) What is the time constant for this circuit?
b) If the switch is closed, how long will it take for the voltage across the capacitor to reach 10% of its original value?
Homework
• Read section 19-6.
• Do problems 50 and 51 on page 550.
• Reminder: our next test will be on Monday.
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Practice Day
• Do problems 20, 70 and 85 on pages 548-553.
• In 20 and 85, try moving the ends and orientations of the resistors to better see the wiring.
• In 70, model your body as a resistor.