19.1 EMF and Terminal Voltage
Analyzing DC circuits
Electric circuit needs a battery or generator (or other
source) to produce a current
– these are called sources of emf.
EMF
Terminal Voltage
emf: “electromotive force” Units:Volts (Its not a force!)
• Resistors in Series and Parallel
• EMFs in Series and Parallel
• Capacitors in Series and Parallel
Kirchhoff’s Rules
• Loop rule
• Junction rule
emf = potential difference across the terminals of
the battery or generator when no current flows to
an external circuit.
• RC Circuits – Resistor and Capacitor in Series
Question...
19.1 EMF and Terminal Voltage
A battery is a nearly constant voltage source, but it does have a small
internal resistance, r that behaves as if it were in series with the emf.
What is the terminal voltage
across the 12 V battery?
If the battery is connected to circuit and a current flows in the normal
direction, there is a voltage drop across the internal resistance and
the terminal voltage is smaller than the emf,
:
9Ω
I
I = 1.2 A
(19-1)
Terminal voltage when current
flows in normal direction (+) → (-)
Formula is different when
I flows in other direction !
1.
2.
3.
4.
5.
9.6 V
10.8 V
12 V
13.2 V
14.4 V
+ r=1Ω
12 V
What is the potential difference
across the 9 Ω resistor?
1
19.2 Resistors in Series and in Parallel
Series: When 2 or more resistors are connected end to end
along a single path they are connected in “series”.
I1
I2
I3
V1
V2
V3
In series the current through
each resistor is the same.
I1 = I2 = I3 = I
19.2 Resistors in Series and in Parallel
Series: From this we can define an equivalent resistance.
This is a single resistance that gives the same current in
the circuit as if the individual resistors were still present.
V = IReq
(same I and V as before)
Which means,
(19-3)
And the voltage across each
resistor depends on its resistance.
[resistors in series]
V1 = IR1 V2 = IR2 V3 = IR3
Finally, the total voltage drop
must equal the battery voltage.
V = V1+V2+V3
(19-2)
Question...
4Ω
2Ω
7Ω
Req = 4 + 2 + 7 = 13 Ω
19.2 Resistors in Series and in Parallel
Parallel: A parallel connection splits the current.
How much power is dissipated
in the 4 Ω resistor?
2Ω
Example:
4Ω
In parallel, the total current is the sum
of the currents across each resistor:
I = I1 + I2 + I3
+ -
1.
2.
3.
4.
5.
2.0 W
6.0 W
16 W
30 W
100 W
12 V
And the voltage across each
resistor is the same:
V = V1 = V2 = V3
V = I1R1
V = I2R2
V = I3R3
2
Please make your selection...
19.2 Resistors in Series and in Parallel
Parallel: We can again calculate an equivalent resistance.
V = IReq
Which circuit draws the most current from the battery?
(same I and V as before)
1Ω
Which means,
1Ω
2Ω
1Ω
2Ω
1Ω
(resistors in parallel 19-4)
30 Ω
Example:
1/Req = 1/20 + 1/30 + 1/50 = 0.1033
Req = 9.7 Ω
1.
2.
3.
Circuit A
Circuit B
They draw the same current
(equivalent resistor is less than smallest individual resistor)
19.3 Kirchhoff’s Rules
Question...
How much current flows through the 2 Ω resistor?
4Ω
0.5 A
0.75 A
1.0 A
1.25 A
1.5 A
For these circuits we use Kirchhoff’s rules.
Junction rule: Sum of currents entering a
junction equals the sum of the currents leaving it.
2Ω
3Ω
1.
2.
3.
4.
5.
Some circuits cannot be broken down
into series and parallel connections.
e.g. at a,
I3 = I1 + I2
4Ω
+ 12 V
3
19.3 Kirchhoff’s Rules
Loop rule: The sum of the changes in
potential around a closed loop is zero.
19.3 Kirchhoff’s Rules
Problem Solving: Kirchhoff’s Rules
1. Label each current
(if you are not sure of the direction, assign one – if you choose correctly
you will calculate a positive current – if not, you will get a negative
value)
2. Identify unknowns
3. Apply junction and loop rules
You will need as many independent equations as there are unknowns.
4. Solve the equations: be careful with signs.
Question... What is I1?
1.
2.
3.
4.
5.
0.25 A
0.49 A
0.77 A
1.3 A
2.8 A
1.0 Ω
= 0.714A
19.4 EMFs in Series and in Parallel; Charging a Battery
EMFs in series - same direction:
total voltage is the sum of the
separate voltages
1.0 Ω
1.0 Ω
r
r
What is the terminal voltage across the 6V battery?
Vca = Vba + Vcb ≅ 3.0V
(ignoring internal resistance of batteries)
4
Question...
19.4 EMFs in Series and in Parallel; Charging a Battery
What is the terminal voltage across
the 12 V battery (Vba)?
EMFs in series - opposite direction:
total voltage is the difference, but the lower-voltage
battery is charged.
r=3Ω
d
+ -
c
6V
Charged by 20 V battery.
I is forced thru battery in
wrong direction.
Vca = Vba + Vcb ≅ 8V (ignoring internal resistance of batteries)
19.4 EMFs in Series and in Parallel; Charging a Battery
EMFs in parallel:
- provides more energy when large
currents are needed.
1.
2.
3.
4.
5.
10.5 V
11.0 V
12.0 V
13.0 V
13.5 V
+ -
b
r=1Ω
12 V
a
What is the terminal voltage
across the 6 V battery (Vdc)?
19.5 Circuits Containing Capacitors in Series and in Parallel
Capacitors in parallel:
Same voltage across each one.
V = V1 = V2 = V3
Total charge is conserved:
Q = Q1 + Q2 + Q3
Equivalent capacitor:
Replace 3 capacitors with a single equivalent capacitor, Ceq.
Also extends the life of the batteries
Ceq is the sum of the
individual capacitances:
Q = CeqV
Ceq = C1 + C2 + C3
(19-5)
5
19.5 Circuits Containing Capacitors in Series and in Parallel
Question...
What is the charge on the 9 µF capacitor?
Capacitors in series
9µF
4µF
Same charge on each when in series.
Q = Q1 = Q2 = Q3
3µF
8V
Total voltage = sum of individual potential differences
V = V1 + V2 + V3
Equivalent capacitor:
Replace 3 capacitors with a single equivalent capacitor, Ceq.
Q = CeqV
1 = 1 + 1 + 1
(19-6)
Ceq C1 C2 C3
Reciprocal of capacitances
add to give reciprocal of Ceq.
19.6 RC Circuits – Resistor and Capacitor in Series
RC circuits contain a resistor, capacitor and emf in series.
- often used as electronic filters.
1.
2.
3.
4.
5.
6 µC
9 µC
12 µC
15 µC
18 µC
19.6 RC Circuits – Resistor and Capacitor in Series
We can calculate how the voltage across the capacitor
changes as a function of time.
At any
moment:
We are interested in their behavior
before they reach a ‘steady state’.
ε = VC+VR
Charging
a capacitor
When the switch is closed in the circuit
the capacitor begins to charge up.
VC
R
ε
ε +-
Charging the
capacitor
C
+
VR
switch
Voltage across the capacitor
increases with time:
Charge on the capacitor
follows a similar curve:
(Max charge)
Time
This curve has a characteristic time constant:
(19-7)
6
19.6 RC Circuits – Resistor and Capacitor in Series
Now, if the charged capacitor is removed from the
battery and connected across a resistor, it discharges
exponentially:
VC = V0e-t/RC
(Max charge)
Summary of Chapter 19
• Emf = potential difference across terminals of a battery when no I flows.
• A battery is a source of emf in series with an internal resistance.
• Resistors in parallel:
• Resistors in series:
Kirchhoff’s rules:
• sum of currents entering a junction equals sum of currents leaving it
• total potential difference around closed loop is zero
• Capacitors in series:
Discharging
a capacitor
• Capacitors in parallel:
RC circuit:
Charging: Voltage across
capacitor increases rapidly.
Discharging: VC decreases exponentially.
VC = V0e-t/RC
Circuit has a characteristic time constant:
7