Direct Current Circuits

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Direct Current Circuits
Two Basic Principles Conservation of
Charge Conservation of Energy Resistance
Networks
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Resistors in series Conservation of Charge I
I1 I2 I3 Conservation of Energy Vab V1
V2 V3
Voltage Divider
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I
Resistors in parallel Conservation of Charge I
I1 I2 I3 Conservation of Energy Vab
V1 V2 V3
a
R1 V1I1R1
R2 V2I2R2
R3 V3I3R3
b
Current Divider
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Example Determine the equivalent resistance of
the circuit as shown. Determine the voltage
across and current through each
resistor. Determine the power dissipated in each
resistor Determine the power delivered by the
battery
R14W
E18V
R36W
R23W
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Kirchoffs Rules
Some circuits cannot be represented in terms of
series and parallel combinations
R1
E1
R1
R2
R5
R2
E2
E
R3
R4
R3
Kirchoffs rules are based upon conservation of
energy and conservation of charge.
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Conservation of Charge Junction rule The
algebraic sum of the currents into any junction
is zero.
R1
I1
I2
R2
I3
R3
S I 0 I1 I2 - I3
S I 0 - I1 - I2 I3
Alternatively The sum of the currents into a
junction is equal to the sum of the currents out
of that junction.
I1 I2 I3
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Conservation of Energy Loop rule The algebraic
sum of the potential differences around any
closed loop is zero.
R1
R2
R3
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Potential Differences in the direction of travel
direction of travel
direction of travel
E
E
-

V E
V - E
direction of travel
direction of travel
V IR

-
V - IR

-
I
I
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E1
R1
I1
I2
E2
R2
I3
R3
I1 I2 - I3 0 -E1 I1 R1 - I2 R2 E2
0 - I3 R3 - I2 R2 E2 0 - I3 R3 - I1 R1
E1 0
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R3 3W
I1 I2 - I3 0 - 12 I1 2 - I2 1 5
0 - I3 3 -I2 1 5 0
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I1 I2 - I3 0 - 12 I1 2 - I2 1 5
0 - I3 3 -I2 1 5 0
Standard linear form 1 I1 1 I2 - 1 I3
0 2 I1 -1 I2 0 I3 7 0 I1 1 I2 3
I3 5
TI-89 2nd MATH 4(selects matrix) 5
(selects simult) Entry line simult(1,1,-12,-1,0
0,1,3,075) I1 3 I2 -1 I3 2
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R1 R2 R3 I 3A (-)1A
2A V P PE1 PE2
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Electrical Instruments
Galvanometer Torque proportional to current
I Restoring torque proportional to angle gt
(Equilibrium) angle proportional to angle
q
G
Maximum Deflection Full Scale Deflection Ifs
Current to produce Full Scale Deflection Rc
Resistance (of coil) Vfs Ifs Rc example Ifs
1.0 mA, Rc 20.0 W Vfs 20 mV .020 V
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Wheatstone Bridge Circuit Balanced Vab 0
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Ammeter
Measures current through device Measure currents
larger than Ifs by bypassing the galvanometer
with another resistor (shunt Resistor Rs). Ia
ammeter full scale
I-Ic
When I Ia , we need Ic Ifs VG Vsh gt Ifs
Rc (Ia -Ifs)Rsh
Design a 50 mA ammeter from example
galvanometer Ifs 1.0 mA, Rc 20.0 W, Vfs 20
mV .020 V
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Voltmeter
Measures potential difference across
device Measure V larger than Vfs by adding a
resistor (Rs) in series to the galvanometer.
Vm Voltmeter full scale
When V Vm , we need Ic Ifs Ic Is , Vm
Vs VG gt Vm Ifs Rs Ifs Rc Rs (Vm /
Ifs )- Rc
Design a 10 V voltmeter from example
galvanometer Ifs 1.0 mA, Rc 20.0 W, Vfs 20
mV .020 V
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Ohmmeter
Measures resistance across terminals Meter
supplies an EMF, resistance is determined by
response.
When R 0 , we need Ic Ifs E (Ifs Rs Ifs
Rc) gt Vm Ifs Rs Ifs Rc Rs (E / Ifs )- Rc
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Resistance Capacitance Circuits Charging Capacitor
Capacitor is initially uncharged. Switch is
closed at t0 Apply Loop rule E -q/C-iR0
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Resistance Capacitance Circuits Discharging
Capacitor
Capacitor has an initial charge Qo. Switch is
closed at t0 Apply Loop rule q/C-iR0
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