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Objective of LectureExplain mathematically how a voltage that is applied
to resistors in series is distributed among the resistors.Chapter 2.5 in Fundamentals of Electric CircuitsChapter 5.7 Electric Circuit Fundamentals
Explain mathematically how a current that enters the a node shared by resistors in parallel is distributed among the resistors.Chapter 2.6 in Fundamentals of Electric CircuitsChapter 6.7 in Electric Circuit Fundamentals
Work through examples include a series-parallel resistor network (Example 4).Chapter 7.2 in Fundamentals of Electric Circuits
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Voltage DividersResistors in series share the same current
Vin
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Voltage DividersResistors in series share the same current
From Kirchoff’s Voltage Law and Ohm’s Law :
22
11
210
IRV
IRV
VVVin
+
V1-
+V2_
Vin
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Voltage DividersResistors in series share the same current
From Kirchoff’s Voltage Law and Ohm’s Law :
in
in
in
VRRRV
VRRRV
RRVV
IRV
IRV
VVV
0
2122
2111
2112
22
11
21
+
V1-
+V2_
Vin
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Voltage DivisionThe voltage associated with one resistor Rn in a chain of multiple resistors in series is:
or
where Vtotal is the total of the voltages applied across the resistors.
totaleq
nn V
R
RV
totalS
ss
nn V
R
RV
1
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Voltage DivisionThe percentage of the total voltage
associated with a particular resistor is equal to the percentage that that resistor contributed to the equivalent resistance, Req.The largest value resistor has the largest
voltage.
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Example 1Find the V1, the voltage
across R1, and V2, the voltage across R2.
+
V1-
+V2_
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Example 1Voltage across R1 is:
Voltage across R2 is:
Check: V1 + V2 should equal Vtotal
+
V1-
+V2_
tVV
tVkkkV
VRRRV
tVV
tVkkkV
VRRRV
total
total
377sin4.11
377sin20 434
377sin57.8
377sin20 433
2
2
2122
1
1
2111
8.57 sin(377t)V + 11.4 sin(377t) = 20 sin(377t) V
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+V1 -
+V2 -
+V3 -
Example 2Find the voltages listed in the
circuit to the right.
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Check: V1 + V2 + V3 = 1V
VV
VV
VV
VV
VV
VV
R
R
eq
eq
143.0
1 700/100
571.0
1 700/400
286.0
1 700/200
700
100400200
3
3
2
2
1
1
+V1 -
+V2 -
+V3 -
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Symbol for Parallel ResistorsTo make writing equations simpler, we use a symbol to indicate that a certain set of resistors are in parallel.
Here, we would write
R1║R2║R3
to show that R1 is in parallel with R2 and R3. This also means that we should use the equation for equivalent resistance if this symbol is included in a mathematical equation.
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Current DivisionAll resistors in parallel share the same voltage
+
Vin
_
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Current DivisionAll resistors in parallel share the same voltage
From Kirchoff’s Current Law and Ohm’s Law :
33
22
11
3210
RIV
RIV
RIV
IIII
in
in
in
in
+
Vin
_
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Current DivisionAll resistors in parallel share the same voltage
+
Vin
_
in
in
in
IRRR
RRI
IRRR
RRI
IRRR
RRI
213
213
312
312
321
321
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Current DivisionAlternatively, you can reduce the number of resistors in parallel from 3 to 2 using an equivalent resistor.
If you want to solve for current I1, then find an equivalent resistor for R2 in parallel with R3.
+
Vin
_
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Current Division
+
Vin
_
ineq
eqeq I
RR
RI
RR
RRRRR
11
32
3232 and where
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Current DivisionThe current associated with one resistor R1 in parallel with one other resistor is:
The current associated with one resistor Rm in parallel with two or more resistors is:
totalm
eqm I
R
RI
totalI
RR
RI
21
21
where Itotal is the total of the currents entering the node shared by the resistors in parallel.
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Current DivisionThe largest value resistor has the smallest
amount of current flowing through it.
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Example 3Find currents I1, I2, and I3 in the circuit to the
right.
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Example 3 (con’t)
Check: I1 + I2 + I3 = Iin
AI
AI
AI
AI
AI
AI
Req
727.0
4 600/109
09.1
4 400/109
18.2
4 200/109
109600140012001
3
3
2
2
1
1
1
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Example 4The circuit to the
right has a series and parallel combination of resistors plus two voltage sources.Find V1 and VpFind I1, I2, and I3
+
V1
_
+
Vp
_
I1
I2 I3
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Example 4 (con’t)First, calculate
the total voltage applied to the network of resistors. This is the addition
of two voltage sources in series.
+
V1
_
+
Vp
_
I1
I2 I3
+
Vtotal
_
)20sin(5.01 tVVVtotal
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Example 4 (con’t)Second, calculate
the equivalent resistor that can be used to replace the parallel combination of R2 and R3.
+
V1
_
+
Vp
_
I1
+
Vtotal
_
80100400
100400
1
1
32
321
eq
eq
eq
R
R
RR
RRR
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Example 4 (con’t)To calculate the
value for I1, replace the series combination of R1 and Req1 with another equivalent resistor.
I1
+
Vtotal
_
280
80200
2
2
112
eq
eq
eqeq
R
R
RRR
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Example 4 (con’t)I1
+
Vtotal
_
)20sin(79.157.3280
)20sin(5.0
280
1280
)20sin(5.01
1
1
1
21
tmAmAI
tVVI
tVVI
R
VI
eq
total
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Example 4 (con’t)To calculate V1,
use one of the previous simplified circuits where R1 is in series with Req1.
+
V1
_
+
Vp
_
I1
+
Vtotal
_
)20sin(357.0714.0
or
1
111
1
11
tVVV
IRV
VRR
RV total
eq
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Example 4 (con’t)To calculate Vp:
Note: rounding errors can occur. It is best to carry the calculations out to 5 or 6 significant figures and then reduce this to 3 significant figures when writing the final answer.
+
V1
_
+
Vp
_
I1
+
Vtotal
_
)20sin(143.0287.0
or
or
1
11
11
1
tVVV
VVV
IRV
VRR
RV
p
totalp
eqp
totaleq
eqp
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Example 4 (con’t)Finally, use the
original circuit to find I2 and I3. +
V1
_
+
Vp
_
I1
I2 I3
)20sin(357.0714.0
or
2
12
12
132
32
tmAmAI
IR
RI
IRR
RI
eq
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Example 4 (con’t)Lastly, the
calculation for I3.+
V1
_
+
Vp
_
I1
I2 I3
)20sin(43.186.2
or
or
3
213
13
13
132
23
tmAmAI
III
IR
RI
IRR
RI
eq
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SummaryThe equations used
to calculate the voltage across a specific resistor Rn in a set of resistors in series are:
The equations used to calculate the current flowing through a specific resistor Rm in a set of resistors in parallel are:
totaln
eqn
totaleq
nn
VG
GV
VR
RV
total
total
I
I
eq
mm
m
eqm
G
GI
R
RI