single loop circuits a single loop circuit is one which has only a single loop. the same current...
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SINGLE LOOP CIRCUITSSINGLE LOOP CIRCUITS
• A single loop circuit is one which has only a single loop.
• The same current flows through each element of the circuit-the elements are in series.
• We will consider circuits consisting of voltage sources and resistors.
VOLTAGE DIVIDERVOLTAGE DIVIDER
Consider two resistors in series with a voltage v(t) across them:
R1
R2
-
v1(t)
+
+
-
v2(t)
+
-
v(t)21
11 )()(
RR
Rtvtv
21
22 )()(
RR
Rtvtv
IMPORTANT VOLTAGEDIVIDER EQUATIONS
+-
+-
+-
+ -
+-
+ -
FIRST GENERALIZATION: MULTIPLE SOURCES
i(t)
KVL
01542321 vvvvvvv RR
Collect all sources on one side
2154321 RR vvvvvvv
21 RReq vvv eqv
1R
2R
Voltage sources in series can be algebraically added to form an equivalent source.
We select the reference direction to move along the path.Voltage drops are subtracted from rises
1R
2R
1Rv
2Rv
1v
2v
3v
4v
5v
SECOND GENERALIZATION: MULTIPLE RESISTORS
APPLY KVLTO THIS LOOP
VOLTAGE DIVISION FOR MULTIPLE RESISTORS
iRv iRi
Multiple Sources/ResistorsMultiple Sources/Resistors
)(1 tv
)(2 tv
)(3 tv
1R
4R
2R
3R+
+
+
_
_
_
_+)(tveq eqR
These two circuits are equivalent where
)()()()( 321 tvtvtvtveq and 4321 RRRRReq
I R1 R2 V
+
-
I1 I2
Single Node Pair CircuitSingle Node Pair Circuit
How do we find I1 and I2?
Apply KCL at the Top NodeApply KCL at the Top Node
I1 + I2 = I
11 R
VI
22 R
VI
R1 R2 V
+
-
I1 I2I
Solve for Solve for VV
2121
11
RRV
R
V
R
VI
21
21
21
111
RR
RRI
RR
IV
Equivalent ResistanceEquivalent Resistance
If we wish to replace the two parallel resistors with a single resistor whose voltage-current relationship is the same, the equivalent resistor has a value of:
21
21
RR
RRReq
21
21
RR
RRIV
21
2
11 RR
RI
R
VI
Now to find Now to find II11
• This is the current divider formula.
• It tells us how to divide the current through parallel resistors.
What is the formula for What is the formula for II22??
Is2 VR1 R2
+
-
I1 I2
More Than One SourceMore Than One Source
How do we find I1 or I2?
Is1
Apply KCL at the Top NodeApply KCL at the Top Node
I1 + I2 = Is1 - Is2
212121
11
RRV
R
V
R
VII ss
21
2121 RR
RRIIV ss
Multiple Current SourcesMultiple Current Sources
• We find an equivalent current source by algebraically summing current sources.
• We find an equivalent resistance.
• We find V as equivalent I times equivalent R.
• We then find any necessary currents using Ohm’s law.
SERIES AND PARALLEL SERIES AND PARALLEL RESISTOR COMBINATIONSRESISTOR COMBINATIONS
• For analysis, series resistors can be replaced by an equivalent resistor.
• Parallel resistors can be replaced by an equivalent resistor/ impedance.
• Complicated networks of resistors can be replaced by a single equivalent resistor.
Equivalent ResistanceEquivalent Resistance
i(t)
+
-
v(t)
i(t)
+
-
v(t)Req
Req is equivalent to the resistor network on the
left in the sense that they have the same i-v characteristics.
Equivalent ResistanceEquivalent Resistance
• The rest of the circuit cannot tell whether the resistor network or the equivalent resistor is connected to it.
• The equivalent resistance cannot be used to find voltages or currents internal to the resistor network.
Series ResistanceSeries Resistance
R1
R3
R2 Req
Req = R1 + R2 + R3
Two elements are in series if the current that flowsthrough one must also flow through the other.
Parallel ResistanceParallel Resistance
Req
1/Req = 1/R1 + 1/R2 + 1/R3
R3R2R1
Two elements are in parallel if they areconnected between the same two nodes.
Circuits with Series and Parallel Circuits with Series and Parallel CombinationsCombinations
• The combination of series and parallel resistances can be used to find voltages and currents in circuits
• Simplification– Resistances are combined to create a simple
circuit (usually one source and one resistance), from which a voltage or current can be found. Start from the furthest branch from the source.
• Backtracking– Once the voltage or current is found, KCL and
KVL, Ohm’s Law, Voltage and Current Dividers are used to work back through the network to find voltages and currents.
FIRST WE PRACTICE COMBINING RESISTORS
6k||3k
(10K,2K)SERIES
SERIESk3
kkk 412||6
k12k3
k5
EXAMPLES COMBINATION SERIES-PARALLEL
k9
kkk 69||18
kkk 1066
AN EXAMPLE WITHOUT REDRAWING
kkk 612||12 kkk 26||3
)24(||6 kkk
RESISTORS ARE IN SERIES IF THEY CARRYEXACTLY THE SAME CURRENT
RESISTORS ARE IN PARALLEL IF THEY ARECONNECTED EXACTLY BETWEEN THE SAME TWONODES
AN “INVERSE SERIES PARALLEL COMBINATION”
AVAILABLE ARERESISTORS ONLY
WHEN600mV BE MUST
1.0
3AIVR
2.03
6.
A
VR REQUIRED 1.01.0R
AVAILABLE ARERESISTORS ONLY
WHEN600mV BE MUST
1.0
9AIVR
0667.09
6.
A
VR REQUIRED
SIMPLE CASE
NOT SO SIMPLE CASE
FIRST REDUCE IT TO A SINGLE LOOP CIRCUITk12kk 12||4
k6
kk 6||6
k
VI
12
121
)12(93
3
aV
SECOND: “BACKTRACK” USING KVL, KCL OHM’S
k
VI a
62 :SOHM'0321 III :KCL
3*3 IkVb :SOHM'
3I
…OTHER OPTIONS...
4
34
*4124
12
IkV
II
b
5
345
*3
0
IkV
III
C
:SOHM'
:KCL