Download Series and Parallel Resistor Combinations (2.5, 8.5)

Series and Parallel Resistor
Combinations (2.5, 8.5)
Dr. Holbert
February 6, 2006
ECE201 Lect-6
1
Introduction
• For analysis, series resistors/impedances
can be replaced by an equivalent resistor/
impedance.
• Parallel resistors/impedances can be
replaced by an equivalent resistor/
impedance.
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Introduction
• Complicated networks of resistors/
impedances can be replaced by a single
equivalent resistor/impedance.
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Equivalent Resistance
i(t)
i(t)
+
+
v(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.
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Equivalent 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.
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Equivalent Impedance
I
I
+
+
V
V
Zeq
–
–
Zeq is equivalent to the network on the left in
the sense that they have the same phasor I-V
characteristics at the frequency w.
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Series Resistance
R1
R2
Req
R3
Req = R1 + R2 + R3
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Series
Two elements are in series if the current that
flows through one must also flow through the
other.
Series
R1
R2
Not Series
R1
R2
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Series Impedance
Z1
Z2
Zeq
Z3
Zeq = Z1 + Z2 + Z3
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Example: Series Inductors
• What is the equivalent impedance of two
series inductors?
L1
L2
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Series Inductors
• The equivalent impedance is
Zeq = Z1 + Z2 = jw(L1+L2)
• Two inductors in series are equivalent to a
single inductor whose inductance is the sum
of the two inductances.
Zeq = jw(L1+L2) = jwLeq
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Parallel Resistance
R1
R2
R3
Req
1
1
1
1
 

Req R1 R2 R3
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Parallel
Two elements are in parallel if they are
connected between (share) the same two
(distinct) end nodes.
R1
Parallel
R2
Not
Parallel
R1
R2
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Parallel Impedance
Z1
Z2
Z3
Zeq
1
1
1
1



Z eq Z1 Z 2 Z 3
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Example: Parallel Capacitors
• What is the equivalent impedance of two
parallel capacitors?
C1
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C2
15
Parallel Capacitors
• The equivalent impedance is
1
1
1


 jw C1  jw C2
Z eq Z1 Z 2
1
Z eq 
jw C1  C2 
• Two capacitors in parallel are equivalent to a
single capacitor whose capacitance is the sum of
the two capacitances.
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Flash Example: Ladder Network
Find the equivalent resistance by making
combinations of series and parallel resistors
until only one resistor is left.
1kW
2kW
1kW
2kW
1kW
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1kW
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Example #1: Bandpass Filter
10W
769pF
159mH
For w = 2.86  106, find the equivalent
impedance.
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Compute Impedances
10W
-j455W
j455W
Now combine series impedances
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Bandpass Filter
10W j455W
-j455W
455.1W  88.7
Now combine parallel impedances
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Bandpass Filter
 j 45510  j 455
 20.7kW  1.3
 j 455  10  j 455
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Example #2: Loaded Bandpass Filter
10W
50kW
769pF
159mH
For w = 2.86  106, find the equivalent
impedance.
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Class Examples
• Learning Extension E2.12
• Learning Extension E2.13
• Learning Extension E8.10
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