Series and Parallel Resistors/Impedances

1
Series and Parallel Resistors/Impedances
2
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.

3
Introduction

• Complicated networks of resistors/ impedances can
  be replaced by a single equivalent
  resistor/impedance.

4
Equivalent Resistance

• Req is equivalent to the resistor network on the
  left in the sense that they have the same i-v
  characteristics.

5
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

• 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.

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?

L2
L1
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Series Inductors

• The equivalent impedance is
• jw(L1L2)
• Two inductors in series are equivalent to a
  single inductor whose inductance is the sum of
  the two inductances.

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Parallel Resistance
R3

• 1/Req 1/R1 1/R2 1/R3

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Parallel

• Two elements are in parallel if they are
  connected between the same two nodes.

Parallel
Not Parallel
R1
R2
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Parallel Impedance
Z3
Zeq

• 1/Zeq 1/Z1 1/Z2 1/Z3

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Example Parallel Capacitors

• What is the equivalent impedance of two parallel
  capacitors?

C1
C2
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Parallel Capacitors

• The equivalent impedance is
• Two capacitors in parallel are equivalent to a
  single capacitor whose capacitance is the sum of
  the two capacitances.

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Example Ladder Network

• Ladder networks are used in analog-to-digital
  converters to provide reference voltages that are
  1/2, 1/4, 1/8, etc. of a source voltage.

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Ladder Network

• Find the equivalent resistance.

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Ladder Network

• Find the equivalent resistance by making
  combinations of series and parallel resistors
  until you have only one resistor left.

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Ladder Network
21
Ladder Network
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Ladder Network
2kW
2kW
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Ladder Network
1kW

• The equivalent resistance of the ladder network
  is 1kW.

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Bandpass Filter

• 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
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Loaded Bandpass Filter

• For w 2.86 ? 106, find the equivalent impedance.