ECE201 Lect-7 1

Series and Parallel Resistor Combinations (2.5, 7.5)

Dr. Holbert

September 11, 2001

ECE201 Lect-7 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.

ECE201 Lect-7 3

Introduction

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

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Equivalent Resistancei(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.

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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 ImpedanceI

+

–

V

I

+

–

VZeq

Zeq is equivalent to the network on the left in the sense that they have the same phasor I-V characteristics at the frequency .

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Series Resistance

R1

R3

R2 Req

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.

R1 R2

Series

R1 R2

Not Series

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Zeq

Series Impedance

Z1

Zeq = Z1 + Z2 + Z3

Z3

Z2

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Example: Series Inductors

• What is the equivalent impedance of two series inductors?

L2L1

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Series Inductors

• The equivalent impedance is

j(L1+L2)

• 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

Req

1/Req = 1/R1 + 1/R2 + 1/R3

R3R2R1

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Parallel

Two elements are in parallel if they are connected between (share) the same two end nodes.

Parallel

Not Parallel

R1

R2

R1

R2

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Parallel Impedance

1/Zeq = 1/Z1 + 1/Z2 + 1/Z3

Z3Z1 Z2 Zeq

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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.

21

1

CCj

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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

1k

1k

2k

1k

2k

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

2k2k

1k

2k

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Ladder Network

1k

1k

2k

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Ladder Network

2k2k

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Ladder Network

1k

The equivalent resistance of the ladder network is 1k

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10

769pF

159H

Example: Bandpass Filter

For = 2.86 106, find the equivalent impedance.

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10

-j455

j455

Compute Impedances

Now combine series impedances

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-j45510 j455

Bandpass Filter

Now combine parallel impedances

455.1 88.7

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Bandpass Filter

3.1k7.2045510455

45510455

jj

jj

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Loaded Bandpass Filter

For = 2.86 106, find the equivalent impedance.

10

769pF

159H

50k

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Class Examples

• Learning Extension E2.12

• Learning Extension E2.13

• Learning Extension E7.10