Lecture 22 1

Series and Parallel Resistors/Impedances

Lecture 22 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.

Lecture 22 3

Introduction

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

Lecture 22 4

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.

Lecture 22 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.

Lecture 22 6

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 .

Lecture 22 7

Series Resistance

R1

R3

R2 Req

Req = R1 + R2 + R3

Lecture 22 8

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

Lecture 22 9

Zeq

Series Impedance

Z1

Zeq = Z1 + Z2 + Z3

Z3

Z2

Lecture 22 10

Example: Series Inductors

• What is the equivalent impedance of two series inductors?

L2L1

Lecture 22 11

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.

Lecture 22 12

Parallel Resistance

Req

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

R3R2R1

Lecture 22 13

Parallel

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

Parallel

Not Parallel

R1

R2

R1

R2

Lecture 22 14

Parallel Impedance

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

Z3Z1 Z2 Zeq

Lecture 22 15

Example: Parallel Capacitors

• What is the equivalent impedance of two parallel capacitors?

C1 C2

Lecture 22 16

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

Lecture 22 17

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.

Lecture 22 18

Ladder Network

1k

1k

2k

1k

2k

Find the equivalent resistance.

Lecture 22 19

Ladder Network

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

Lecture 22 20

Ladder Network

2k2k

1k

2k

Lecture 22 21

Ladder Network

1k

1k

2k

Lecture 22 22

Ladder Network

2k2k

Lecture 22 23

Ladder Network

1k

The equivalent resistance of the ladder network is 1k

Lecture 22 24

10

769pF

159H

Bandpass Filter

For = 2.86 106, find the equivalent impedance.

Lecture 22 25

10

-j455

j455

Compute Impedances

Now combine series impedances

Lecture 22 26

-j45510 j455

Bandpass Filter

Now combine parallel impedances

455.1 88.7

Lecture 22 27

Bandpass Filter

3.1k7.2045510455

45510455

jj

jj

Lecture 22 28

Loaded Bandpass Filter

For = 2.86 106, find the equivalent impedance.

10

769pF

159H

50k