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ECE 3336 Introduction to Circuits & Electronics

Note Set #3

Equivalent Circuits and Tools

Spring 2015,TUE&TH 5:30-7:00 pmDr. Wanda Wosik

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Series and Parallel Resistors Equivalent Circuits

• Equivalent circuit is used to simplify the original circuit but at the terminals it maintains the exact same parameters: ex. voltage and current.

• Example: Elements A||B in the circuit below are replaced by C. • Currents iA||B=iC are the same & voltage V2 is the same

A

B

iA||B

C

iCEquivalent circuit

A

B

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Equivalent Circuits

Example: The same circuit but different equivalent circuit at different points

A

B

i0

VD

}

This part of the circuit must not “notice” any change on the right.

D

iD

VD

}Equivalent

circuit

• All elements to the right of VS2 are replaced by equivalent circuit D.

• Currents i0=iD are the same

• Voltages V2&V3 lost their meanings but VD is the same.

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Equivalent Circuits Summing up: Basic Requirements

• Equivalent circuits as being equivalent in terms of terminal properties.

• The properties (voltage, current, power) within the equivalent circuit may be different.

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Series Connections of Elements• Two parts of a circuit are in series if the same current flows

through both of them.• It means there is no charge accumulation in the circuit.

• A hydraulic analogy: Two water pipes in series - the same flow.

Connections may be not obvious:

• the red part and the blue part of the pipes are in series

• but the blue part and the green part and black are not in se | ries.

current

Series Connections of Elements

We will substitute the chain of resistors by one equivalent resistor REQ

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Parallel Connection of Circuit Elements

• Parts of a circuit are in parallel if the same voltage is across both of them.

• The same exact voltage across each part of the circuit means that the two end points must be connected together.

V1

V2

circuit

circuit

voltage

+ -

A hydraulic analogy

The analogy is between voltage and height

Hight

Voltage

Parallel Resistors and KCL

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Similarly, we will substitute the resistors in parallel by one equivalent resistor REQ

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Series Resistors Equivalent Circuits

• Series resistors, R1 and R2, can be replaced with an equivalent circuit (with respect to the rest of the circuit) with a single resistor REQ, as long as

vR1

vR2

+

+

-

-

iR1=iR2

Because:

vREQ

+

-

i

No VR1 and VR2

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More than 2 Series Resistors

• In case of N series resistors we have

Any voltage drop on individual resistor in the equivalent circuit will be “lost”

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The Resistors Must be in Series

• Resistors R1 and R2 cannot be replaced with a single resistor REQ

R1 and R2 are not in series here.

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Parallel Resistors Equivalent Circuits

Here:

Parallel resistors, R1 and R2, can be replaced with an equivalent circuit with a single resistor REQ.

iR1iR2

vR1=vR2

vREQ

+

-

i=iR1+iR2

Notation R1||R2

Note that individual currents do not exist now

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Two and More Parallel Resistors

N parallel resistors will have an equivalent value:

Notation: R1||R2||R3||…||RN

REQ for 2 parallel resistors:

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The Resistors NOT in Parallel

R1 and R2, can be replaced with REQ

NOT PARALLEL

Wheatstone Bridge Circuits

Important Applications of Series and Parallel

Connections

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Warning

• Orientation and position of the resistors in circuits may be misleading when they just look like being connected in parallel or in series BUT THEY ARE NOT.

Voltage Divider and Current Divider Rules

These rules give us tools for important simplifications in solutions

of circuits to find fractions either of the whole• VDR Voltage that will drop only on selected element(s)

connected in series • CDR Current that will flow only through selected element(s)

connected in parallel

These rules are very useful but have to be carefully used:

directions and signs (YES: polarity) of current and voltages will

be critical

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Voltage Divider Rule (VDR)• The Voltage Divider Rule involves the voltages

across series resistors.

• We find the voltage on one element ex. VR1 (or VR2)

that is the fraction of the total voltage VTOTAL.

ix

For R2

For R1

But also

VR2

Note the voltages polarities of in VDR

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Voltage Divider Rule (VDR)Negative Polarity

• The Voltage Divider Rule involves the voltages across series resistors.

• We find the voltage on one element ex. VR1 (or VR2)

that is the fraction of the total voltage VTOTAL.

ix

For R2

For R1

VR2

Note the voltage polarity of in VDR;NOW THEY ARE CHANGED

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Current Divider Rule (CDR)

This is our Second Circuit Analysis Tool to make circuit analysis quicker and easier.

If the current iTOTAL entering the node at two

resistors is known we can find the currents

through each of the resistors (R1&R2)

vx

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Current Divider Rule For Each Resistor

Note the polarities of all currents and the voltage.

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The Current Divider Rule

Direct write-up for the Current Divider Rule (CDR).

This is: voltage divided by resistancevx/R1

/R1( )

R1

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Negative Signs in the Current Divider Rule

Change of the sign of the current iQ in resistor R1 to have relative polarity opposite to iTOTAL.

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Polarities Voltage Divider and Current Divider Rules

• Correct polarities are critically important for correct solutions of the circuits.

• VDR and CDR confirm the importance of reference polarities.

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Example Problem