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1 ECE 3336 Introduction to Circuits & Electronics Note Set #3 Equivalent Circuits and Tools Spring 2015, TUE&TH 5:30-7:00 pm Dr. Wanda Wosik

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2 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 i A||B =i C are the same & voltage V 2 is the same A B i A||B C iCiC Equivalent circuit A B

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3 Equivalent Circuits Example: The same circuit but different equivalent circuit at different points A B i0i0 VDVD } This part of the circuit must not “notice” any change on the right. D iDiD VDVD } Equivalent circuit All elements to the right of V S2 are replaced by equivalent circuit D. Currents i 0 =i D are the same Voltages V 2 &V 3 lost their meanings but V D is the same.

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4 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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5 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

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Series Connections of Elements We will substitute the chain of resistors by one equivalent resistor R EQ

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7 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. V1V1 V2V2 circuit voltage +- A hydraulic analogy The analogy is between voltage and height Hight Voltage

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Parallel Resistors and KCL 8 Similarly, we will substitute the resistors in parallel by one equivalent resistor R EQ

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9 Series Resistors Equivalent Circuits Series resistors, R 1 and R 2, can be replaced with an equivalent circuit (with respect to the rest of the circuit) with a single resistor R EQ, as long as v R1 v R2 + + - - i R1 =i R2 Because: v REQ + - i No V R1 and V R2

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10 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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11 The Resistors Must be in Series Resistors R 1 and R 2 cannot be replaced with a single resistor R EQ R 1 and R 2 are not in series here.

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12 Parallel Resistors Equivalent Circuits Here: Parallel resistors, R 1 and R 2, can be replaced with an equivalent circuit with a single resistor R EQ. i R1 i R2 v R1 =v R2 v REQ + - i=i R1 +i R2 Notation R 1 ||R 2 Note that individual currents do not exist now

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13 Two and More Parallel Resistors N parallel resistors will have an equivalent value: Notation: R 1 ||R 2 ||R 3 ||…||R N R EQ for 2 parallel resistors:

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14 The Resistors NOT in Parallel R 1 and R 2, can be replaced with R EQ NOT PARALLEL

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Wheatstone Bridge Circuits Important Applications of Series and Parallel Connections

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16 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.

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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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18 Voltage Divider Rule (VDR) The Voltage Divider Rule involves the voltages across series resistors. We find the voltage on one element ex. V R1 (or V R2 ) that is the fraction of the total voltage V TOTAL. ixix For R 2 For R 1 But also V R2 Note the voltages polarities of in VDR

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19 Voltage Divider Rule (VDR) Negative Polarity The Voltage Divider Rule involves the voltages across series resistors. We find the voltage on one element ex. V R1 (or V R2 ) that is the fraction of the total voltage V TOTAL. ixix For R 2 For R 1 V R2 Note the voltage polarity of in VDR; NOW THEY ARE CHANGED

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20 Current Divider Rule (CDR) This is our Second Circuit Analysis Tool to make circuit analysis quicker and easier. If the current i TOTAL entering the node at two resistors is known we can find the currents through each of the resistors (R 1 &R 2 ) vxvx

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21 Current Divider Rule For Each Resistor Note the polarities of all currents and the voltage.

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22 The Current Divider Rule Direct write-up for the Current Divider Rule (CDR). This is: voltage divided by resistance v x /R 1 /R 1 ( ) R1R1

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23 Negative Signs in the Current Divider Rule Change of the sign of the current i Q in resistor R 1 to have relative polarity opposite to i TOTAL.

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24 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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26 Example Problem

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