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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 pm Dr. Wanda Wosik 1 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 iA||B A B A Equivalent circuit iC C B 2 Equivalent Circuits Example: The same circuit but different equivalent circuit at different points • 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. Equivalent circuit i0 iD D A VD B VD } } This part of the circuit must not “notice” any change on the right. 3 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. 4 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. current • 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. 5 Series Connections of Elements We will substitute the chain of resistors by one equivalent resistor REQ Parallel Connection of Circuit Elements • A hydraulic analogy Parts of a circuit are in parallel if the same voltage is across both of them. Voltage V1 • The same exact voltage across each part of the circuit means that the two end points must be connected together. voltage + circuit - Pipe Section 1 Hight Pipe Section 2 circuit The analogy is between voltage and height V2 7 Parallel Resistors and KCL Similarly, we will substitute the resistors in parallel by one equivalent resistor REQ 8 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 REQ = R1 + R2 . + i vR1 - + R1 iR1=iR2 + Rest of the Circuit vREQ REQ Rest of the Circuit Because: vR1 = iR1R1 vR2 = iR2 R2 vR2 R2 No VR1 and VR2 - vREQ = vR1 + vR2 so vREQ =i(R1 + R2 ) = iREQ 9 More than 2 Series Resistors • In case of N series resistors we have R1 REQ = R1 + R2 + ... + RN . Any voltage drop on individual resistor in the equivalent circuit will be “lost” Rest of the Circuit REQ Rest of the Circuit R2 10 The Resistors Must be in Series R1 and R2 are not in series here. • Resistors R1 and R2 cannot be replaced with a single resistor REQ R1 REQ ¹ R1 + R2 . iX + vR2 Rest of the Circuit REQ Rest of the Circuit R2 11 Parallel Resistors Equivalent Circuits Here: Parallel resistors, R1 and R2, can be replaced with an equivalent circuit with a single resistor REQ. vREQ = vR1 = vR 2 Note that individual currents do not exist now vR1=vR2 iR2 iR1 = vR1 / R1 R2 iR2 = vR2 / R2 + i=iR1+iR2 iR1 R1 vREQ Rest of the Circuit REQ Rest of the Circuit iEQ = iR1 + iR2 so iEQ =vREQ / (R1 + R2 ) = vREQ / REQ 1 1 1 = + REQ R1 R2 Notation R1||R2 - 12 Two and More Parallel Resistors REQ for 2 parallel resistors: 1 1 1 = + REQ R1 R2 REQ = R1 || R2 = R1 R2 R1 + R2 N parallel resistors will have an equivalent value: R2 R1 Rest of the Circuit REQ Rest of the Circuit 1 1 1 1 = + + ... + . REQ R1 R2 RN Notation: R1||R2||R3||…||RN 13 The Resistors NOT in Parallel R1 and R2, can be replaced with REQ 1 1 1 ¹ + . REQ R1 R2 iR2 R2 R1 Rest of the Circuit REQ Rest of the Circuit NOT PARALLEL 14 Important Applications of Series and Parallel Connections Wheatstone Bridge Circuits 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. 16 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 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. vTOTAL iX = . R1 + R2 Note the voltages polarities of in VDR But also For R1 For R2 vR1 vR2 ix = = R1 R2 vR1 = vTOTAL R1 R1 + R2 vR2 = vTOTAL R2 R1 + R2 Other Parts of the Circuit + VR2 R2 ix vTOTAL + vR1 R1 - Other Parts 18 of the Circuit 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. Note the voltage polarity of in VDR; NOW THEY ARE CHANGED For R1 For R2 vR1 = vTOTAL R1 R1 + R2 vR2 = -vTOTAL Other Parts of the Circuit + VR2 R2 ix vTOTAL R2 R1 + R2 + vR1 R1 - Other Parts 19 of the Circuit Current Divider Rule (CDR) This is our Second Circuit Analysis Tool to make circuit analysis quicker and easier. Other Parts of the Circuit If the current iTOTAL entering the node at two resistors is known we can find the currents through each of the resistors (R1&R2) æ RR ö v X = iTOTAL çç 1 2 ÷÷. è R1 + R2 ø v X = iR1R1 iR1 = iTOTAL iTOTAL iR1 R2 R1 + R2 ( v X = iTOTAL R1 || R2 R1 v x Other Parts of the Circuit R2 20 ) Current Divider Rule For Each Resistor iR1 = iTOTAL iR 2 = iTOTAL R2 . R1 + R2 Other Parts of the Circuit iTOTAL R1 . R1 + R2 Note the polarities of all currents and the voltage. + iR1 R1 vX iR2 R2 Other Parts of the Circuit 21 The Current Divider Rule Other Parts of the Circuit Direct write-up for the Current Divider Rule (CDR). iTOTAL This is: voltage divided by resistance vx/R1 + R2 ) R1 /R1 iR = iTOTAL( 1 R1 + R2 iR1 R1 R2 vX - Other Parts of the Circuit 22 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. iQ = -iTOTAL Other Parts of the Circuit R2 . R1 + R2 iTOTAL iQ R1 Other Parts of the Circuit R2 23 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. 24 25 Example Problem 37[kW] 2.2[kW] 4.7[kW] + 8.2[kW] 5[mA] 27[kW] 6.2[kW] iX 3.3[kW] vW - Find iX and vW. 26