ECE 3336 Introduction to Circuits & Electronics

1. ECE 3336 Introduction to Circuits & Electronics Note Set #3 Equivalent Circuits and Tools Fall 2013, TUE&TH4:00-5:300 pm Dr. Wanda Wosik

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 iA||B=iC are the same & voltage V2 is the same A Equivalent circuit iC iA||B C B

3. 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=iDare the same • Voltages V2&V3 lost their meanings but VD is the same. Equivalentcircuit iD i0 A D B VD VD } } This part of the circuit must not “notice” any change on the right.

4. Equivalent Circuits Summing up: Basic Requirements • Equivalent circuits as being equivalent in terms ofterminal properties. • The properties (voltage, current, power) within the equivalent circuit may be different.

5. Series Connections of Elements current • 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 greenpart and black are not in se | ries.

6. Series Connections of Elements We will substitute the chain of resistors by one equivalent resistor REQ

7. Parallel Connection of Circuit Elements A hydraulic analogy Voltage • 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 connectedtogether. V1 Hight voltage + - circuit The analogy is between voltage and height circuit V2

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

9. 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 + i + vR1 iR1=iR2 vREQ - + Because: vR2 No VR1 and VR2 - -

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”

11. 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

12. ParallelResistorsEquivalent Circuits Note that individual currents do not exist now Here: Parallel resistors, R1 and R2, can be replaced with an equivalent circuit with a single resistor REQ. i=iR1+iR2 vR1=vR2 + iR1 iR2 vREQ - Notation R1||R2

13. Two and More Parallel Resistors REQ for 2 parallel resistors: N parallel resistors will have an equivalent value: Notation: R1||R2||R3||…||RN

14. The Resistors NOT in Parallel R1 and R2, can be replaced with REQ NOT PARALLEL

15. Important Applications of Series and Parallel Connections Wheatstone Bridge Circuits

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.

17. 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 and Current Divider Rules

18. 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. VR2 ix But also Note the voltages polarities of in VDR For R1 For R2

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. VR1 (or VR2)that is the fraction of the total voltage VTOTAL. VR2 ix Note the voltage polarity of in VDR; NOW THEY ARE CHANGED For R1 For R2

20. 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

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

22. The Current Divider Rule Direct write-up for the Current Divider Rule (CDR). This is: voltage divided byresistance vx/R1 /R1 R1 ( )

23. 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.

24. PolaritiesVoltage 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.

26. Example Problem