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

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
iA||B A
Equivalent circuit C iC A
You can click on the links to jump to the subject that you want to learn about now. B 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=iD are the same
Voltages V2&V3 lost their meanings but VD is the same. D iD VD } i0 VD
Equivalent circuit A
You can click on the links to jump to the subject that you want to learn about now. B }
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 of terminal properties.
The properties (voltage, current, power) within the equivalent circuit may be different.

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.
current
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.

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
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.
Voltage V1
circuit
voltage + -
Hight
The analogy is between voltage and height 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. Parallel Resistors Equivalent 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. 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

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 by resistance
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. 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.

26. Example Problem