1. Chp 2.[5-7] Series-Parallel Resistors
Engineering 43
Chp 2.[5-7] Series-Parallel Resistors
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer

2. Series Parallel
Up To Now We Have Studied Circuits That Can Be Analyzed With One Application Of KVL (Single Loop) Or KCL (Single Node-pair)
We Have Also Seen That In Some Situations It Is Advantageous To Combine Resistors To Simplify The Analysis Of A Circuit
Now We Examine Some More Complex Circuits Where We Can Simplify The Analysis Using Techniques:
Combining Resistors
Ohm’s Law

3. Resistor Equivalents Series Parallel
Resistors Are In Series If They Carry Exactly The Same Current
Parallel
Resistors Are In Parallel If They have Exactly the Same Potential Across Them

4. Combine Resistors Example: Find RAB
SERIES
6k||3k = 2k
(10K,2K)SERIES
(4K,2K)SERIES
(3K,9K)SERIES

5. More Examples
Step-1: Series Reduction
Step-2: Parallel Reduction

6. Example w/o Redrawing Step-1: 4k↔8k = 12k Step-2: 12k12k = 6k
Step-4: 6k(4k↔2) = 3k = RAB

7. Inverse Series Parallel Combo
Find R: Simple Case
Constraints
VR = 600 mV
I = 3A
Only 0.1Ω R’s Available
Recall R = V/I
Since R>Ravail, Then Need to Run in Series

8. More Complex Case Find R for Constraints
VR = 600 mV
I = 9A
Only 0.1Ω Resistors Available
Either of These 0.1Ω R-Networks Will Work
33.33 mΩ
0.1║(0.1↔0.1)

9. Effect Of Resistor Tolerance
The R Spec: 2.7k, ±10%
What Are the Ranges for Current & Power?
I = V/R
P = V2/R
For Both I & P the Tolerance.: -9.1%, +11.1%
Asymmetry Due to Inverse Dependence on R

10. Series-Parallel Resistor Circuits
Combing Components Can Reduce The Complexity Of A Circuit And Render It Suitable For Analysis Using The Basic Tools Developed So Far
Combining Resistors In SERIES Eliminates One NODE From The Circuit
Combining Resistors In PARALLEL Eliminates One LOOP From The Circuit

11. S-P Circuit Analysis Strategy
Reduce Complexity Until The Circuit Becomes Simple Enough To Analyze
Use Data From Simplified Circuit To Compute Desired Variables In Original Circuit
Hence Must Keep Track Of Any Relationship Between Variables

12. Example – Ladder Network
Find All I’s & V’s in Ladder Network
1st: S-P Reduction
2nd: S-P Reduction
Also by Ohm’s Law

13. Ladder Network cont. Final Reduction; Find Calculation Starting Points
Now “Back Substitute” Using KVL, KCL, and Ohm’s Law
e.g.; From Before

14. More Examples
Voltage Divider
Current Divider

15. “BackSubbing” Example
Given I4 = 0.5 mA, Find VO

16. BackSubbing Strategy Always ask: What More Can I Calculate?
In the Previous Example Using Ohm’s Law, KVL, KCL, S-P Combinations - Calc:

17. Final Example Find VS Straight-ForwardVB
Find VS
Straight-Forward IB IS
Or, Recognize As Inverse V-Divider

18. Final Example cont VB
Inverse Divider Calculation IS IB

19. Wye↔ Transformations
This Circuit Has No Series or Parallel Resistors
If We Could Make The Change Below Would Have Series- Parallel Case

20. Y↔ Xforms cont
Then the Circuit Would Appear as Below and We Could Apply the Previous Techniques

21. Wye↔ Transform Eqns →Y (pg. 51) ←Y (pg. 51) →Y ←Y
Rac =R1||(R2↔R3)
Rab = Ra + Rb
←Y
→Y (pg. 51)
←Y (pg. 51)

22. ↔Y Application Example
 Connection
Find IS
Use the →Y Eqns to Arrive at The Reduced Diagram Below
Calc IS
Req

23. Another ↔Y Example For this Ckt Find Vo Keep This Node-Pair
Convert this Y to Delta

24. Another ↔Y Example cont
Notice for Y→Δ in this Case Ra = Rb = Rc = 12 kΩ
Only need to Calc ONE Conversion
The Xformed Ckt
||-R’s Form a Current Divider

25. Another ↔Y Example cont
The Ckt After ||-Reductions
Can Easily Calc the Current That Produces Vo
Then Finally Vo by Ohm’s Law

26. WhiteBoard Work Done Previously
Let’s Use KCL to Derive the Req for N Parallel Resistors
Done Previously