1. ECE 3301 General Electrical Engineering
Presentation 03
Simple Resistive Circuits

2. Resistivity Length l (m) I 𝑅=𝜌 𝑙 𝐴 Resistivity ρ (Ω-m)
Cross-sectional Area A (m2 )
Assuming the Current is uniformly distributed across the Cross-Sectional Area.

3. Simple Resistive Circuits
The simplest circuit is a voltage source connected across a resistance as demonstrated below. VS IS R
VR= VS

4. Simple Resistive Circuits
The voltage source VS is placed across the resistance R and the current IS flows in the closed loop. VS IS R
VR= VS

5. Simple Resistive Circuits
Assuming VS and R are known, the current is given by Ohm’s Law.
+ 𝑉 𝑆 − 𝑉 𝑅 =0
𝑉 𝑆 − 𝐼 𝑆 𝑅=0
𝐼 𝑆 𝑅= 𝑉 𝑆
𝐼 𝑆 = 𝑉 𝑆 𝑅 VS IS R
VR= VS

6. Resistances in Series VS IS V1 V2 R1 R2 I1 I2
A slightly more complicated resistive circuit has two resistors in series.

7. Resistances in Series
Note that resistors R1 and R2 are connected end-to-end.
In this configuration they share the same current
IS = I1 = I2 VS IS V1 V2 R1 R2 I1 I2

8. Series Connection
When the same continuous current flows through circuit elements, those circuit elements are connected in series.

9. Resistances in Series VS IS V1 V2 R1 R2
The voltage source VS is placed across the series combination of R1 and R2.
The current IS is established in the closed loop.

10. Resistances in Series VS IS V1 V2 R1 R2
There will be a voltage V1 “dropped” across resistor R1
There will be a voltage V2 “dropped” across resistor R2

11. Resistances in Series VS IS V1 V2 R1 R2
Applying Kirchhoff’s Voltage Law to this closed loop gives the equation
+ 𝑉 𝑆 − 𝑉 1 − 𝑉 2 =0
𝑉 𝑆 = 𝑉 1 + 𝑉 2

12. Resistances in Series VS IS V1 V2 R1 R2
Using Ohm’s Law, we may express the voltage across each resistance by the product of the current and resistance, IR.
This gives the equation
𝑉 𝑆 = 𝐼 𝑆 𝑅 1 + 𝐼 𝑆 𝑅 2

13. Resistances in Series Factoring IS reveals 𝑉 𝑆 = 𝐼 𝑆 𝑅 1 + 𝑅 2VS IS V1 V2 R1 R2
Factoring IS reveals
𝑉 𝑆 = 𝐼 𝑆 𝑅 1 + 𝑅 2
The quantity in parentheses represents the equivalent resistance of the series connection of R1 and R2.

14. Resistances in Series 𝑅 eq = 𝑅 1 + 𝑅 2
The current in the loop is given by
𝐼 𝑆 = 𝑉 𝑆 𝑅 eq VS
Req

15. The same current, I, flows through the resistors.
Vab I V1 V2 R1 R2 Vk Rk a b
This same procedure may be applied to any number of resistors connected in series.
The same current, I, flows through the resistors.
The total voltage from point a to point b is Vab.

16. Vab I V1 V2 R1 R2 Vk Rk a b
Applying Kirchhoff’s Voltage Law to this combination of resistors leads to the equation:
𝑉 ab = 𝑉 1 + 𝑉 2 + ∙∙∙+ 𝑉 k

17. Using Ohm’s Law, the same voltage may be expressed
Vab I V1 V2 R1 R2 Vk Rk a b
Using Ohm’s Law, the same voltage may be expressed
𝑉 ab =𝐼 𝑅 1 +𝐼 𝑅 2 + ∙∙∙+𝐼 𝑅 k

18. Factoring I from each term in the summation reveals
Vab I V1 V2 R1 R2 Vk Rk a b
Factoring I from each term in the summation reveals
𝑉 ab =𝐼 𝑅 1 + 𝑅 2 + ∙∙∙+ 𝑅 k
The term in parentheses represents the equivalent resistance of the series combination of resistances.

19. I a
𝑅 eq = 𝑅 1 + 𝑅 2 + ∙∙∙+ 𝑅 k
𝑅 eq = 𝑛=1 𝑘 𝑅 𝑛
Vab
Req b

20. Resistances in Series
For resistors in series, the equivalent resistance is equal to the sum of the resistances of the individual resistors.
For resistances in series, the equivalent resistance is always larger than the largest resistance in the group.

21. Resistances in Parallel
Consider the resistive circuit below: VS IS V1 R1 V2 R2 I1 I2
Node

22. Resistances in Parallel
Note that resistances R1 and R2 are connected side-by side.
The voltage source VS is placed across both R1 and R2. VS IS V1 R1 V2 R2 I1 I2
Node

23. Resistances in Parallel
+ 𝑉 𝑆 − 𝑉 1 =0
𝑉 𝑆 = 𝑉 1
+ 𝑉 𝑆 − 𝑉 2 =0
𝑉 𝑆 = 𝑉 2
+ 𝑉 1 − 𝑉 2 =0
𝑉 1 = 𝑉 2 VS IS V1 R1 V2 R2 I1 I2
Node
𝑉 𝑆 = 𝑉 1 = 𝑉 2

24. Parallel Connection
When the same voltage appears across circuit elements, those circuit elements are connected in parallel.

25. Resistances in Parallel
Applying Kirchhoff’s Current Law to the node results in the equation:
𝐼 𝑆 − 𝐼 1 − 𝐼 2 =0
𝐼 𝑆 = 𝐼 1 + 𝐼 2 VS IS R1 R2 I1 I2

26. Resistances in Parallel
By applying Ohm’s Law, we can replace the current in each resistance by the quotient of the voltage divided by the resistance, V/R. VS IS R1 R2 I1 I2

27. Resistances in Parallel
𝐼 S = 𝑉 S 𝑅 𝑉 S 𝑅 2 VS IS R1 R2 I1 I2

28. Resistances in Parallel
The term VS can be factored out of both terms.
𝐼 S = 𝑉 S 1 𝑅 𝑅 2 VS IS R1 R2 I1 I2

29. Resistances in Parallel
The term in brackets has the units of conductance, the reciprocal of resistance. This suggests an equivalent conductance:
𝐺 eq = 1 𝑅 eq = 1 𝑅 𝑅 2
𝐺 eq = 𝐺 1 + 𝐺 2

30. Resistances in Parallel
Putting the term in brackets over a common denominator leads to the equation:
1 𝑅 eq = 𝑅 2 + 𝑅 1 𝑅 1 𝑅 2
𝑅 eq = 𝑅 1 𝑅 2 𝑅 1 + 𝑅 2

31. Resistances in Parallel
This connection occurs frequently, so it is worthwhile to remember that for two resistors in parallel :
𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒= 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠 𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠

32. Resistances in Parallel
An arbitrary number of resistances connected in parallel.
Vab I R1 R2 I1 I2 Rk Ik
∙∙∙
Node

33. Resistances in Parallel
The same voltage Vab appears across each of the resistances.
Vab I R1 R2 I1 I2 Rk Ik
∙∙∙
Node

34. Resistances in Parallel
Applying Kirchhoff’s current Law at the node results in the equation:
𝐼= 𝐼 1 + 𝐼 2 + ∙∙∙+ 𝐼 𝑘
Vab I R1 R2 I1 I2 Rk Ik
∙∙∙
Node

35. Resistances in Parallel
Applying Ohm’s Law, we can replace the current through each resistance with the quotient of the voltage divided by the resistance, V/R.
Vab I R1 R2 I1 I2 Rk Ik
∙∙∙

36. Resistances in Parallel
𝐼= 𝑉 ab 𝑅 𝑉 ab 𝑅 2 + ∙∙∙+ 𝑉 ab 𝑅 𝑘
Factoring Vab from each term leads to:
𝐼= 𝑉 𝑎𝑏 1 𝑅 𝑅 2 + ∙∙∙+ 1 𝑅 𝑘

37. Resistances in Parallel
𝐼= 𝑉 𝑎𝑏 1 𝑅 𝑅 2 + ∙∙∙+ 1 𝑅 𝑘
The term in brackets has the units of conductance and can be expressed
𝐺 eq = 1 𝑅 eq = 1 𝑅 𝑅 2 + ∙∙∙+ 1 𝑅 𝑘

38. Resistances in Parallel
1 𝑅 eq = 1 𝑅 𝑅 2 + ∙∙∙+ 1 𝑅 𝑘
Vab I R1 R2 I1 I2 Rk Ik
∙∙∙

39. Resistances in Parallel
1 𝑅 eq = 1 𝑅 𝑅 2 + ∙∙∙+ 1 𝑅 𝑘
1 𝑅 eq = 𝑛=1 𝑘 1 𝑅 𝑛
𝐺 eq = 𝑛=1 𝑘 𝐺 𝑛
Vab I
Req

40. Resistances in Parallel
For resistors in parallel, the equivalent conductance is equal to the sum of the conductances of the individual resistors.
For resistances in parallel, the equivalent resistance is always smaller than the smallest resistance in the group.

41. Resistances in Parallel
Warning: Do not make these common mistakes!
𝑅 eq ≠ 1 𝑅 𝑅 2 + ∙∙∙+ 1 𝑅 𝑘
𝑅 eq ≠ 𝑅 1 𝑅 2 𝑅 3 𝑅 1 + 𝑅 2 + 𝑅 3
These are wrong!

42. Simplifying Resistive CircuitsVS R2 R4
Are R1 and R3 connected in series?
No! They do not share the same current!

43. Simplifying Resistive CircuitsVS R2 R4
Are R2 and R4 connected in parallel?
No! They do not share the same voltage!

44. Simplifying Resistive CircuitsVS R2 R4
Are R1 and R2 connected in series?
 No! They do not share the same current!
 Are R1 and R2 connected in parallel?
No! They do not share the same voltage!

45. Simplifying Resistive CircuitsVS R2 R4
Are R3 and R4 connected in series?
Yes! They share the same current!

46. Simplifying Resistive CircuitsVS R2
R3 + R4

47. Simplifying Resistive Circuits
R2 || (R3 + R4) VS

48. Simplifying Resistive Circuits
R1 + R2 || (R3 + R4) VS
𝑅 eq = 𝑅 1 + 𝑅 2 𝑅 3 + 𝑅 4 𝑅 2 + 𝑅 3 + 𝑅 4
𝑅 eq = 𝑅 1 𝑅 2 + 𝑅 3 + 𝑅 4 + 𝑅 2 𝑅 3 + 𝑅 4 𝑅 2 + 𝑅 3 + 𝑅 4

49. Simplifying Resistive CircuitsVS R1 R2 R3 R4 R5

50. Simplifying Resistive CircuitsVS
R2 + R3 R1 R4

51. Simplifying Resistive CircuitsVS
R5 || (R2 + R3) R1 R4

52. Simplifying Resistive CircuitsVS
R4 + R5 || (R2 + R3) R1

53. Simplifying Resistive CircuitsVS
R1||{R4 + R5 || (R2 + R3)}
𝑅 eq = 𝑅 1 𝑅 4 + 𝑅 5 𝑅 2 + 𝑅 3 𝑅 5 + 𝑅 2 + 𝑅 𝑅 1 + 𝑅 4 + 𝑅 5 𝑅 2 + 𝑅 3 𝑅 5 + 𝑅 2 + 𝑅 3

54. Series or Parallel? R1 R2 R3 R4 R5 R6 R7 R8 VS

55. Series or Parallel? R2 R3 R1 R4 R7 R8 R5 VS R6