Simple Circuits & Kirchoff’s Rules

1. Simple Circuits&Kirchoff’s Rules

2. Simple Series Circuits • Each device occurs sequentially. • The light dilemma: If light goes all of them go .

3. Simple Series Circuit - Conservation of Energy • In a series circuit, the of the is equal to . Vsource + Where we consider the source voltage to be and the voltage drops of each device to be . Vsource = Since V = (): Vsource =

4. R1 + V R2 R3 Simple Series Circuit - Conservation of Charge • In a series circuit, the same amount of passes through each device. IT =

5. Simple Series Circuit – Determining Requivalent • What it the total in a series circuit? • Start with of Vsource = Vsource = • Due to conservation of charge, ITotal = I1 = I2 = I3, we can factor out I such that Vsource = • Since Vsource = : RTotal = REq =

6. Simple Parallel Circuit • A parallel circuit exists where components are connected across the same . • Parallel circuits are similar to those used in . + V

7. Simple Parallel Circuits • Since each device is connected across the : Vsource = + V

8. Simple Parallel Circuits AnalogyHow Plumbing relates to current • In parallel circuits, the is equal to the of the through each individual leg. • Consider your home plumbing: • Your water comes into the house under pressure. • Each faucet is like a that occupies a leg in the circuit. You turn the valve and the water flows. • The drain reconnects all the faucets before they go out to the septic tank or town sewer. • All the water that flows through each of the faucets adds up to the total volume of water coming into the house as well as that going down the drain and into the sewer. • This analogy is similar to current flow through a parallel circuit.

9. + V Simple Parallel Circuits – Conservation of Charge & Current • The total from the voltage source(pressurized water supply) is equal to the sum of the (flow of water through faucet and drain) in each of the (faucets) ITotal =

10. Simple Parallel Circuit – Determining Requivalent • What it the total resistance in a parallel circuit? • Using conservation of charge ITotal = or • Since Vsource = V = V = Vwe can substitute Vsource in (1) as follows

11. Simple Parallel Circuit – Determining Requivalent • What it the total resistance in a parallel circuit (cont.)? • However, since ITotal = /substitute in (2) as follows • Since Vsource cancels, the relationship reduces to Note: Rtotal has been replaced by .

12. Kirchoff’s Rules • Loop Rule (Conservation of ): • The sum of the ( )equals the sum of the () around a closed loop. • Junction Rule (Conservation of Electric ): • The sum of the magnitudes of the going into a junction equals the sum of the magnitudes of the leaving a junction.

13. ΣV R1 + V R2 R3 Rule #1: Voltage Rule (Conservation of ) Vsource – V1 – V2 – V3 =

14. Rule #2: Current Rule (Conservation of Electric ) I1 I2 I3 I1 + I2 + I3 =

15. I1 I2 R1 = 5Ω R3 = 5Ω R2 = 10Ω + + 1 = 3V 2 = 5V I3 Example Using Kirchoff’s Laws • Create individual loops to analyze by Kirchoff’s . • Arbitrarily choose a direction for the current to flow in each loop and apply Kirchoff’s .

16. Ex. (cont.) • Apply Kirchoff’s Current Rule ( = ): I1 + I2 = (1) • Apply Kirchoff’s Voltage Rule to the left loop (Σv = ): 1 – – = 1– – = • Substitute (1) to obtain: 1 – – ( ) = (2)

17. Ex. (cont.) • Apply Kirchoff’s Voltage Rule to the right loop: 2 – – = 2– – = • Substitute (1) to obtain: 2 – – ( ) = (3)

18. I1 = Ex. (cont.) • List formulas to analyze. I1 + I2 = (1) 1 – – ( ) = (2) 2 – – ( ) = (3) • Solve 2 for I1 and substitute into (3) 1 – – – = – – = – 1 I1 ( ) = 1 - 1 - ( )

19. 2 – – + = [ ( ) [ 2 – – – = Ex. (cont.) [ () ( ) • Plug in known values for R1, R2, R3, 1 and 2 and then solve for I2 and then I3. [ Multiply by (R1 + R2) to remove from denominator. 2( ) – ( ) – + –( ) = 0 I2 = A

20. 1 - ( ) 3V –( )( ) ( + ) I1 = I1 = Ex. (cont.) • Plug your answer for I2 into either formula to find I1 • 1 – – ( ) = • What does the tell you about the current in loop 1? I1 =

21. Ex. (cont.) • Use formula (1) to solve for I3 • I1 + I2 = • + =

22. R1 I3 e 2 I2 I1 R2 R3 e 1 How to use Kirchhoff’s Laws A two loop example: • Analyze the circuit and identify all circuit nodes and use KCL. • (1)I1 = I2 + I3 • Identify all independent loops and use KVL. (2)e1-I1R1-I2R2 = 0 (3) e1-I1R1-e2-I3R3 = 0 (4) I2R2-e2-I3R3 = 0

23. R1 I3 I2 I1 e 2 R2 R3 e 1 From eqn. (2) From eqn. (3) Þ How to use Kirchoff’s Laws • Solve the equations for I1, I2, and I3: • Firstfind I2 and I3 in terms of I1: Now solve for I1using eqn. (1):

24. R1 I3 I2 I1 e 2 R2 R3 e 1 Let’s plug in some numbers e1 = 24 V e 2 = 12 V R1= 5W R2=3W R3=4W Then,and I1=2.809 A I2= 3.319 A, I3= -0.511 A