Physics 212 Lecture 10 Kirchhoff’s Rules

= Last Time Resistors in series: Current through is same. Voltage drop across is IRi Resistors in series: Voltage drop across is same. Current through is V/Ri Resistors in parallel: Solved Circuits V R1 R2 R4 R3 V R1234 I1234 = 5

= New Circuit How Can We Solve This One? THE ANSWER: Kirchhoff’s Rules 5

Kirchoff’s Voltage Rule Kirchoff's Voltage Rule states that the sum of the voltage changes around a circuit must be zero. WHY? The potential difference between a point and itself is zero ! 1 2 3

The voltage “gains” , ( V2-V1) + (V3-V2) + (V1-V3) = 0 True for any loop Add the voltage “drops” ( V1-V2) + (V2-V3) + (V3-V1) = 0 or The voltage “gains” , ( V2-V1) + (V3-V2) + (V1-V3) = 0

Kirchoff’s Current Rule Kirchoff's Current Rule states that the sum of all currents entering any given point in a circuit must equal the sum of all currents leaving the same point. WHY? Electric charge is conserved and cannot accumulate at nodes. Node I1 I2 I4 I3

Checkpoint 2 With the current VOLTAGE DROP Against the current In the following circuit, consider the loop abc. The direction of the current through each resistor is indicated by black arrows. DROP GAIN If we are to write Kirchoff's voltage equation for this loop in the clockwise direction starting from point a, what is the correct order of voltage gains/drops that we will encounter for resistors R1, R2 and R3? A. drop, drop, drop B. gain, gain, gain C. drop, gain, gain D. gain, drop, drop E. gain, gain, drop A B C D E With the current VOLTAGE DROP Against the current VOLTAGE GAIN

In this circuit, assume Vi and Ri are known. Calculation V1 I1 I3 I2 (1) Label and pick directions for each current R1 + - - + For resistors, the “upstream” side is +. + - This is easy for batteries. In this circuit, assume Vi and Ri are known. What is I2 ? R2 V2 R3 V3 (2) Label the + and – side of each element The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Now write down loop and node equations.

In this circuit, assume Vi and Ri are known. Calculation V1 R1 I1 - + + - R2 V2 I2 In this circuit, assume Vi and Ri are known. What is I2? + - + - R3 V3 I3 - + + - How many equations do we need to write down in order to solve for I2? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Why? We have 3 unknowns: I1, I2, and I3 We need 3 independent equations to solve for these unknowns (3) Choose loops and directions The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

In this circuit, assume Vi and Ri are known. Calculation V1 R1 I1 - + + - R2 V2 I2 In this circuit, assume Vi and Ri are known. What is I2 ? + - + - R3 V3 I3 - + + - Which of the following equations is NOT correct? I2 = I1 + I3 + V1 - I1R1 + I3R3 - V3 = 0 + V3 - I3R3 - I2R2 - V2 = 0 + V2 + I2R2 - I1R1 + V1 = 0 Node Outer loop Bottom loop The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Top loop Why is (D) wrong ? Start at negative terminal of V2 and go clockwise around top loop: (+V2) + (+I2R2) + (+I1R1) + (-V1) = 0 This is not the same as answer (D).

In this circuit, assume Vi and Ri are known. Calculation V1 R1 I1 R2 V2 I2 In this circuit, assume Vi and Ri are known. What is I2 ? R3 V3 I3 We have the following 4 equations: We need 3 equations: Which 3 should we use? 1. I2 = I1 + I3 2. + V1 - I1R1 + I3R3 - V3 = 0 3. + V3 - I3R3 - I2R2 - V2 = 0 4. + V2 + I2R2 + I1R1 - V1 = 0 Node Outer Bottom Top A) Any 3 will do B) 1, 2, and 4 C) 2, 3, and 4 Do the addition of equations Why? We need 3 INDEPENDENT equations Equations 2, 3, and 4 are NOT INDEPENDENT Eqn 2 + Eqn 3 = - Eqn 4 We must choose Equation 1 and any two of the remaining ( 2, 3, and 4)

In this circuit, assume Vi and Ri are known. Calculation V1 R1 I1 In this circuit, assume Vi and Ri are known. What is I2? R2 V2 I2 R3 V3 I3 We have 3 equations and 3 unknowns. I2 = I1 + I3 V1 - I1R1 + I3R3 - V3 = 0 V2 + I2R2 + I1R1 - V1 = 0 2V R 2R V I1 I3 I2 (6) Solve the equations The solution will get very messy! Simplify: assume V2 = V3 = V V1 = 2V R1 = R3 = R R2 = 2R The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Calculation: Simplify In this circuit, assume V and R are known. What is I2 ? 2V R 2R V I1 I3 I2 We have 3 equations and 3 unknowns. I2 = I1 + I3 +2V - I1R + I3R - V = 0 (outside) +V + I2(2R) + I1R - 2V = 0 (top) With this simplification, you can verify: I2 = ( 1/5) V/R I1 = ( 3/5) V/R I3 = (-2/5) V/R current direction The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Follow-Up 2V R 2R V I1 I3 I2 We know: I2 = ( 1/5) V/R I1 = ( 3/5) V/R I3 = (-2/5) V/R a b Suppose we short R3: What happens to Vab (voltage across R2?) Vab remains the same Vab changes sign Vab increases Vab goes to zero 2V R 2R V I1 I3 I2 a b c d Why? Redraw: The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. (Va - Vb ) + V - V = 0 Bottom Loop Equation: Vab = 0

V Is there a current flowing between a and b ? a b Yes No Current flows from battery and splits at A Some current flows right Some current flows down V R R Is there a current flowing between a and b ? Yes No No current flows between A & B A & B have the same potential

Checkpoint 3a Consider the circuit shown below. Note that this question is not identical to the similar looking one you answered in the prelecture. I2 2I I I I 2I Which of the following best describes the current flowing in the blue wire connecting points a and b? 1. Same voltage across R, 2R in parallel so IR = 2 I2R . 2. R, 2R combinations are identical so voltage drop in each case is V/2. Therefore total current in top and bottom must be 3 I. 3. Satisfy Kirchhoff current law at node a so Iab = I

What is different? Current flowing from a to b Prelecture Checkpoint What is the same? Current flowing in and out of the battery 2R 3 What is different? Current flowing from a to b

I 2/3I 1/3 I R 2R V 2/3I a b V/2 R 2/3I 2R I 1/3

Checkpoint 3b Current will flow from left to right in both cases Consider the circuit shown below. Checkpoint 3b IA IB Current will flow from left to right in both cases In both cases, Vac = V/2 c In which case is the current flowing in the blue wire connecting points a and b the largest? A. Case A B. Case B C. They are both the same I2R = 2I4R IA = IR – I2R = IR – 2I4R IB = IR – I4R

+ r r V0 VL R R VL V0 Model for Real Battery: Internal Resistance Usually can’t supply too much current to the load without voltage “sagging”

r V0 R VL Voltage divider I = V0 / (r + R) VL = I R VL = V0 R / (r + R) VL V0 R >> r