Parallel Circuits. Objectives

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1 Parallel Circuits Objectives Identify a parallel circuit Determine the voltage across each parallel branch Apply Kirchhoff s Current Law Determine total parallel resistance Apply Ohm s law in a parallel circuit Use a parallel circuit as a current divider Determine power in a parallel circuit 2 February 2005 Parallel Circuits 2 Professor Andrew H. Andersen 1

2 Characteristics of the Parallel Circuit The voltage across each component (branch) is the same everywhere in the circuit. This means that wherever I try to measure the voltage, I will obtain the same reading, and this is the supply voltage. Each branch has an individual current path. We may calculate the branch current using Ohm's Law if we know the voltage across the component and the resistance. Kirchoff's Current Law Applies. This means that the sum of all the currents entering a node is equal to the sum of all the currents leaving the node IT = I1 + I2 + I IN The inverse of the total resistance in the circuit is equal to inverse the sum of the inverse of the individual resistances = R T R 1 R 2 R 3 R N The sum of the power supplied by the source is equal to the sum of the power dissipated in the components. PT = P1 + P2 + P PN 2 February 2005 Parallel Circuits 3 Identifying Parallel Circuits There is more than one current path (branch) as we move from one source terminal to the other (between two separate points) The voltage between these two points also appears across each of the branches, then there is a parallel circuit between those two points Each current path is called a branch A parallel circuit is one that has two or more branches 2 February 2005 Parallel Circuits 4 Professor Andrew H. Andersen 2

3 Application of a Parallel Circuit All lights and appliances in a home are wired in parallel The switches are located in series with the lights 2 February 2005 Parallel Circuits 5 Application of a Parallel Circuit An advantage of a parallel circuit over a series circuit is that when one component (branch) of the circuit opens the other branches are not affected 2 February 2005 Parallel Circuits 6 Professor Andrew H. Andersen 3

4 Voltage in a Parallel Circuit 2 February 2005 Parallel Circuits 7 Voltage in Parallel Circuits The voltage across any branch of a parallel circuit is equal to the voltage across all of the other branches in parallel 2 February 2005 Parallel Circuits 8 Professor Andrew H. Andersen 4

5 Parallel Circuit 2 February 2005 Parallel Circuits 9 Voltage in a Parallel Ciruit 2 February 2005 Parallel Circuits 10 Professor Andrew H. Andersen 5

6 Determining the Resistance on a Printed Circuit Board 2 February 2005 Parallel Circuits 11 Kirchhoff s Current Law Kirchhoff s current Law (KCL) can be stated as: ΣI = 0 The algebraic sum of all the currents entering and leaving a node is equal to zero ΣIin = ΣIout The algebraic sum of all the currents entering a node is equal to the algebraic sum of all the currents leaving a node I = 400mA 2 February 2005 Parallel Circuits 12 Professor Andrew H. Andersen 6

7 Kirchhoff s Current Law (KCL) The sum of the currents entering a node (total current in) is equal to the sum of the currents leaving that node (total current out) ΣIIN = ΣIOUT IIN1 + IIN IINn = IOUT1 + IOUT IOUTn IT = I1 + I2 + I3 + + In 2 February 2005 Parallel Circuits 13 KCL at Node A ΣIIN = ΣIOUT IT = I1 + I2 + I3 KCL KCL at Node B ΣIIN = ΣIOUT I1 + I2 + I3 = IT 2 February 2005 Parallel Circuits 14 Professor Andrew H. Andersen 7

8 KCL 2 February 2005 Parallel Circuits 15 Find IT 2 February 2005 Parallel Circuits 16 Professor Andrew H. Andersen 8

9 Find I2 2 February 2005 Parallel Circuits 17 What is the Reading of Meters A3 and A5 2 February 2005 Parallel Circuits 18 Professor Andrew H. Andersen 9

10 What is the Reading of Meters A3 and A5 IA3 = 3.5mA IA5 = 2.5mA At X: IT IR1 IA3 = 0 IA3 = IT IR1 = 5mA 1.5mA IA3 = 3.5mA At Y: IA3 IR2 IA5 = 0 IA5 = IA3 IR2 = 3.5mA 1mA IA3 = 2.5mA 2 February 2005 Parallel Circuits 19 Total Parallel Resistance When two or more resistors are connected in parallel, the total resistance of the circuit (REQ) decreases The total resistance of a parallel circuit is always smaller than the value of the smallest resistor The equation to find the equivalent (total) resistance of a parallel circuit is: = R R R R R EQ N R EQ = R R R R N 2 February 2005 Parallel Circuits 20 Professor Andrew H. Andersen 10

11 Shorthand Notation for Parallel Resistors A quick way to indicate 5 resistors connected in parallel, is R1//R2//R3//R4//R5 Conductance G <Siemens> Inverse of resistance 1 G = R G T = R 1 R 2 R N 2 February 2005 Parallel Circuits 21 Resistors in Parallel The total (equivalent) resistance for two resistors in parallel is equal to the product of the two resistors divided by the sum of the two resistors = + R R R RR 1 2 R 1+ R2 EQ 1 2 R EQ = The total (equivalent) resistance for three resistors in parallel is: = + + R EQ R 1 R 2 R 3 RR RR RR R + R + R R + R + R R + R + R R EQ = February 2005 Parallel Circuits 22 Professor Andrew H. Andersen 11

12 Resistors in Parallel The total (equivalent) resistance for two resistors in parallel is equal to the product of the two resistors divided by the sum of the two resistors = + R R R RR 1 2 R EQ = R 1+ R2 If R1 = R2 then: EQ R EQ = = R EQ = RR R + R R 1 2 R 2R 2 February 2005 Parallel Circuits 23 Find REQ 2 February 2005 Parallel Circuits 24 Professor Andrew H. Andersen 12

13 Find REQ Since all resistors are the same value: R R T = ; where N is the number of resistors N 100Ω R T = 5 R T = 20Ω 2 February 2005 Parallel Circuits 25 A parallel circuit acts as a current divider because the current entering the junction of parallel branches divides into several individual branch currents The total current divides among parallel resistors into currents with values inversely proportional to the resistance values Current Dividers 2 February 2005 Parallel Circuits 26 Professor Andrew H. Andersen 13

14 General Current-Divider Formula The current (IX) through any branch equals the total parallel resistance (RT) divided by the resistance (RX) of that branch, and then multiplied by the total current (IT) into the junction of the parallel branches I R = I R X X T T I X= I T R R T X 2 February 2005 Parallel Circuits 27 Current-Divider Formulas for Two Branches For two resistors in parallel, the current-divider formulas for the two branches are: R 1R 2 I 1R 1= I T R 1+ R 2 R 2 I 1= I T R 1+ R 2 R 1R 2 I 2R 2= I T R 1+ R 2 R 1 I 2= I T R 1+ R 2 I 1R 1= I 2R2 R2 I 1= I 2 R 1 2 February 2005 Parallel Circuits 28 Professor Andrew H. Andersen 14

15 Find All Currents 2 February 2005 Parallel Circuits 29 Find All Branch Currents 2 February 2005 Parallel Circuits 30 Professor Andrew H. Andersen 15

16 Find All Currents Using Current Divider 2 February 2005 Parallel Circuits 31 Find All Currents Using Current Divider 2 February 2005 Parallel Circuits 32 Professor Andrew H. Andersen 16

17 Find All Currents Using Current Divider 2 February 2005 Parallel Circuits 33 Power in a Parallel Circuit The total amount of power in a series resistive circuit is equal to the sum of the powers in each resistor in series PS = P1 + P2 + P PN VS I = V1I + V2I+ V3I VNI I 2 RT = I 2 R1 + I 2 R2 + I 2 R I 2 RN 2 February 2005 Parallel Circuits 34 Professor Andrew H. Andersen 17

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