ENGR. VIKRAM KUMAR B.E (ELECTRONICS) M.E (ELECTRONICS SYSTEM ENGG:) MUET JAMSHORO 1 KIRCHHOFF’S VOLTAGE LAW

BRANCHES AND NODES Branch: Elements connected end-to-end, nothing coming off in between (in series) OR A circuit element between two nodes 2

 Node: Place where elements are joined—entire wire OR Any point where 2 or more circuit elements are connected together - Wires usually have negligible resistance -Each node has one voltage (w.r.t. ground) 3

Loop – a collection of branches that form a closed path returning to the same node without going through any other nodes or branches twice 4

Example How many nodes, branches & loops? + - Vs Is R1R1 R2R2 R3R3 + Vo - 5

Example Three nodes + - Vs Is R1R1 R2R2 R3R3 + Vo - 6

Example 5 Branches + - Vs Is R1R1 R2R2 R3R3 + Vo - 7

Example Three Loops, if starting at node A + - Vs Is R1R1 R2R2 R3R3 + Vo - AB C 8

Kirchhoff’s Voltage Law: Kirchhoff’s voltage law tells us how to handle voltages in an electric circuit. Kirchhoff’s voltage law basically states that the algebraic sum of the voltages around any closed path (electric circuit) equal zero. The secret here, as in Kirchhoff’s current law, is the word algebraic.    There are three ways we can interrupt that the algebraic sum of the voltages around a closed path equal zero. This is similar to what we encountered with Kirchhoff’s current law. Basic Laws of Circuits 9

Kirchoff’s Voltage Law (KVL) The algebraic sum of voltages around each loop is zero Beginning with one node, add voltages across each branch in the loop (if you encounter a + sign first) and subtract voltages (if you encounter a – sign first) Σ voltage drops - Σ voltage rises = 0 Or Σ voltage drops = Σ voltage rises 10

KIRCHOFF’S VOLTAGE LAW (KVL) The sum of the voltage drops around any closed loop is zero. We must return to the same potential (conservation of energy). + - V 2 Path “rise” or “step up” (negative drop) + - V 1 Path “drop” Closed loop: Path beginning and ending on the same node Our trick: to sum voltage drops on elements, look at the first sign you encounter on element when tracing path 11

KVL EXAMPLE Path 1: Path 2: Path 3: vcvc vava ++ ++  vbvb v3v3 v2v2 + - Examples of three closed paths: a b c 12

Kirchhoff’s Voltage Law: Basic Laws of Circuits Consideration 1: Sum of the voltage drops around a circuit equal zero. We first define a drop. We assume a circuit of the following configuration. Notice that no current has been assumed for this case, at this point _ _ _ _ v1v1 v2v2 v4v4 v3v3 13

Basic Laws of Circuits Kirchhoff’s Voltage Law: Consideration 1. We define a voltage drop as positive if we enter the positive terminal and leave the negative terminal. + _v1v1 The drop moving from left to right above is + v 1. + _ v1v1 The drop moving from left to right above is – v 1. 14

Basic Laws of Circuits Kirchhoff’s Voltage Law: _ _ _ _ v1v1 v2v2 v4v4 v3v3 Consider the circuit of following Figure once again. If we sum the voltage drops in the clockwise direction around the circuit starting at point “a” we write: - v 1 – v 2 + v 4 + v 3 = 0 - v 3 – v 4 + v 2 + v 1 = 0 “a”  drops in CW direction starting at “a”  drops in CCW direction starting at “a” 15

Basic Laws of Circuits Kirchhoff’s Voltage Law: Consideration 2: Sum of the voltage rises around a circuit equal zero. We first define a drop. We define a voltage rise in the following diagrams: + _v1v1 + _v1v1 The voltage rise in moving from left to right above is + v 1. The voltage rise in moving from left to right above is - v 1. 16

Basic Laws of Circuits Kirchhoff’s Voltage Law: Consider the circuit of Figure 3.7 once again. If we sum the voltage rises in the clockwise direction around the circuit starting at point “a” we write: __ _ v1v1 v2v2 v4v4 v3v3 “a” + v 1 + v 2 - v 4 – v 3 = 0 + v 3 + v 4 – v 2 – v 1 = 0  rises in the CW direction starting at “a”  rises in the CCW direction starting at “a” _ 17

Basic Laws of Circuits Kirchhoff’s Voltage Law: Consideration 3: Sum of the voltage rises around a circuit equal the sum of the voltage drops. Again consider the circuit of following Figure in which we start at point “a” and move in the CW direction. As we cross elements 1 & 2 we use voltage rise: as we cross elements 4 & 3 we use voltage drops. This gives the equation, _ _ _ _ v1v1 v2v2 v4v4 v3v3 v 1 + v 2 = v 4 + v p 18

Basic Laws of Circuits Kirchhoff’s Voltage Law: Comments. We note that a positive voltage drop = a negative voltage rise. We note that a positive voltage rise = a negative voltage drop. We do not need to dwell on the above tongue twisting statements. There are similarities in the way we state Kirchhoff’s voltage and Kirchhoff’s current laws: algebraic sums … However, one would never say that the sum of the voltages entering a junction point in a circuit equal to zero. Likewise, one would never say that the sum of the currents around a closed path in an electric circuit equal zero. 19

Example Kirchoff’s Voltage Law around 1 st Loop + - Vs Is R1R1 R2R2 R3R3 + Vo - AB C I2I2 I1I1 + I2R2I2R2 - + I 1 R 1 - Assign current variables and directions Use Ohm’s law to assign voltages and polarities consistent with passive devices (current enters at the + side) 20

Example Kirchoff’s Voltage Law around 1 st Loop + - Vs Is R1R1 R2R2 R3R3 + Vo - AB C I2I2 I1I1 + I 2 R I 1 R 1 - Starting at node A, add the 1 st voltage drop: + I 1 R 1 21

Example Kirchoff’s Voltage Law around 1 st Loop + - Vs Is R1R1 R2R2 R3R3 + Vo - AB C I2I2 I1I1 + I 2 R I 1 R 1 - Add the voltage drop from B to C through R 2 : + I 1 R 1 + I 2 R 2 22

Example Kirchoff’s Voltage Law around 1 st Loop + - Vs Is R1R1 R2R2 R3R3 + Vo - AB C I2I2 I1I1 + I 2 R I 1 R 1 - Subtract the voltage rise from C to A through Vs: + I 1 R 1 + I 2 R 2 – Vs = 0 Notice that the sign of each term matches the polarity encountered 1st 23

Kirchhoff’s Voltage Law: Further details. For the circuit of Figure there are a number of closed paths. Three have been selected for discussion v1v1 v2v2 v4v4 v3v3 v 12 v 11 v9v9 v8v8 v6v6 v5v5 v7v7 v Figure Multi-path Circuit. Path 1 Path 2 Path 3 24

Kirchhoff’s Voltage Law: Further details. For any given circuit, there are a fixed number of closed paths that can be taken in writing Kirchhoff’s voltage law and still have linearly independent equations. We discuss this more, later. Both the starting point and the direction in which we go around a closed path in a circuit to write Kirchhoff’s voltage law are arbitrary. However, one must end the path at the same point from which one started. Conventionally, in most text, the sum of the voltage drops equal to zero is normally used in applying Kirchhoff’s voltage law. 25

Kirchhoff’s Voltage Law: Illustration from Figure v1v1 v2v2 v4v4 v3v3 v 12 v 11 v9v9 v8v8 v6v6 v5v5 v7v7 v “a” Blue path, starting at “a” - v 7 + v 10 – v 9 + v 8 = 0 “b” Red path, starting at “b” +v 2 – v 5 – v 6 – v 8 + v 9 – v 11 – v 12 + v 1 = 0 Yellow path, starting at “b” + v 2 – v 5 – v 6 – v 7 + v 10 – v 11 - v 12 + v 1 = 0 Using sum of the drops = 0 26

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