Basic Electrical - Kirchhoff's Laws and Voltage Division (PDF)

# Kirchhoff's Voltage Law Kirchhoff's voltage law states that the algebraic sum of all branch voltages around any closed path in a circuit is always zero at all instants of time. When the current passes through a resistor, there is a loss of energy and, therefore, a voltage drop. In any element, th...

# Kirchhoff's Voltage Law Kirchhoff's voltage law states that the algebraic sum of all branch voltages around any closed path in a circuit is always zero at all instants of time. When the current passes through a resistor, there is a loss of energy and, therefore, a voltage drop. In any element, the current always flows from higher potential to lower potential. ## Example Circuit Consider the circuit in the diagram below. It is customary to take the direction of current *I* as indicated in the figure, i.e. it leaves the positive terminal of the voltage source and enters into the negative terminal.  As the current passes through the circuit, the sum of the voltage drop around the loop is equal to the total voltage in that loop. Here the polarities are attributed to the resistors to indicate that the voltages at points *a*, *c* and *e* are more than the voltages at *b*, *d* and *f*, respectively, as the current passes from *a* to *f*. # Kirchhoff's Current Law Kirchhoff's current law states that the sum of the currents entering into any node is equal to the sum of the currents leaving that node. The node may be an interconnection of two or more branches. In any parallel circuit, the node is a junction point of two or more branches. The total current entering into a node is equal to the current leaving that node. ## Example Circuit For example, consider the circuit shown in the diagram below, which contains two nodes *A* and *B*. The total current *IT* entering node *A* is divided into *I1*, *I2* and *I3*. These currents flow out of node *A*. According to Kirchhoff's current law, the current into node *A* is equal to the total current out of node *A*: *IT = I1 + I2 + I3*.  If we consider node *B*, all three currents *I1*, *I2*, *I3* are entering *B*, and the total current *IT* is leaving node *B*, Kirchhoff's current law formula at this node is therefore the same as at node *A*: *I1 + I2 + I3 = IT*. ## General Case In general, the sum of the currents entering any point, node or junction, is equal to the sum of the currents leaving from that point, node or junction, as shown in the diagram below.  If all of the terms on the right side of the equation below are brought over to the left side, their signs change to negative and a zero is left on the right side. *I1 + I2 + I4 + I7 - I3 - I5 - I6 = 0* This means that the algebraic sum of all the currents meeting at a junction is equal to zero. # Voltage Division The series circuit acts as a voltage divider. Since the same current flows through each resistor, the voltage drops are proportional to the values of the resistors. Using this principle, different voltages can be obtained from a single source, called a voltage divider. For example, the voltage across a 40-ohm resistor is twice that of a 20-ohm resistor in a series circuit. ## General Case In general, if the circuit consists of a number of series resistors, the total current is given by the total voltage divided by the equivalent resistance. This is shown in the diagram below.  The current in the circuit is given by *I = V / (R1 + R2 + ... + Rn)*. The voltage across any resistor is nothing but the current passing through it, multiplied by that particular resistor: *Vr1 = IR1* *Vr2 = IR2* *Vr3 = IR3* or *VRm = IRm = Vs (Rm) / (R1 + R2 + ... + Rm)* From the above equation, we can say that the voltage drop across any resistor, or a combination of resistors, in a series circuit is equal to the ratio of that resistance value to the total resistance, multiplied by the source voltage, i.e. *Vm = (Rm / RT) Vs* Where *Vm* is the voltage across mth resistor, *Rm* is the resistance across which the voltage is to be determined and *RT* is the total series resistance. # Current Division In a parallel circuit, the current divides in all branches. Thus, a parallel circuit acts as a current divider. The total current entering into the parallel branches is divided into the branches currents according to the resistance values. The branch having higher resistance allows lesser current, and the branch with lower resistance allows more current. ## Example Circuit Let us find the current division in the parallel circuit shown in the diagram below.  The voltage applied across each resistor is *Vs*. The current passing through each resistor is given by: *I1 = Vs / R1* *I2 = Vs / R2* If *RT* is the total resistance, which is given by *R1R2 / (R1 + R2)*, Total current *IT = Vs / RT = Vs / (R1R2 / (R1 + R2))* or *IT = I1 R1 / (R1 + R2) since Vs = I1 R1* *I1 = IT R2 / (R1 + R2)* *I2 = IT R1 / (R1 + R2)* ## General Case Similarly, from the above equations, we can conclude that the current in any branch is equal to the ratio of the opposite branch resistance to the total resistance value, multiplied by the total current in the circuit. In general, if the circuit consists of m branches, the current in any branch can be determined by: *Ii = (RT / (Ri + RT)) IT* where * Ii represents the current in the ith branch * Ri is the resistance in the ith branch * RT is the total parallel resistance to the ith branch and * IT is the total current entering the circuit.