First of all let us introduce with some common elements of a circuit. It is a most common problem that an Electrical Engineer can understand these facts, but could not able to give proper definition of these. So, let see how to define them properly.
Figure 1: An electric network showing nodes, branches, elements and loops.
Electric Network A connection of various circuit elements can be termed as an electric network. The circuit diagram shown in Figure 1 is an electric network.
Electric Circuit A connection of various circuit elements of an electric network forming a closed path is called an electric circuit. The closed path is commonly termed as either loop or mesh. In Figure 1, meshes BDEB, ABCA and BCDB are electric circuits because they form a closed path. In general, all circuits are networks but not all networks are circuits.
Node A connection point of several circuit elements is termed as a node. For instance, A, B, C, D and E are five nodes in the electric network of Figure 1. Please note that there is no element connected between nodes A and C and therefore can be regarded as a single node.
Branch The path in an electric network between two nodes is called a branch. AB, BE, BD, BC, CD and DE are six branches in the network of Figure 1.
Now, come to the analytical part of circuit. There are three primary laws of solving a DC circuit: Ohm’s Law, Kirchoff’s Voltage Law (KVL) and Kirchoff’s Current Law (KCL). We all already have much better knowledge about Ohm’s Law, which is “V = IR”. It does not mean that we don’t know about KVL and KCL. Obviously we do, but I just want to re-install these in your mind with definitions and applications.
Kirchoff’s Voltage Law (KVL):
“The sum of all the voltages (rises and drops) around a closed loop is equal to zero”
In other words, the algebraic sum of all voltage rises is equal to the algebraic sum of all the voltage drops around a closed loop. In figure 1, consider mesh BEDB, then according to KVL, V3 = V4 + V5
Example: In each of the circuit diagram in Figure 2, write the mesh equations using KVL.
Figure 2: Circuit diagrams to demonstrate the application of KVL in the above example.
Figure 2(a) contains a single loop hence a single current, I is flowing around it. Therefore a single equation will result as given below,
Vs = IR1+IR2 ……………………………………………………………… (1.1)
If Vs, R1, R2 are known, then I can be found.
Figure 2(b) contains two meshes with currents I1 and I2 hence there will be two equations as shown below. Note that the branch containing R2 is common to both meshes with currents I1 and I2 flowing in opposite directions.
Left Loop: Vs = I1R1+(I1–I2)R2
Vs = (R1+R2)I1–R2I2 ……………………………. (1.2)
Right Loop: 0 = (I2–I1)R2+I2R3
0 = –R2I1+(R2+R3)I2 …………………………..…. (1.3)
Given Vs, R1, R2 and R3, equations 1.2 and 1.2 can be solved simultaneously to evaluate I1 and I2.
For the figure 2(c), three equations need to be written as follows. Also note that there is no circuit element shared between loops 2 & 3 hence I2 and I3 are independent of each other.
Left Bottom Loop: Vs = (I1-I3)R1+(I1-I2)R2
Vs = (R1+R2)I1-R2I2-R1I3 …………………. (1.4)
Right Bottom Loop: 0 = (I2-I1)R2+I2R3
0 = -R2I1+(R2+R3)I2 ………………….. (1.5)
Upper Loop: 0 = (I3-I1)R1+I3R4
0 = -R1I1+(R1+R4)I3 ..….……………… (1.6)
If Vs and resistors’ values are known, the mesh currents can be evaluated by solving equations 1.4, 1.5 and 1.6 simultaneously.
Resistors in Series: Consider figure 3 with one voltage source and two resistors connected in series to form a single mesh with current I.
According to KVL, Vs = V1+V2
Using Ohm’s Law (V = IR),
IReq = IR1+IR2
Req = R1+R2 ………………………… (1.7)
Where, Req = combined or equivalent resistance of the series network. In general, for n number of serial resistors, Req is given by,
Req = R1+R2+R3+….+Rn ………………………………….. (1.8)
Voltage Divider Rule (VDR): VDR provides a useful formula to determine the voltage across any resistor when two or more resistors are connected in series with a voltage source. In figure 3, the voltage across the individual resistors can be given in terms of the supply voltage and the magnitude of individual resistances as follows,
…………………………….. (1.9)
………………………………… (1.10)
In general, for n number of resistors connected in series, the voltage across the i th resistor can be specified as,
…………………………….. (1.11)
This is everything till now. Next, I’ll discuss about KCL and more circuit analysis methods.
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