KIRCHOFF’S LAWS
Electric circuits can be analyzed using Kirchoff’s Laws. It is known that a circuit is made up of
elements. Two elements are introduced in this chapter in order to facilitate explanation of what
a circuit is. A commonly used element is a voltage source and a battery is a voltage source. It
can be connected to a bulb, as shown in Fig. 1 to form a circuit. Even this simple circuit
illustrates what a circuit aims to do. The battery is represented by its symbol and the bulb by its
model, a resistor. In circuit theory, the circuit diagram models the elements in the circuit and
then the circuit is analyzed.
In Fig. 1, a and b are the terminals of the battery and the voltage of the battery is E volts. We can state that
Some of the conventions used can be explained with the help of equation (3.1). Normally a capital letter such V or I is used to indicate a voltage or a current which does not vary with time. In addition, the voltage is indicated as Vab. This convention defines the voltage or the potential difference between terminals a and b. If the potential at a is positive with respect to that at b, then Vab is positive. Here the battery voltage is stated to be E. For example if a 6 V battery is used, then E = 6 V .
When the bulb is connected across the battery, there is current through the bulb. If the resistance of the bulb is R W, current I through the bulb is obtained using the Ohm’s law. According to the Ohm’s law,
Then the current I is:
Current I flows out of the positive terminal as shown in Fig. 2. As shown in Fig. 2, current leaves the battery via its positive terminal and returns to it via its negative terminal. This current passes through the bulb and causes the potential at c to be positive with respect to that at d. Then
Then power, PR, absorbed by the bulb is given by
Power, PB, delivered by the battery is:
The convention followed is made explicit by this example. Power absorbed, here in this case by the bulb modeled as a resistor, is positive. Power delivered by the battery is negative. It is seen that
This equation shows that there is power balance within a circuit. The algebraic sum of all powers in a circuit is zero. The use of term algebraic is significant. It means that the allocation of power in each element of a circuit includes specifying its sign and the allocation of sign should be based on a convention that is consistently followed. This aspect would be explained again later.
If there exists a closed path or a loop for current to flow in a circuit, then the algebraic sum of voltages in the loop is zero at any instant according to Kirchoff’s voltage law(KVL). Kirchoff’s Voltage Law can be seen to be based on the axiom of conservation of energy, since voltage is defined as energy (or work) per unit charge. If there be n elements in a loop, then Kirchoff’s voltage law states that
where the polarity of voltage of an element is judged in relation to the assigned direction of loop current.
Again, the term algebraic is used. It means that the summation is to be performed taking into account the polarity of voltages. Hence in order to use KVL, it is necessary to follow a consistent notation. The current referred to in this text is the conventional current and it flows out of the positive terminal of a source through the external circuit connected to it and returns to the source via its negative terminal. The direction of conventional current is opposite to the flow of electrons constituting current. In an electrical system, the current is due to the flow of electrons and the electrons emanate from the negative terminal of the source and flow towards the positive terminal. Hence the electron current flows in a direction opposite to that of the conventional current. But in this text, we make no further mention of electron flow and we use the term current to refer to the conventional current.
A circuit is shown in Fig. 3C to illustrate Kirchoff’s Voltage Law. Current I flows due to the battery and this current flows through the three resistors connected in series. Based on Ohm’s Law, we get that
Current I causes a voltage drop across each resistor, as marked in Fig. 3C and each of these voltages is positive. According to Kirchoff’s Voltage Law,
In equation (3.8), the sign for voltages across resistors is positive, whereas we have a negative sign in front of the battery voltage. This is because the current passes through a resistor by entering via its positive terminal and leaving it via its negative terminal. But current passes through the battery by leaving via its positive terminal and entering via its negative terminal and hence a negative sign is affixed in front of E in equation (3.8). We can multiply equation (3.8) by I. Then
The above equation illustrates the power balance that exists in a circuit. The power delivered by the source equals the power consumed by the three resistors. It is seen that the power consumed by each resistor is positive whereas the power delivered has a negative sign.
Based on Kirchoff’s Voltage Law, two deductions can be made. The first deduction is that two arbitrary voltage sources should not be connected in parallel. When two non-ideal voltage sources are connected as shown in Fig. 4, the circulating current can be expressed as follows:
where the internal resistance of voltage source E1 is R1 and that of E2 is R2. When the sources are ideal, both R1 and R2 are of zero value and then the circulating current is infinite. Even in the case of two non-ideal voltage sources, the circulating current can be high enough to damage the sources, since a practical voltage source such as a battery tends to have a fairly low internal resistance.
The second deduction is that an ideal voltage source with zero voltage can be replaced by a short circuit. From Fig. 4, equation (3.11) can be formed. Assume here that the sources are ideal and the resistors shown in Fig. 4 are external to the sources. If voltage source E2 is of zero value, then current I is:
Figure 4B has been obtained by replacing source E2 by a short circuit, and current I for this circuit is the same as given by equation (3.11). If the source with zero voltage is not ideal, only the ideal part of the source should be replaced by a short circuit leaving the internal resistance in circuit. This conclusion can be reached if resistance R2 is considered as the internal resistance of source E2.
When a voltage source has zero value, its contribution to the loop equation based on KVL is zero, but current may flow through it. A short circuit satisfies these requirements and hence the ideal voltage source with zero volts is replaced by a short circuit.
Kirchoff’s Current Law states that the algebraic sum of currents in elements incident at a node is zero at any instant. If there be n elements incident at a node, Kirchoff’s current law states that
where current in an element is considered to be positive if it is flowing away from the node and it is negative if it is flowing towards the node. Usually, the KCL equation at a node is formed on the basis of assigned direction of current in elements.
The KCL is applied to currents incident at a node. In Fig. 2, the terminals of battery are marked as a and b and resistor R is connected to terminal a. The point at which two or more circuit elements are connected is usually referred to as a node in circuit theory. Then there are two elements incident at node a in Fig. 2.
In Fig. 5, a part of a circuit is shown and the node a in this circuit has five elements connected to it. The currents in the elements have also been marked. We can then apply Kirchoff’s Current Law to the currents in the elements incident at node a in Fig. 5. Note again that the term algebraic is used in the definition of the KCL. It means that current in each element should have a sign associated with it and again a consistent notation should be followed. When a KCL equation is formed for the currents incident at a node, currents marked as flowing towards the node would carry a negative sign and currents marked as flowing away from the node would carry a positive sign.
From Fig. 5,
It is seen that sign affixed for I1 and I4 is negative because both I1 and I4 flow towards node a, whereas the sign affixed for I2, I3 and I5 is positive because these currents flow away from the node. It should be noted that the assigned direction of current in an element need not be the same as the actual direction of current. For example, the assigned direction of is I2 away from node a. If value of I2 is - 2 Amps, it means that the actual direction of is I2 towards node a. Here the negative sign of I2 indicates that the assigned direction and the actual direction are opposite of each other.
WORKED EXAMPLE 3.1
A network is shown above.. Find the values of V3, R1, R2 and R3.
Solution:
Using KVL, an equation can be formed for the circuit shown above..
Hence V3 = 5 V. We have that I = 2 A. It is given that
WORKED EXAMPLE 3.2
Figure 3.7 shows node a with five elements incident at it. Find I1 and I5, given that I2 = 2 A, I3 = - 4 A, and I4 = 6 A, and that I1 =2 A
Solution:
Using KCL, an equation can be formed at node a.
Using the relation that I1 =2 I5, the above equation becomes:
On substituting the given values,
Then I1 = - 16 A. The sketch in Fig. 3.14 can be re-drawn as shown in Fig. 3.8. In this example, the assigned direction of currents I1 and I5 is opposite to the actual direction of those currents.
Current Sources In Series
Based on Kirchoff’s Current Law, two deductions can be made. The first deduction is that two arbitrary current sources should not be connected in series. As shown in Fig. 6, when two non-ideal current sources are connected in series, KCL has to be satisfied at the node Y. In Fig. 6, R1 is the internal resistance of current source IA and R2 is the internal resistance of current source IB. Applying KCL at node Y, we get that
Since internal resistances of current sources are high, one of the sources can have a very high voltage across it, leading to its damage. Hence two arbitrary current sources should not be connected in series. If the two sources are ideal, R1 = R2 = ∞ and then KCL at node Y is not satisfied.
The second deduction is that an ideal current source supplying zero current behaves like an open circuit. This zero current source makes no contribution to the algebraic sum of currents to the nodes to which it is connected. If it makes no contribution, it is valid to represent it as an open circuit. If the source is non-ideal, its internal resistance should be left in the circuit, but the current source is replaced by an open circuit.
Conservation of charge is a fundamental law of physics and this law asserts that electric charge cannot be created or destroyed, but is conserved. Kirchoff's current law is an expression of this law interpreted for a circuit with lumped elements. We can draw a closed gaussian surface around the node at which two or more elements meet. Since charge cannot be created or destroyed, charge entering the node through the gaussian surface should equal the charge leaving the surface. Net charge transported into the region enclosed by the gaussian surface has to be zero, for otherwise charge accumulation within the region would occur. Since current is a derivative of charge, the net current which is the algebraic sum of currents leaving the gaussian surface around the node, is zero at any instant. The current leaving the surface can be assigned positive sign whereas the current entering the surface can be assigned a negative sign. Then the algebraic sum of currents leaving the surface is zero.
In circuit theory, the definition of KCL presented at the beginning of this section is more useful.
3.5 TELLEGEN'S THEOREM
Tellegen's theorem is based on the fundamental law of conservation of energy and is a logical outcome of Kirchoff's laws. It is a general and useful theorem. It states that the algebraic sum of power absorbed by all elements in a circuit is zero at any instant. Note that power absorbed by a resistor is always positive, whereas a source may deliver power and the power absorbed by the source is negative. A proof for this theorem can be presented based on graph theory. Since graph theory is not covered in this text, an insight into why this theorem has to be true can be obtained by analyzing a simple circuit.
A simple circuit has been shown in Fig. 10. This circuit has four nodes, inclusive of the datum node. Though the voltage at datum node is zero, its voltage has been designated to be V0. Then V0 = 0 Volt. We can form one equation based on KCL at each node. From Fig. 10,
The above equation can be presented as follows.
Each pair of voltages represents the voltage across an element carrying the current indicated and hence the above equation expresses the algebraic sum of power absorbed by elements in the circuit shown in Fig. 3.10. It is seen that the algebraic sum of power in elements in this circuit is zero. It can also be understood why this has to be true for any circuit.
This chapter has explained and provided some insight into Kirchoff's laws. Tellegen's theorem has also been introduced in this chapter. Kirchoff’s laws form the basis for circuit analysis.
E3.1: Find the currents I2, I4, I5, I6 and I9 in the branches of the network shown in Fig. E3.1.
E3.2: Find voltage Vad for the circuit shown in Fig. E3.2.
