2 . 1 I NTRODUCTION
Kirchhoffs postulated two hasic laws way hack in
1845
which are used for writing network equations. These laws concern the algebraic sum of voltages around a loop and currents entering or leaving a node. The word algebraic is used to indicate that summation is carried out taking into account the polarities of voltages and direction of currents. While traversing a loop we will take voltage drops as positive and voltage rises as negative. Also while considering currents at a node, the currents entering the node will be taken as positive and those leaving would be taken as negative.
2.2 KIRCHHOFF'S LAWS
Kirchhoffs voltage law usually abbreviated as KVL is stated as follows :
The algebraic sum of all branch voltage around any closed loop of a network is zero at all instants of time. Alternatively, Kirchhoffs voltage law can be stated in terms of voltage drops and rises as follows: The sum of voltage rises and drops in a closed loop at any instant of time are equal. KVL is a consequence of law of conservation of energy as voltage is energy or work per unit charge. If we start from one node in a loop and move along the closed loop and come
back to the same node, obviously the total potential difference or sum of potential rises must equal the total sum of potential falls. Just as, if we start from one point on the surface of the earth and after travelling through valleys and hills come back to the same point, the total displacement is zero. We talk elevation's and depression on the earth with respect to the sea level. Similarly, in case of voltages we take ground as the reference which is shown in Fig.
2. l(a). Here potential of node A is above the ground and that of B is below the ground poten
tial.
We know that electronic current flows form negative potential to positive potential, the conventional current flows from positive potential to negative potential. Therefore, when cur
rent i flows in the circuit of Fig. 2.l(b) it produces voltages polarities in various elements as shown in the Fig. Applying Kirchhoff's voltage law
VI + V2 + v3 -V = 0
Or in terms of voltage drops and voltage rises
VI + V2 + v3 = V ... (2. 1)
1 1 9
A +
B +
(a) Ground
(b) Fig. 2.1 (a) Potential reference, (b) KVL application .
Kirchhoffs current law states that the algebraic sum of all currents terminating at a node equals zero at any instant of time. Alternatively, this states that sum of all currents entering a node equals the sum of currents leaving the same node at any instant of time.
Fig. 2.2 One node with terminating branches.
Fig. 2.2 shows one node along with various branches terminating in it, in a network and the currents with directions in the various branches are shown by the arrows. Applying Kirchhoffs current law abbreviated by KCL we get
i1 + i4 -i2 -i3 -i5 =
0
or
i1
+ i4 = i2 + i3 + i5 • • • (2.2)KCL is a consequence of law of conservation of charge. The charge that enters a node, must leave that node as it can't be stored there. Since the algebraic sum of charges at a node must be zero, it's time derivative must also be zero at any instant of time.
The two basic laws by Kirchhoffs can be applied to solve any network irrespective of its complexity. One typical application could be to find out equivalence between two networks or given a network in some configurations, how to find its equivalent so that it could be used more conveniently.
By definition, two networks are said to be equivalent at a pair of terminals if the volt
age current relationships for the two networks are identical at these terminals. Consider a network having two resistances in series as shown. The objective is to replace it by a single equivalent resistance
From Fig. 2.3(a), we have
v = v 1 + v2 = ir 1 + ir2 and from Fig. 2.3(b) we have
Therefore,
(a) (b)
Fig. 2.3 (a) Series connection of resistances, (b) Equivalent of (a).
... (2.3)
Similarly if there are n. number of resistances in series, the equivalent single resistance would be
r = r1 + r2 + ... + rn ... (2.4)
Similarly if the elements are inductances and vl' v2 are the drops across L1, L2 respec
tively, and i is the current then applying Kirchhoffs voltage law we have
Therefore,
di di
v = V1 + V2 = Ll dt + L2 dt
di di
= (L1 + L2) dt = L dt L = L1 + L2
If there are n. inductors in series it can be shown that single equivalent inductor
L = L1 + L2 + ... + Ln ... (2.5)
If two capacitors are connected in series then
Therefore,
v = v + v i 2 C1 =
__!_ I
i dt +__!_
C2I
i dt=
(__!_
C1 C2 +__!_) I
i dt = ]_ CJ
i dt 1 1 1- =-+
-c C1 C2 ... (2.6)
Similarly by using KCL, law of parallel combination of these elements can be obtained.
We will try here for inductors. Say there are two inductors connected in parallel.
From Fig. 2.4(a)
�L
i(a) (b)
Fig. 2.4 (a) Inductor in parallel (b) Equivalent of (a).
Therefore, ... (2. 7) Similarly it can be proved that if there are n resistors connected in parallel its single equivalent resistance is given by
1 1 1 1
-=
-+ - + ... + - ... (2.8)
r r1 r2 r11
and if n number of capacitors are connected in parallel, then single equivalent capacitance is given as
... (2.9)
2.3 STAR-D ELTA TRANSFORMATION
By using the same definition of equivalence of two networks i.e. the two networks are equiva
lent between two terminals if the v-i relations are same i.e. the impedances as seen between the two terminals are same.
(a)
{b)Fig. 2.5 (a) Star, (b) Delta.
Suppose we are given star connected elements as shown in Fig. 2.5(a) and we have to find it's equivalent delta connected elements then we should equal the equivalent resistance between the same two terminals.
Between a and b
... (2. 10) Between b and
c
... (2. 1 1)
or or
Between c and a
Re(RA + RB) = r + r RA + RB + Re e a
Similarly shorting be and measuring resistance between a and be, we have
rbrc
+
--RA + Re a
rb +
'"cShorting ca and resistance between b and ca