# An introduction to probability and random processes in circuit theory from a pedagogical viewpoint

## Circuit theory concepts from field theory concepts

In order to learn the fundamentals of electrical and electronics engineering, the student just when he enters into college should first brush up his fundamentals of electromagnetic field theory, the reason being that electric and magnetic fields form the fundamental building blocks of this branch of engineering. The voltage difference between two terminals on a bread -board on which a circuit has been built is simply the line integral of the electric field along any path between the two points. The current flowing through a wire having a small cylindrical cross section is the surface integral of the current density over a cross section of the wire and this current density *J* can be calculated in principle once the electric and magnetic fields are known using the Maxwell equation

*curlH = J + e-----*(1)

*dt*

Kirchhoff’s current law (KCL) states that the sum of currents emanating from a given node in a circuit is zero. This is provided that the node cannot accumulate charges. In the language of field theory, the KCL is expressed by the restricted charge conservation equation

*divJ* = 0---(2)

which when integrated over a closed surface containing the node yields after applying the Gauss’ divergence theorem *J.ndS* = 0 where *S* is a closed surface enclosing the node and this is precisely the KCL. If the node is big, it can accumulate charges and then the generalization of the KCL would read

*div J* + — =0---(3)

*dt ^{v}*

which in integral form is

*I J.ndS =- ^{d}-^>---*(4)

where Q(t) is the charge contained in the node at time *t.* This charge conservation equation can in turn be derived from the Maxwell equations:

*dE*

*curlH = J + e-&-,divE = p/e---*(5)

On taking the divergence of the first and using the second, we obtain (3). Likewise, the Kirchhoff voltage law which states that the algebraic sum of voltage around a closed loop of a circuit is zero can be derived by starting with Faraday’s law of electromagnetic induction

*dH curlE = —u-=— ^{h} dt*

which gives on integrating over the flat surface *S* whose boundary T is the given circuit and applying Stokes’ theorem,

where '!> is the magnetic flux through *S* and its contribution comes only from the inductors in the circuit. Thus, *Vp + Vc* + K = /_{r} *E.dr* represents the algebraic sum of voltage drops in the circuit T coming from resistors, capacitors and voltage sources while 14, = — ^ is the algebraic sum of the emf/voltage drops across all the inductors in the circuit. This equation therefore reads

Vr + Vc + Vi + Vl = 0

which is the KVL. The question which naturally arises in ones mind is that why do we have to introduce such field theoretic concepts which are harder to grasp first rather than directly talk about resistors, capacitors, inductors, voltage and current sources and the relationship between current and voltage for these elements along with the KCL and KVL. The reason being that the circuitsassembled by us on Bread-boards are lumped parameter circuits, ie they consist of discrete distributions of circuit element while on the other hand, when we go by train we observe massive transmission lines carrying current attached to tall pylons and also we observe large antenna dishes for our television and internet basis being fed in by transmission lines. These objects, ie, transmission lines, waveguides and antennas are examples of distributed parameter networks, ie, we have a continuous distribution of resistances, capacitances and inductances which may be functions of the spatial location along the network. These networks are important in power transmission from the generator to our houses, receiving image signals on our television screen and in transmitting messages encoded as electromagnetic fields from our antennas into space whose signals we collect using receiver antennas. They also form a major part of electrical and electronics engineering and can be modeled and understood only using the Maxwell field theory. Thus a thorough understanding of the Maxwell equations enables us to grasp both lumped parameter circuits and distributed parameter circuits in one stroke. This is the precise reason for introducing field theoretic concepts right at the beginning of our electronics engineering curriculum.

Properties of resistances, capacitances and inductances also follow from field theoretic analysis naturally. For example, the field theoretic version of Ohm’s law J = and the definition of voltage as the line integral of the electric field, the definition of current as the surface integral of the current density and the definition of resistance as the line integral J_{1} dr/crA(r) where dr is the line element, A(r) is the cross sectional area at r and a is the conductivity which may also vary with r, r being the distance parameter along the wire, all are natural consequence of classical field theory and lead to the familiar Ohm’s law V = IR. Likewise, the Q = CV relation between charge on the plates and voltage between the plates is a consequence of the one dimensional version of Gauss’ law d[eE_{z})/dz = 0 in between the plates and the boundary condition derived from Gauss’ law that eE_{z} on the surface of the plates gives us the surface charge density. The field theoretic viewpoint thus enables us to solve more complex problems like the determination of the resistance or capacitance between two surfaces of arbitrary shape when the medium inside has non-uniform conductivity and non-uniform resistivity. Coming over now to inductance, the starting point for determining the inductance of a coil is Ampere’s law in integral form /_{r} H.dr = I which in differential form reads curlH = J. This states that the line integral of the magnetic field H around a closed loop equals the net current flowing through the loop. This equation therefore yields the magnetic field within the loop as a linear function of I. The magnetic flux density B = pH where p may even be field dependent or even have memory as in Hysteresis effects is then computed and its flux through the loop surface is calculated yielding the magnetic flux $ as a function of the current which when combined with Faraday’s law of induction (once again a field equation) in integral form yields the EMF of an inductor E = —d as being proportional to the current if the permeability is a constant or it may even be a complicated nonlinear functional of the current if the medium has Hysteresis.

All this discussion shows us that to clearly formulate the circuit equations KCL, KVL and obtain the current voltage relations for resistances, capacitances and inductances, we require a thorough grinding in electromagnetic field theory.