Circuit & Field Theory - short answer questions from AMIE exams (Winter 2020)

Explain the following terms in brief

Final Value theorem

In the solution of Networks, Transient, and Systems sometimes we may not be interested in finding out the entire function of time f(t) from its Laplace Transform F(s), which is available for the solution. It is very interesting to find that we can find the first value or last value of f(t) or its derivatives without having to find out the entire function f(t).

If F(s) is given, we would like to know what is F(∞), Without knowing the function f(t), which is Inverse Laplace Transformation, at time t→ ∞. This can be done by using the property of Laplace Transform known as the Final Value Theorem. The final value theorem and initial value theorem are together called the Limiting Theorems.

If f(t) and f'(t) both are Laplace Transformable and sF(s) has no pole in jw axis and in the R.H.P. (Right half Plane) then,

Tellegen theorem

According to the Tellegen theorem, the summation of instantaneous powers for the n number of branches in an electrical network is zero. 

Let’s explain. Suppose n number of branches in an electrical network have i₁, i₂, i₃, …………. in respective instantaneous currents through them. These currents satisfy Kirchhoff’s Current Law.

Again, suppose these branches have instantaneous voltages across them are v₁, v₂, v₃, ………..  respectively. If these voltages across these elements satisfy Kirchhoff Voltage Law then,

Magnetic dipole moment

The magnetic dipole moment, often simply called the magnetic moment, may be defined then as the maximum amount of torque caused by magnetic force on a dipole that arises per unit value of surrounding magnetic field in vacuum.

Causes of Eddy currents

Eddy currents are generated when a conductor moves through a magnetic field or when the magnetic field surrounding a stationary conductor is varied. Thus, eddy currents may be created whenever a conductor experiences a change in the intensity or direction of a magnetic field.

Moving a powerful magnet through a copper pipe induces circulating electrical currents inside the electrically conducting pipe.

Potential and Potential difference

Electric Potential

It is the work done per unit charge in order to bring that charge from infinity to a point in the electrostatic field against the field force. Electric Potential is also referred as Voltage. The SI unit for Electric Potential or Electric Potential difference is Voltage or Volts. Electric potential is a scalar quantity.

Electric Potential Difference

The energy possessed by Electric charges is known as electrical energy. A charge with higher potential will have more potential energy and the charge with lesser potential will have less potential energy. The current always moves from higher potential to lower potential. The difference in these energies per unit charge is known as the electric potential difference.

Displacement Current

Displacement current is a quantity appearing in Maxwell’s Equation. It is defined in terms of the rate of change of the electric displacement field. The units of displacement current are the same as that of electric current density. Displacement current is not an electric current caused due to moving of charges(called conventional current), but it is caused by a time-varying electric field.

If we consider a capacitive circuit, we can see there is current flowing through the circuit. But if you look at the capacitor plates, there is a small empty region in between them.  Then how the circuit is completed?  The circuit is completed despite the small space because there is displacement in that region which is developed as a consequence of the electric field in between the plates.

The curl of a vector

See this link

Superposition theorem

Superposition theorem states that in any linear, bilateral network where more than one source is present, the response across any element in the circuit, is the sum of the responses obtained from each source considered separately while all other sources are replaced by their internal resistance. Superposition theorem is a circuit analysis theorem that is used to solve the network where two or more sources are present and connected.

The application of this theorem is both the AC & DC circuits where it assists to build the circuits like “Norton” as well as “Thevenin” equivalent circuits.

See example and circuit diagram

Equipotential surface with an example

Electric potential is the potential energy per unit charge. Potential on an equipotential surface is constant through the surface. The direction of the electric field is perpendicular to the equipotential surface.

At every point on the equipotential surface, electric field lines are perpendicular to the surface. This is because potential gradient along any direction parallel to the surface is zero i.e.,

E = - (dv/dr) = 0

so component of electric Field parallel to equipotential surface is zero.

Transfer function of a system with its limitation.

A transfer function is expressed as the ratio of the Laplace transform of output to the Laplace transform of input assuming all initial conditions to be zero.

Consider a system whose time-domain block diagram is

Time-domain

where, r(t) = input and c(t) = output

Now taking Laplace transform then

Laplace domain

Now the transfer function G(s) is given by

G(s) = C(s)/R(s)

Disadvantages of Transfer function

• The transfer function does not take into account the initial conditions.
• The transfer function can be defined for linear systems only.
• No inferences can be drawn about the physical structure of the system.

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