Circuit & Field Theory - short answer questions from AMIE exams (AMIE Winter 2017)

Answer the following in brief

What are complex frequency and real frequency? Give their applications.
A type of frequency that depends on two parameters; one is the σ which controls the
magnitude of the signal and the other is “w”, which controls the rotation of the signal; is known as “complex frequency”.

s = σ + jw

"s" is a generalized frequency variable whose real part a describes growth and decay of the amplitudes of signals, and whose imaginary part jω is the angular frequency in the usual sense.

From the pole-zero diagram of a network function, how can you understand whether the network is stable or not?
Given a polynomial N(s) = P(s) / Q(s)
By factorizing the numerator and denominator polynomials, we can easily show that the polynomial becomes zero when the ‘s’ terms in the numerator polynomial have the values s=0, -z₁,-z₂…….-zₙ, thus the roots of numerator define the zeros. Also, the roots of the denominator polynomial define the poles of the function. Zeros are marked by ‘O’ on the s-plane while a pole is indicated by an X.

A pole-zero diagram is shown below

A system that has poles on the right hand of the s-plane is basically unstable because such a system gives rise to exponentially increasing transient responses.

The necessary condition for the stability of the network function therefore are:
  • Any function F(s) cannot have poles on the right hand of the s-plane.
  • There should not be multiple poles on the jω axis
  • The degree of the numerator polynomial cannot exceed the denominator polynomial by more than one. As, if n - m > 1, mean a pole at s = ∞ would impair the stability of the system. 
State Norton’s theorem.
Norton’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. 
Following circuit 

can be converted in Norton's circuit

Compare between a mesh and a loop. Explain with necessary figures.
Loops and mesh are two terms used in the circuit analysis and refer to the topology of the circuits. A loop is any closed path in a circuit, in which no node is encountered more than once. A mesh is a loop that has no other loops inside of it.

A loop can be found by starting from a point and travelling through a path, to finish at the same point such that the same node is not traversed twice (except the starting point).

Meshes are used to analyse planar circuits.

In the above diagram, the path (A>B>F>G>C>D>A) is a loop, and there are other closed paths inside. For example, (B>F>G>C>B) is another loop. Path (A>B>C>E>A) is a closed path where there are no smaller closed paths inside. Therefore, it is a mesh.

 A loop is a closed path in a circuit where two nodes are not traversed twice except the initial point, which is also the final one. But in a loop, other paths can be included inside.

A mesh is a closed path in a circuit with no other paths inside it. In other words, a loop with no other loops inside it.

Define the time constant of a circuit. 
The time required for a current turned into a circuit under a steady electromotive force to reach (e-1)/e or 0.632 of its final strength (where e is the base of natural logarithms) is called the time constant.
The time constant is commonly used to characterize the response of an RLC circuit.
Consider the following figure

The time constant will be 𝛕 = RC.
In a complex RC circuit, the time constant will be the equivalent resistance and capacitance of the circuit.

Define with a neat diagram plane of incidence. 
When a ray of light is incident upon a plane surface separating two mediums (e.g., air and glass), it is partly reflected (thrown back into the original medium) and partly refracted (transmitted into the other medium). The laws of reflection and refraction state that all the rays (incident, reflected, and refracted) and the normal (a perpendicular line) to the surface lie in the same plane, called the plane of incidence. Angles of incidence and reflection are equal; for any two mediums, the sines of the angles of incidence and refraction have a constant ratio, called the mutual refractive index. All these relations can be derived from the electromagnetic theory of Maxwell, which constitutes the most important wave theory of light. 

State right-hand rule in finding the direction of the cross product of two vectors
  • A cross product, or vector product, is created when an ordered operation is performed on two vectors, a and b. The cross product of vectors a and b, is perpendicular to both a and b and is normal to the plane that contains it. Since there are two possible directions for a cross product, the right-hand rule should be used to determine the direction of the cross product vector.
  • For example, the cross product of vectors a and b can be represented using the equation: a x b = c
  • To apply the right-hand rule to cross products, align your fingers and thumb at right angles. Then, point your index finger in the direction of vector a and your middle finger in the direction of vector b. Your right thumb will point in the direction of the vector product, a x b (vector c).

What is the time-harmonic field? Explain.
A time-harmonic field is one that varies periodically or sinusoidally with time. 

Write the value of absolute permittivity and permeability of free space.
  • Permittivity.  Permittivity can be explained as the ratio of electric displacement to the electric field intensity. It is the property of a material to measure the opposition generated by the material during the electric current development. The permittivity of a material is represented by the symbol ε. The SI unit of permittivity is Farad per meter. The approximate value for permittivity is 8.85 X 10⁻¹² Faraday/meter, which is found in a vacuum medium.
  • Permeability.  The property of the material which supports the formation of magnetic flux when passed through a magnetic field is known as permeability. It is affected by the field frequency, temperature, field strength, and humidity. It is represented by μ. The permeability of a material is defined as the ratio of flux density to the field strength of a material. It is also directly proportional to the conduction of magnetic lines of force. The permeability of free space is also known as the permeability constant and is represented by μ0, which is approximately equal to 4π x 10⁻⁷ Henry/meter.
State Biot Savart Law.
Biot-Savart’s law is an equation that gives the magnetic field produced due to a current-carrying segment. This segment is taken as a vector quantity known as the current element. The Biot Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb’s law in electrostatics.

It is given by the following equation

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