Test: Properties of Systems - Electrical Engineering (EE) MCQ

Now, y(t) = 2x(t) => x(t) = 0.5y(t)
Thus, reversing x(t) <-> y(t), we obtain the inverse system: y(t) = 0.5x(t)

Concept:

• Properties
• All measurable characteristics of a system are known as properties.
• Eg. Pressure, volume, temperature, internal energy, density etc.
• There are two types of properties:

Explanation:
Pressure:

• Intensive properties are properties that do not depend on the quantity of matter. For example, pressure and temperature are intensive properties so the correct answer is option 3.

Entropy:

• Entropy in classical thermodynamics is an extensive quantity, which like energy, volume, or particle number, is additive when systems in equivalent thermodynamic states are aggregated.

Volume:

• Volume is the amount of space an object takes up, it is denoted by V.
• Volume depends on the mass of the substance as the formula for volume is :

• We see that volume is the ratio of two intensive properties. It is thus an extensive property.

Additional Information

Enthalpy:

• It is defined as the amount of heat change during a chemical reaction denoted by 'H'.
• It is given by:

H = U + PΔ V, where H = Enthalpy, U = Internal Energy, P = Pressure, V = Volume.

• The unit of enthalpy is Joule/mole, which signifies that it depends on the amount of substance present.
• Thus, Enthalpy is an extensive property.

Energy:

• Internal energy is the sum of all the types of energy present in the system, such as kinetic energy, potential energy, vibrational energy, rotational energy, etc.
• It is denoted by U and given by the formula:

​dU = q + w, where q is the heat absorbed and w is work done.

• The unit of Internal energy is Joule/mole, which signifies that it depends on the amount of substance present.
• Thus, Energy is an Extensive property.

Temperature:

• Temperature(T) is the measurement of the heat content of a body.
• It,s units are Celcius (0C), Kelvin (K), Farhenheit (0F).
• The temperature of a body does not depend on the amount of mass of a substance. If gas has say temperature 288K, it will mean that every particle of the gas is at a temperature of 288K. It is thus an intensive property.

Hence, the intensive property is Temperature.

• Additivity:
  If you input the sum of two signals, the output equals the sum of each individual output, satisfying the additivity property.

• Homogeneity (Scalability):
  Scaling the input by a constant factor results in the output being scaled by the same factor, fulfilling the homogeneity condition.

• Linearity Conclusion:
  Since both additivity and homogeneity are satisfied, the system is linear.

• Final Answer Choice:
  d) Linear

For positive time, the system may seem to be causal. However, for negative time, the output depends on time at a positive sign, thus being in the future, enforcing non causality.

While evaluating the integral, it becomes imperative to know the values of x[t] from 0 to t, thus making the system requiring memory.

We cannot determine the sign of the input from the second function, thus, the output doesn’t replicate the input. Thus, the second function is not an inverse of the first one.

The output at any time t = A, requires knowing the input at an earlier time, t = A – 1, hence making the system require memory aspects.

Concept: Linearity: Necessary and sufficient condition to prove the linearity of the system is that linear system follows the laws of superposition i.e. the response of the system is the sum of the responses obtained from each input considered separately.

y{ax1[n] + bx2[t]} = a y{x1[n]} + b y{x2[n]}

Conditions to check whether the system is linear or not.

1. The output should be zero for zero input
2. There should not be any non-linear operator present in the system

Causality: A system is causal, if the output of the system does not depend on future inputs, but only on past input.

Time-Invariance: If the input to a time-invariant system is shifted in time, its output remains the same signal, but is shifted equally in time.

If the output for an input x(t) is y(t), then for a time shift of t0 in the input gives the t0 shift in the output.

x(t) → y(t), then x(t – t0) → y(t – t0)

Application:

(i) y(n) = n x(n)

Here in the system, there is a nonlinear operator and hence it is linear.

(ii) y(n) = e^x(n)

Here there is a non-linear operator (exponential) and hence it is nonlinear.

Properties of Systems 
Intensive Property: These are the properties of system which are independent of mass under consideration. For e.g. Pressure (Pa), Temperature (K), density (kg/m³), specific enthalpy (kJ/kg)
Extensive Properties: The properties which depend on the mass of system under consideration.
For e.g. Internal Energy (kJ), Enthalpy (kJ), Volume (m³), Entropy (kJ)

Note: All specific properties are intensive properties. For e.g. specific volume, specific entropy, specific enthalpy etc.
Thus, Specific enthalpy (enthalpy per unit mass) is an intensive property and its unit is kJ/kg.

If we write for n-1, n-2, .. we will obtain y[n] = x[n] + x[n-1] + x[n-2] …,
thus obtaining an adder system.

Sampling  indicates the equality between the area under the product of function with shifted impulse and the value of function located at unit impulse instant

 For a time instant existing between 0 and 1, it would depend on the input at a time in the future as well, hence being non causal.

A system possessing no memory has its output depending upon the input at the same time instant, which is prevalent only in option b.

Here, when we put k=2

=> y(t) = x(t) + x(2t) – x(2t) + x(t-1)

=> y(t) = x(t) + x(t-1)

As we need to eliminate the term x(2t) to make the system time invariant, the value of k must be 2.

Therefore, the correct answer is B.

In each of a, b and c there is a negative sign of t involved, which means a backward shift of t-0 in time, would mean a forward shift in each of them. However, only in d, the backward shift will remain as backward, and undiminished.

Concept:

The time scaling property of convolution is given by

Solution:

as we can see are having a scaling factor of a = 3

∴ h(t) = [(3t)² + (3t)]

=[9t² + 3t]

The impulse response is the output of a system when the input is an impulse, i.e.

The impulse response of a system can be used to evaluate various system properties. Causality is one such property, that states, “if the output of the system at any time depends only on the past and present values of the input, the system is said to be causal.”
If the impulse response is known, the system is said to be causal, if h(t) = 0 for t < 0.
Examples:

System (1) is causal

System (2) is non-causal, as h(t) ≠ 0 for t < 0.

Concept:

An LTI Causal System is BIBO Stable if an only if all the poles are strictly in Left hand side of s-plane.
Calculation:
The impulse function can be written as:

The transfer function has poles at s = -3 and s = 2.
In order to make the system causal and stable, the right-hand side pole must be removed.
Thus, H₁ (s) = s - 2 needs to be added in cascade to make the system stable and causal.

Concept:

For a system to be causal, the Present output should depend on the present or past input only.

For a system to be Stable, a Bounded input should produce a Bounded Output.

Analysis:

For 0 ≤ n ≤ 10,

y(n) = n|x(n)|

Let x(n) is bounded, i.e. for -∞ ≤ n ≤ ∞, x(n) ≤ M, where M is finite.

So, for -∞ ≤ n ≤ ∞, |n.x(n)| will also approach a finite value. Hence in this interval y(n) is bounded. i.e. the system is stable.

Since the output y(n) is depending on the present value of the input only, the system in this interval is also causal.

For n < 0 and n > 10:

y(n) = x(n) - x(n-1).

Let input x(n) is bounded. ∴ y(n) = x(n) - x(n-1) will also be bounded, i.e. it will go to a finite value.

Hence in this interval, the system is said to be stable.

Also, because the output y(n) depends on the present and the past input values only, the system is causal as well in this interval.

∴ After considering both the intervals we can conclude that the system is both stable and causal.

The Characteristics equation is an essential tool for analyzing different system parameters

• It is obtained by equating the denominator of the closed-loop transfer function to 0
• It helps in the determination of system stability
• Also provides information about the system bandwidth, speed, response time, etc. 

From solving the characteristics equation we could infer that

• The roots of the characteristic equation give the location of the Closed-loop poles of the system
• These roots if strictly lie on the left half of the S-plane, then the system is stable
• If these roots have at least one root on the imaginary axis or on the right half of S-plane then the system is unstable
• The system transfer function solely determines the characteristic equation
• It is independent of the input applied
• Therefore system stability is independent of input

Hence statement a and c are correct,

Statement d is incorrect

If the system is practically realizable (stable) then

• For bounded input, it must give bounded output
• If finite magnitude input is applied then the system must also produce finite magnitude output
• The above is known as (BIBO) stable

Hence statement b is correct

Therefore statements a b and c are correct

The correct answer is option C.