Test: Networks- 1 - Electrical Engineering (EE) MCQ

The correct option is A, Z/√3
Z_Δ = 3Zᵧ
⇒ √3Z_Δ = 3Zᵧ
Zᵧ = Z_Δ / √3

To calculate the rise time of the RC circuit:

• Time constant (τ):
  τ = R × C = 5 kΩ × 1 μF = 5 ms
• Rise time (tₗᵢₛₑ):
  The rise time is approximately 2.2τ for the capacitor to charge from 10% to 90%.
  tᵣᵢₛₑ = 2.2 × 5 ms = 11 ms

Correct answer: (a) 11 ms.

Concept:
Thevenin's Theorem:
Any two-terminal bilateral linear DC circuits can be replaced by an equivalent circuit consisting of a voltage source and a series resistor.

To find V_ac:
Calculate the open-circuit voltage across load terminals. This open-circuit voltage is called Thevenin's voltage (V_th).

To find I_sc:
Short the load terminals and then calculate the current flowing through it. This current is called Norton current (Iₙₒ) or short-circuit current.

To find R_th:
Since there are independent sources in the circuit, we can’t find Rᵗʰ directly. We will calculate Rᵗʰ using Vᵥᵃ and Iₙₒ and it is given by
R_th = V_ac / I_sc

Application:

Given: Load line equation = V + i = 100

To obtain open-circuit voltage (V_th), put i = 0 in the load line equation
⇒ V_th = 100 V

To obtain short-circuit current (i_sc), put V = 0 in the load line equation
⇒ i_sc = 100 A

So,
Rth = V_th / i_sc = 100 / 100 = 1 Ω

Equivalent circuit is

The equivalent network is

The given circuit is an RLC circuit with components:

• Resistance (R) = 1Ω
• Inductance (L) = 2H
• Capacitance (C) = 1/2F

To determine whether the circuit is over damped, under damped, critically damped, or un-damped, we need to use the condition for damping in an RLC circuit, which is based on the discriminant of the characteristic equation.

The condition for the damping factor is given by:

1. Overdamped: If the damping factor is greater than 1.
2. Underdamped: If the damping factor is less than 1.
3. Critically damped: If the damping factor is exactly 1.
4. Undamped: If there is no resistance.

The discriminant of the equation is determined by:

Damping factor (ζ) = R / 2 * √(L/C)

Now let's calculate it step by step:

1. R = 1Ω
2. L = 2H
3. C = 1/2F

So, the damping factor (ζ) = 1 / (2 * √(2 * 1/2)) = 1 / (2 * √(1)) = 1 / 2.

Since the damping factor is less than 1, the circuit is underdamped.

Therefore, the correct answer is B: Under damped.

The components in the circuit are:

1. Resistor: 5Ω (top branch)
2. Inductor: 3H (top branch)
3. Inductor: 2H (bottom branch)

Impedance for each component in the frequency domain:

• Resistor (R): Z_R = 5
• Inductor 1 (3H): Z_L = jω × 3
• Inductor 2 (2H): Z_L = jω × 2

The impedance matrix for a two-port network is written as:

Where:

• Z₁₁ is the impedance seen at port 1 due to a current injected at port 1 (open at port 2).
• Z₁₂ is the impedance seen at port 1 due to a current injected at port 2 (open at port 1).
• Z₂₁ is the impedance seen at port 2 due to a current injected at port 1 (open at port 2).
• Z₂₂ is the impedance seen at port 2 due to a current injected at port 2 (open at port 1).

From the circuit, the impedance matrix is:

Therefore, the correct answer is A.

Voltage across C₁ under steady state = (50 × 5) / (5 + 5) = 25V

Voltage across C₂ is voltage across 31d / resistance = (25 × 3) / (2 + 3) = 15V

Voltage across C₃ is equal to voltage across C₂ under steady state = 15V

Cut set is minimum number of branches to be removed to break graph into two parts.
Adding a link to existing tree results in one positive set

 From the circuit applying Kirchhoff’s voltage law, we can write 50= 15+ 10+ 15+V_x => V_x= 10V.

To find the dual of the given network, we apply the following principles:

1. Voltage sources become current sources and vice versa.
2. Resistors become capacitors and vice versa.
3. Series connections become parallel connections and vice versa.
4. Parallel connections become series connections and vice versa.

Let's analyze the given network:

• The original circuit contains a 3V voltage source in series with inductors (8H, 6H), and resistors (4Ω, 5Ω).

Applying the dual principles:

1. The 3V voltage source changes to a current source of 3A.
2. The inductors (8H and 6H) become capacitors with the same values, i.e., 8F and 6F.
3. The resistors (4Ω and 5Ω) become capacitors with values 4F and 5F.
4. The series combination of elements becomes a parallel combination and vice versa.

Now, looking at the options:

• Option B correctly shows the dual network with a 3A current source, 4F and 6F capacitors, and 5F and 8F capacitors in parallel.

Thus, the correct answer is B.

The transient response of a series RLC circuit can exhibit different behaviors based on the values of resistance (R), inductance (L), and capacitance (C). If the resistance is low relative to the inductance and capacitance, the circuit will be underdamped and exhibit oscillations. If the resistance is high, the response will be overdamped, where the circuit reaches steady state without oscillations. If the resistance is just right, the circuit will be critically damped, where it reaches steady state as quickly as possible without oscillating. Therefore, the correct answer is A.

Applying KVL in the loop, 5 = 2kI + 2k(I + 10^-3)
5 = I(4k) + 2 * 10³ * 1 * 10^-3
I(4k) = 5 - 2
I = 3/(4k)
I = 0.75 * 10^-3 * A
I = 0.75mA