Network Theory Test 1

Calculation:

Consider the circuit given

5 A current is entering at the +ve terminal and 4 A current is lowing at the +ve terminal. Hence Total 5 - 4 = 1 A current will be entering at +ve terminal.

Since the equivalent current is entering at +ve terminal hence it will be charging.

Therefore, the correct option will be 4.

The circuit is redrawn as:

Concept:

R_AB = (12× 6) / 18

= 4 Ω

Calculation:

The given circuit is

From loop i₁

-12 + 6 i1 + 6 i1 = 0

i₁ = 1 A

From loop i2

-12 + 6i₂ + 6i₂ - 12 = 0

12i₂ = 24

i₂ = 2 A

From loop i3

-12 + 6[i₃ + i₄] + 12i₃ = 0

18i₃ + 6i₄ = 12      .......(1)

From loop i4

6[i3 + i4] + 12i4 = 0

12i₄ + 6i₄ + 6i₃ = 0

6i₃ + 18i₄ = 0

i₃ = -3i₄     .......(2)

from eq 1

18(-3i₄) + 6i₄ = 12

- 54i₄ + 6i₄ = 12

- 48i₄ = 12

Calculation:

Concept:

Energy is given by:

E = I²Rt

where, E = energy

I = current

R = resistance

t = time

Calculation:

Concept:

Tellegen’s theorem:

• According to Tellegen’s theorem, the summation of instantaneous powers for the n number of branches in an electrical network is zero.
• Let n number of branches in an electrical network have I₁, I₂, I₃, ….. In respective instantaneous currents through them.
• These branches have instantaneous voltages across them are V₁, V₂, V₃, ….. Vn respectively.

• ∑ power delivered = ∑ power absorbed
• It is based on the conservation of energy.
• It is applicable to both linear and non-linear circuits.

Calculation:

Power delivered by 15 V source = 15 × 5.75 = 86.25 W

Power delivered by 20 V source = 20 × 9 = 180 W

By Tellegon's theorem 

Total power consumed = total power delivered = 180 + 86.25 = 266.25 W

Concept:

According to Thevenin’s theorem, any linear circuit across a load can be replaced by an equivalent circuit consisting of a voltage source Vth in service with a resistor Rth as shown:

To evaluate the Thevenin Resistance, we consider the following two cases:

Case 1: Circuits with independent Sources only

If the network has no dependent sources, we turn off all independent sources. Rth is the input resistance of the network looking between terminals a and b.

Case 2: Circuit with Both Dependent and independent sources

We may use one of the two methods:

Using a Test Source:

We apply a voltage source V0 at terminals a and b and determine the resulting current I₀.

Then R_th = V₀/I₀

Application:

Since there are no independent sources present in the network, the Thevenin’s voltage will simply be zero i.e.

V_TH = 0 V

Now, because of the pressure of a dependent source, we will be a test charge to evaluate the Thevenin equivalent resistance, i.e.

To find the thevenin resistance, open circuit the current source and short circuit the voltage source.

Concept:

Thevenin's Theorem states that “In any linear, bidirectional circuit having more than one independent source, having more number of active and passive elements, it can be replaced by a single equivalent circuit consisting of equivalent voltage source V_Th and series with equivalent circuit resistance RTh“.

Where

V_Th = Thevenin Voltage or open circuit voltage

R_Th = Thevenin Resistance 

For RTh All independent sources replaced by their internal resistance (ideal voltage source replaced by a short circuit and ideal current source replaced by an open circuit.

V_th is open circuit voltage

Calculation:

The Thevenin’s resistance can be find by neglecting the voltages source.

Concept:

Any linear circuit can be replaced by its Thevenin equivalent representation. This is explained with the help of the following circuits:

V_th = Open Circuit Voltage (Thevenin Voltage)

R_th = Thevenin Equivalent Resistance

Analysis:

The unknown circuit can be converted to an equivalent circuit as shown:

Concept:

Superposition Theorem:

• The superposition theorem states that "in a linear circuit with several sources, the current and voltage for any element in the circuit is the sum of the currents and voltages produced by each source acting independently."
• The superposition theorem applies only when all the components of the circuit are linear, which is the case for resistors, capacitors, and inductors it is not applicable to networks containing nonlinear elements.
• Voltage Source  →   short
• Current source   → open
• Do not disturb the dependent source present in the network.

Calculation:

Apply super position theorem,

1) when 6 A source is active

Additional Information

Limitations of SPT:

• This theorem cannot be used to measure power.
• This theorem is not applicable to unbalanced bridge circuits.
• Applicable only to linear circuits not for nonlinear circuits.
• Applicable only for the circuits having more than one source.

 Application of SPT:

 Superposition Theorem is applied to determine the current in one particular branch of a network containing several voltage source and/or current source.

Concept:

Superposition Theorem:

• It is state that in any linear, active, bilateral network having more than one source, the response across any element is the Algebraic sum of the response obtained from each source considered separately and all other sources are replaced by their internal resistance.
• The principle of the superposition theorem is based on Linearity.
• Voltage Source  →   short
• Current source   →  open
• Do not disturb the dependent source present in the network.

Step to solving Network by superposition theorem

• Step 1 – Take only one independent source of voltage or current and deactivate the other sources.
• Step 2 – If there is a voltage source then short circuit it and if there is a current source then just open circuit it.
• Step 3 – Thus, by activating one source and deactivating the other source find the current in each branch of the network.
• Step 4 – Now to determine the net branch current utilizing the superposition theorem, add the currents obtained from each individual source for each branch.
• Step 5 – If the current obtained by each branch is in the same direction then add them and if it is in the opposite direction, subtract them to obtain the net current in each branch.

Calculation:

Now taking 1 V voltage source

Deactivate other sources like Voltage source short circuit and current source open circuit.

Here,

2 V source → short circuited

1 A source → open circuited

The correct answer is 1/15, 4/30, -1/30

Solution:

Apply superposition theorem apply one source at a time 

Case:1 Apply I₁ current source and short circuit voltage source and open circuit I₂

Case2: Apply I₂ current source and short circuit the voltage source and open circuit the I₁ current source.

Concept:

Maximum power transfer theorem(MPPT)

Condition for maximum power:

R_L = R_th

Maximum power across load during MPPT:

Shortcut Trick

During MPPT voltage across the load is always half the value of the supply voltage.

Concept:

Maximum Power Transfer Theorem(MPPT):

The maximum power will flow across load terminal AB when the value of load resistance is equal to equivalent resistance from the load terminal point.

Calculation:

While calculating equivalent resistance all voltage and current sources must be replaced by short and open circuits respectively.

By making voltage source short circuit, 3Ω resistance also gets short-circuited as both are in parallel.

Concept:

Maximum power occurs in the load if load impedance complex conjugate of source impedance i.e,

Z­_L = Z_S*

And the maximum power is calculated by:

So, the time taken for the current to rise from half of the peak value to the peak value is given by:

To consider voltage as a reference phasor, The First step would be to make Cosine as Sine by adding 90º to its phase.

Phasor diagram for above two equations can drown as below.

Given that,

V = 200 Sin (2000t + 50°) V

i = 4 cos (2000t + 13.2°) A

We know that,

sin (90 + ϕ) = cos ϕ 

⇒ V = 200 sin (2000t + 50°) V  --------- (i)

⇒ i = 4 sin (2000t + 13.2°+ 90°) A

⇒ i = 4 sin (2000t + 103.2°) A    ------- (ii)

From, (i) and (ii) we can observe that,

Current 'i' is leading voltage V by 53.2°

Hence, the element is practical capacitor

Concept:

The resistance offered by a capacitor when an alternating current flows through it is called capacitive reactance (X_C) and is given by

Here, f is the frequency

L is the inductance

C is the capacitance

For a DC supply, ω = 0.

∴ The capacitive reactance becomes infinity, i.e. a capacitor acts as an open circuit for DC supplies.

Calculation:

At a steady-state of t = ∞, the capacitor becomes open.

As there is no closed loop in the circuit. So again there will be no flow of current in the circuit, i.e. the final current = 0 A

Additional Information

The resistance offered by an inductor when an alternating current flows through it is called inductive reactance (X_L) and is given by:

At steady state i.e., as t → ∞, the capacitor behaves as an open circuit. The circuit at steady state is as shown:

Apply voltage division across R₁ = 2 kΩ and R₂ = 1 kΩ, we get: