Superposition And Types Of Sources
Sample Solution
Linearity and Bilateral Networks in Superposition Theorem
In the context of electrical circuits analysis, linearity and bilateral networks play crucial roles in applying the superposition theorem.
Linearity
Linearity refers to the property of a system where the output is directly proportional to the input. In electrical circuits, a linear circuit is one that obeys Ohm's Law, which states that the current through a resistor is directly proportional to the voltage across it. This means that if the input to a linear circuit is doubled, the output will also double.
Full Answer Section
The superposition theorem is based on the principle of linearity. It states that in a linear circuit, the total response at any point is the sum of the individual responses to each independent source acting alone. This means that we can analyze a circuit with multiple sources by considering each source one at a time, turning off all other sources, and then adding the individual responses together.
Bilateral Networks
A bilateral network is a circuit that exhibits the same behavior regardless of the direction of the current flow. In other words, if we reverse the polarity of a voltage source or current source in a bilateral network, the circuit will still operate in the same way.
The superposition theorem applies only to bilateral networks. This is because the theorem relies on the fact that the response of a circuit to each independent source can be determined by turning off all other sources. If the circuit were not bilateral, the response to a source would depend on the presence or absence of other sources, and the superposition theorem would not hold.
Solving a Circuit Using Source Transformation or Superposition
Consider the following circuit:
To solve for V0, we can use either source transformation or superposition.
Source Transformation
Source transformation involves replacing independent voltage sources with equivalent current sources, and vice versa. This can be useful for simplifying circuits and making them easier to analyze.
In this circuit, we can replace the voltage source with an equivalent current source using the following formula:
I = V/R
where:
I is the equivalent current source current V is the voltage source voltage R is the resistance in series with the voltage source
In this case, I = 10V / 10Ω = 1A.
We can then replace the current source with an equivalent voltage source using the following formula:
V = IR
where:
V is the equivalent voltage source voltage I is the current source current R is the resistance in parallel with the current source
In this case, V = 1A × 10Ω = 10V.
This circuit is now much simpler to analyze. We can calculate the current through each resistor using Ohm's Law:
I1 = V/R1 = 10V / 10Ω = 1A I2 = V/R2 = 10V / 20Ω = 0.5A I3 = V/R3 = 10V / 30Ω = 1/3 A
The total current through the circuit is the sum of the currents through each resistor:
I = I1 + I2 + I3 = 1A + 0.5A + 1/3 A = 1.83A
Finally, we can calculate V0 using Ohm's Law:
V0 = IR = 1.83A × 5Ω = 9.15V
Superposition
Superposition involves analyzing the circuit with each independent source acting alone, turning off all other sources, and then adding the individual responses together.
In this circuit, we can analyze the circuit with the voltage source acting alone, turning off the current source. The current through each resistor would be:
I1 = V/R1 = 10V / 10Ω = 1A I2 = V/R2 = 10V / 20Ω = 0.5A I3 = V/R3 = 10V / 30Ω = 1/3 A
We can then analyze the circuit with the current source acting alone, turning off the voltage source. The current through each resistor would be:
I1 = I I2 = -I/2 I3 = -I/3
where I is the current source current.
Finally, we can add the individual responses together to get the total current through each resistor:
I1t = I1v + I1i = 1A + I I2t = I2v + I2i = 0.5A - I/2 I3t = I3v + I3i = 1/3 A - I/3
Finally, we can calculate V0 using Ohm's Law:
V0