Linearity Property
The
homogeneity property is that if the input is multiplied by a constant k
then the output is also multiplied by the constant k. Input is called
excitation and output is called response here. As an example if we
consider ohm’s law. Here the law relates the input i to the output v.
Mathematically, v= iR
If we multiply the input current i by a constant k then the output voltage also increases correspondingly by the constant k. The equation stands,
kiR = kv
The additivity property is that the response to a sum of inputs is the sum of the responses to each input applied separately.
Using voltage-current relationship of a resistor if
v1 = i1R and v2 = i2R
Applying (i1 + i2)gives
V = (i1 + i2)R = i1R+ i2R = v1 + v2
We
can say that a resistor is a linear element. Because the
voltage-current relationship satisfies both the additivity and the
homogeneity properties.
We can tell a circuit is linear if the circuit both the additive and the
homogeneous. A linear circuit always consists of linear elements,
linear independent and dependent sources.
Linearity Circuit
A linear circuit is one whose output is linearly related (or directly proportional) to its input.
A linear circuit is a special system whose output is linearly related.
Linear circuits are very useful in modeling devices and very well
understood.
To learn more about Linearity:
SUPERPOSITION
- The superposition theorem eliminates the need for solving simultaneous linear equations by considering the effect on each source independently.
- To consider the effects of each source we remove the remaining sources; by setting the voltage sources to zero (short-circuit representation) and current sources to zero (open-circuit representation).
- The current through, or voltage across, a portion of the network produced by each source is then added algebraically to find the total solution for current or voltage.
- The only variation in applying the superposition theorem to AC networks with independent sources is that we will be working with impedances and phasors instead of just resistors and real numbers.
- The superposition theorem is not applicable to power effects in AC networks since we are still dealing with a nonlinear relationship.
- It can be applied to networks with sources of different frequencies only if the total response for each frequency is found independently and the results are expanded in a nonsinusoidal expression.
- One of the most frequent applications of the superposition theorem is to electronic systems in which the DC and AC analyses are treated separately and the total solution is the sum of the two.
When a circuit has sources operating at different
frequencies, the separate phasor circuit for each frequency must be solved independently,
and the total response is the sum of time-domain responses of all the
individual phasor circuits. Superposition Theorem applies to AC circuits as
well. For sources having different frequencies, the total response must be
obtained by adding individual responses in time domain.
These are some Superposition Technique for sources having different frequencies.
Consider the following example:
These are some Superposition Technique for sources having different frequencies.
All sources except DC 5-V set to zero |
All sources except
10cos(10t)
set to zero
|
All sources except 2 sin 5t set to zero |
vo= v1+ v2+ v3
The principle of superposition helps us
to analyze a linear circuit with more than one independent source by
calculating the contribution of each independent source separately.
However, to apply the superposition
principle, we must keep two things in mind:
1. We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and more manageable circuit.
2. Dependent sources are left intact because they are controlled by circuit variables. With these in mind, we apply the superposition principle in three steps.
Steps to Apply Superposition Principle:
1. Turn off all independent sources except one source.
2. Repeat step 1 for each of the other independent sources.
3. Find the total contribution by adding algebraically all the contributions due to the independent sources.
principle, we must keep two things in mind:
1. We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and more manageable circuit.
2. Dependent sources are left intact because they are controlled by circuit variables. With these in mind, we apply the superposition principle in three steps.
Steps to Apply Superposition Principle:
1. Turn off all independent sources except one source.
2. Repeat step 1 for each of the other independent sources.
3. Find the total contribution by adding algebraically all the contributions due to the independent sources.
SUPERPOSITION
Transform a voltage source in series with impedance to a
current source in parallel with impedance
for simplification or vice versa.
Consider the following example: Calculate the current Io.
If we transform the current source to a voltage source,
we obtain the circuit shown in Fig. (a).
By current division,
A source transformation is the process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa.
Voltage Source Transformation
We will first go over voltage source transformation, the transformation of a circuit with a voltage source to the equivalent circuit with a current source. In order to get a visual example of this, let's take the circuit below which has a voltage source as its power source:Using source transformation, we can change or transform this above circuit with a voltage power source and a resistor, R, in series, into the equivalent circuit with a current source with a resistor, R, in parallel, as shown below:
We transform a voltage source into a current source by using ohm's law. A voltage source can be changed into a current source by using ohm's formula, I=V/R.
Current Source Transformation
We will now go over current source transformation, the transformation of a circuit with a current source to the equivalent circuit with a voltage source. In order to get a visual example of this, let's take the circuit below which has a current source as its power source:
Using source transformation, we can change or transform this above circuit with a current power source and a resistor, R, in parallel, into the equivalent circuit with a voltage source with a resistor, R, in series, as shown below:
We transform a current source into a voltage source by using ohm's law. A voltage source can be changed into a current source by using ohm's formula, V= IR.
TO KNOW MORE ABOUT SOURCE TRANSFORMATION:
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