Monday, October 14, 2013
Introduction
A major advantage of analyzing circuits using Kirchhoff's laws like we did in Chapter 3 is that we can analyze a circuit without modifying the original configuration but the major disadvantage is that it will take some time to solve it because of the long and complex circuitry.
Linear Property
Linear property is the linear relationship between cause and effect of an element. This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additivity property are both the combination of 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 of 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.
What is linear circuit?
A circuit is linear if the output is linearly related with its input.
The relation between power and voltage is nonlinear. So this theorem cannot be applied in power.
See a circuit in figure 1. The box is linear circuit. We cannot see any independent source inside the linear circuit.
Figure 1
The linear circuit is excited by another outer voltage source vs. Here the voltage source vs acts as input. The circuit ends with a load resistance R. we can take the current I through R as the output.
Suppose vs = 5V and i = 1A. According to linearity property if the voltage is multiplied by 2 then the voltage vs = 10V and then the current also will be multiplied by 2 hence i = 2A.
The power relation is nonlinear. For example, if the current i1 flows through the resistor R, the power p1 = i12R and when current i2 flows through the resistor R then power p2 = i22R.
If the current (i1 + i2) flows through R resistor the power absorbed
P3 = R(i1 + i2)2 = Ri12 + Ri22 + 2Ri1i2 ≠ p1 + p2
So the power relation is nonlinear. Circuit solution method superposition is based on linearity property.
Superposition
The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or current through) that element due to each independent source acting alone.
To apply this principle for analysis, we must follow these procedures:
1. Turn off all independent sources except one. Find the output (voltage or current) due to that source.
2. Repeat Step 1 for each independent source.
3. Add the contribution of each source to find the total output.
Also:
*Superposition is based on circuit linearity
*Must analyze as many circuits as there are independent sources
*Dependent sources are never turned off
*As with examples, is usually more work than combining resistors, the node voltage analysis, or mesh current analysis
*Is an important idea
*If you want to consider a range of values for an independent source, is sometimes easier than these methods
*Although multiple circuits must be analyzed, each is simpler than the original because all but one of the independent sources is turned off
*Will be necessary when we discuss sinusoidal circuit
Source Transformation is the replacement of a voltage in series with a resistor by a current source in parallel with a resistor or vice versa. the two circuits are equivalent if they have the same current-voltage relationship at their terminals
*A two terminal circuit element is defined by its voltage-current relationship
*Relationship can be found by applying a voltage source to the element and finding the relationship to current
*Equivalently, can apply a current source and find relationship to voltage
*If two elements have the same, they are interchangable
Thevenin's theorem: a linear two-terminal circuit is electrically equivalent to a voltage source in series with a resistor
Norton's theorem: a linear two-terminal circuit is electrically equivalent to a current source in parallel with a resistor
Suppose we have a voltage source or battery whose internal electrical resistance is Ri and a load resistance RL is connected across this battery. Maximum Power Transfer Theorem determines the value of resistance RL for which the maximum power will be transferred from source to it. Actually the maximum power, drawn from the source, depends upon the value of the load resistance. There may be some confusion let us clear it.
Power delivered to the load resistance,
To find the maximum power, differentiate the above expression with respect to resistance RL and equate it to zero. Thus
Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery.
Maximum Power Transfer Theorem can be applicable in complex network as follows
A resistive load in a resistive network will abstract maximum power when the load resistance is equal to the resistance viewed by the load as it looks back to the network. Actually this is nothing but the resistance presented to the output terminals of the network. This is actually Thevenin equivalent resistance as we explained in Thevenin's theorem if we consider the whole network as a voltage source. Similarly if we consider the network as current source, this electrical resistance will be Norton equivalent resistance as we explained in Norton theorem.
Source Transformation
Source Transformation is the replacement of a voltage in series with a resistor by a current source in parallel with a resistor or vice versa. the two circuits are equivalent if they have the same current-voltage relationship at their terminals
*A two terminal circuit element is defined by its voltage-current relationship
*Relationship can be found by applying a voltage source to the element and finding the relationship to current
*Equivalently, can apply a current source and find relationship to voltage
*If two elements have the same, they are interchangable
Thevenin's Theorem
Thevenin's theorem: a linear two-terminal circuit is electrically equivalent to a voltage source in series with a resistor
Norton's Theorem
Norton's theorem: a linear two-terminal circuit is electrically equivalent to a current source in parallel with a resistor
Maximum Power Transfer
Suppose we have a voltage source or battery whose internal electrical resistance is Ri and a load resistance RL is connected across this battery. Maximum Power Transfer Theorem determines the value of resistance RL for which the maximum power will be transferred from source to it. Actually the maximum power, drawn from the source, depends upon the value of the load resistance. There may be some confusion let us clear it.
Power delivered to the load resistance,
To find the maximum power, differentiate the above expression with respect to resistance RL and equate it to zero. Thus
Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery.
Maximum Power Transfer Theorem can be applicable in complex network as follows
A resistive load in a resistive network will abstract maximum power when the load resistance is equal to the resistance viewed by the load as it looks back to the network. Actually this is nothing but the resistance presented to the output terminals of the network. This is actually Thevenin equivalent resistance as we explained in Thevenin's theorem if we consider the whole network as a voltage source. Similarly if we consider the network as current source, this electrical resistance will be Norton equivalent resistance as we explained in Norton theorem.
