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Cause and effect for linear systems are related by linear differential equations. note that for the exponential function ,

 and

where is also a constant. Similar equations hold for derivatives also.

Consider a function . If on operation by a system, the function is only multiplied by a constant , the function is said to be the eigenfunction of the given system, as shown in 8.1.
We know that:

 (8.1)

Consider a linear system, as in 8.2. The functions and are eigen functions of the linear system. If sum of these two functions is input to the system, the output can be predicted easily, by superposition theorm. Thus knowing the output to eigenfunctions helps us in predicting output of several other functions.

Now we look at how output to sinusoidal excitations of linear circuits can be determined. Consider or .

Note that . Thus, the phase and associated constant changes when a sinusoid is passed through a differentiator.

Similarly, .

Now consider the RLC circuit shown in 8.3.

 If i(t) is sinusoidal, say cos(wt), then where,

the last equality follows using

, as shown in 8.4.

Analysis can be done simply using sin and cos terms.

But can't be further simplified using imaginary quantities

As in the diagram shown in 8.5, In figure 8.6, the complex values shown are rotating with time. The actual value at any time is the projection on the real axis.

Note that and have constant separation with each other. All entities in general will have same relative separation.

Consider figure 8.7. Let us rotate the frame of reference (or axis) also with speed rad/sec. Then the angles of and with respect to the axis become constants. The figure 8.7 is a phasor diagram, showing current and voltage phasors, and the phase difference between the two.

We show the application of phasors in circuit analysis by the circuit shown in figure 8.8, a simple inductor circuit excited by a sinusoidal voltage source.

The associated phasor diagram is shown in figure 8.9. It can be seen that the phase difference is radians. Voltage leads the current by radians.

Similarly, consider a resistance excited by a sinusoidal votage source, as shown in figure 8.10. Here, , and is in phase with the voltage. The same is shown in the phasor diagram 8.11.

Now consider a resistor and an inductor in series with an AC voltage source, as in 8.12.

 where

The resulting phasor diagram is plotted in figure 8.13.

This gives an inkling to a general result: phasors can be added/subtracted just like vectors. Resulting magnitude and phase would come out to be the same. See the hint below.

In phasor terms: Voltage across inductor: , Voltage across resistor. Now adding the two vectors,

Compairing with Ohm's law, , the previous equation , the complex term can be taken to be similar to resitance. This is called impedance. Inverse of impedance is called admittance, complex analog of conductance. In the above circuit,

Similar to the above analysis, we now work with the capacitor. See figure 8.14.

Thus, lags by w.r.t. , as shown in phasor diagram 8.15. The phasors are typically written in capital letters, whereas their continuous time counterparts in small letters. In phasor diagram, . Thus, impedance is .

Now we analyse circuit shown in in figure 8.16, . We wish to calculate and .

 (phasor in rectangular coordinates x and y) for first loop:  ...(*) for second loop:  ...(**) From (**):    From (*) :

Solving these two for and , we get:

 which after rationalization, gives:

For sinusoidal forcing functions, we can use the same techniques, but with complex variables

A sinusoid passed through a linear system with transfer function , the output would be .

Thus, for a sum of sinusoids of different frequencies, using superposition principle, the output for would be:

 (8.2)

Next: Power Supply Up: Introduction to Electronics Previous: R-C Circuits   Contents
ynsingh 2007-07-25