Application of Complex variables in AC circuits

Presentation by NoorQadar

Reg#0454

Complex numbers in electronics

Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier

Simple approach

To help you get a clear picture of how they're used and what they mean we can look at a mechanical example...

Simple approach

The above animation shows a rotating wheel. On the wheel there is a blue blob which goes round and round. When viewed 'flat on' we can see that the blob is moving around in a circle at a steady (if you computer is working OK!) rate. However, if we look at the wheel from the side we get a very different picture.

Contd…

From the side the blob seems to be oscillating up and down. If we plot a graph of the blob's position (viewed from the side) against time we find that it traces out a sine wave shape which oscillates through one cycle each time the wheel completes a rotation. Here, the sine-wave behavior we see when looking from the side 'hides' the underlying behavior which is a continuous rotation.

Now AC

We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as 'side views' of something which is actually 'rotating' at a steady rate. We can only see the 'real' part of this, of course, so we have to 'imagine' the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the 'real' portion' of a 'complex' quantity that also has an 'imaginary' part. At any instant what we see is determined by a phase angle which varies smoothly with time

The smooth rotation 'hidden' by our sideways view means that this phase angle varies at a steady rate which we can represent in terms of the signal frequency, 'f'. The complete complex version of the signal has two parts which we can add together provided we remember to label the imaginary part with an 'i' or 'j' to remind us that it is imaginary. Note that, as so often in science and engineering, there are various ways to represent the quantities we're talking about here. For example: Engineers use a 'j' to indicate the square root of minus one since they tend to use 'i' as a current.

Mathematicians use 'i' for this since they don't know a current from a hole in the ground! Similarly, you'll sometimes see the signal written as an exponential of an imaginary number, sometimes as a sum of a cosine and a sine. Sometimes the sign on the imaginary part may be negative. These are all slightly different conventions for representing the same things. (A bit like the way 'conventional' current and the actual electron flow go in opposite directions) The choice doesn't matter so long as you're consistent during a specific argument

KVL allows addition of complex voltages. The polarity marks for all three voltage sources are

oriented in such a way that their stated voltages should add to make the total voltage across the load resistor. Notice that although magnitude and phase angle is given for each AC voltage source, no frequency value is specified. If this is the case, it is assumed that all frequencies are equal, thus meeting our qualifications for applying DC rules to an AC circuit (all figures given in complex form, all of the same frequency). The setup of our equation to find total voltage appears as such:

The sum of these vectors will be a resultant vector originating at the starting point for the 22 volt vector (dot at upper-left of diagram) and terminating at the ending point for the 15 volt vector (arrow tip at the middle-right of the diagram):

Resultant is equivalent to the vector sum of the three original voltages.

In order to determine what the resultant vector's magnitude and angle are without resorting to graphic images, we can convert each one of these polar-form complex numbers into rectangular form and add. Remember, we'readding these figures together because the polarity marks for the three voltage sources are oriented in an additive manner:

In polar form, this equates to 36.8052 volts -∠20.5018o. What this means in real terms is that the voltage measured across these three voltage sources will be 36.8052 volts, lagging the 15 volt (0o phase reference) by 20.5018o. A voltmeter connected across these points in a real circuit would only indicate the polar magnitude of the voltage (36.8052 volts), not the angle. An oscilloscope could be used to display two voltage waveforms and thus provide a phase shift measurement, but not a voltmeter. The same principle holds true for AC ammeters: they indicate the polar magnitude of the current, not the phase angle.

Application to Electrical Engineering:

First, set the stage for the discussion and clarify some vocabulary.   Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics.  Scalar numbers are the kind of numbers students use most often.  In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.

When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance

Application to Electrical Engineering: (called impedance in AC) were not the familiar one-dimensional scalar

quantities that are used when measuring DC circuits.  These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.

In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities.  In order to accomplish this task, scalar numbers were abandoned and complex numbers were used to express the two dimensions of frequency and phase shift at one time.

In mathematics, i is used to represent imaginary numbers.  In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current.  It is also customary for scientists to write the complex number in the form a + jb.

Introduce the formula E = I • Z  where E is voltage, I is current, and Z is impedance. 

Examples

The impedance in one part of a series circuit is 2 + j8 ohms, and the impedance in another part of the circuit is 4 - j6 ohms.  Find the total impedance in the circuit.  Answer:  6 + j2 ohms 

The voltage in a circuit is 45 + j10 volts and the impedance is 3 + j4 ohms.  What is the current?Answer:E = I • Z45 + j10 = I • (3 + j4)

7 - j6 amps

Conclusion

Firstly, it helps us understand the behaviour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals. Secondly, it gives us a new way to think about oscillations.