IMAGINARY AC CIRCUITS AREN’T really complex

If you have ever read advanced textbooks or papers about electronics, you may have been amazed to see the use of complex numbers used in the analysis of AC circuits. A complex number has two parts: a real part and an imaginary part. I’ve often thought that a lot of books and classes just kind of gloss over what this really means. What part of electrical power is imaginary? Why do we do this?

The short answer is phase angle: the time delay between a voltage and a current in a circuit. how can an angle be a time? That’s part of what I’ll need to explain.

First, consider a resistor. If you apply a voltage to it, a certain current will flow that you can identify by Ohm’s law. If you know the immediate voltage across the resistor, you can derive the current and you can find the power–how much work that electrical power will do. That’s fine for DC current through resistors. but components like capacitors and inductors with an AC current don’t obey Ohm’s law. Take a capacitor. current only flows when  the capacitor is charging or discharging, so the current through it relates to the rate of change of the voltage, not the immediate voltage level.

That implies that if you plot the sine wave voltage against the current, the top of the voltage will be where the current is minimal, and the top current will be where the voltage is at zero. You can see that in this image, where the yellow wave is voltage (V) and the green wave is current (I). See how the green top is where the yellow curve crosses zero? and the yellow top is where the green curve crosses zero?

These linked sine and cosine waves might remind you of something — the X and Y coordinates of a point being swept around a circle at a constant rate, and that’s our connection to complex numbers. By the end of the post, you’ll see it isn’t all that complicated and the “imaginary” quantity isn’t imaginary at all.

Simplifying Assumptions

Start with an audio signal of someone speaking and feed that into your circuit. It is awash with different frequencies that change constantly. If you had a circuit with only resistors in it, you could pick a point in time, find all the frequency components present or the immediate amplitude, derive the immediate currents, and you could use conventional techniques on it. You’d just have to do it over and over and over again. If the circuit involves inductors or capacitors, whose behavior depends on much more than just the voltage across them, this becomes very challenging very quickly.

Instead, it is simpler to start with a sine wave at a single frequency and assume that a complex signal of numerous different frequencies is just the sum of numerous single sines. One way to think of a capacitor is to consider it a resistor that has higher resistance at lower frequencies. An inductor acts like a resistor that gets larger at higher frequencies. because we are only considering a single frequency, we can convert any capacitance and inductance values to an impedance: a resistance that is only good at the frequency of interest. What’s much more is that we can represent impedance as a complex number so that we can track the phase angle of the circuit, which directly relates to a particular time delay between voltage and current.

For a true resistor, the imaginary part is 0. That makes sense because the voltage and current are in phase and for that reason there is no time delay at all. For a pure capacitor or inductor, the real part is zero. real circuits will have combinations and thus will have a combination of real and imaginary parts. Numbers like that are complex numbers and you can write them in several different ways.

Complex Review

The first thing to remember is that the word imaginary is just an arbitrary term. maybe it is better to forget the normal implying of the word imaginary. These imaginary quantities are not some kind of magic electricity or resistance. We use imaginary numbers to represent time delays in circuits. Det er alt.

There is a long story about what imaginary numbers imply in pure math and why they are called imaginary. You can look that up if you are a math-head, but you ought to know that math books use the symbol i for the imaginary part of a complex number. However, because electrical engineers use i for current, we use j instead. You just have to remember when reading math books, you’ll see i and it isn’t a current, and it is the same as j in electrical books.

There are several ways to represent a complex number. The simplest way is to write the real part and the imaginary part as being added together along with j. So consider this:

5 + 3j

We say the real part is 5 and the imaginary part is 3. Numbers written in this form are in rectangular format. You can plot it on the number lines like this:

That leads to the second way to write a complex number: polar notation. If the point on the graph is 5 + 3j, you can note that a vector can represent the same point. It will have a length or magnitude and an angle (the angle it makes with the X-axis of the graph). In this case, the magnitude is 5.83 (about) and the angle is just a little under 31 degrees.

This is interesting because it is a vector and there are a lot of good math tools to manipulate vectors. It is going to become really essential in a minute because the angle can correspond to a phase angle in a circuit and the magnitude has a direct physical relationship, as well.

Fasevinkel

Remember that I said we do an AC analysis at a single frequency? If you plot the AC voltage across and the current going through a resistor at some frequency, the two sine waves will line up exactly. That’s because a resistor doesn’t time delay anything. We’d say the phase angle across the resistor is zero degrees.

However, for a capacitor, the current will appear to rise before the voltage by some amount of time. This makes sense if you think about your intuition about capacitors at DC. When a capacitor is discharged, it has no voltage across it, but it will consume a lot of current — it temporarily looks like a short circuit. As the charge builds, the voltage rises but the current drops, until the capacitor is fully charged. At that point, the voltage is at a maximum, but the current is zero, or nearly so.

Inductors have the opposite arrangement: voltage leads current, so the curves would look the same but the V curve is now the I and the I curve is now the V. You can remember that with the easy mnemonic ELI the ICE man, where E is voltage just like in Ohm’s law. When you talk about phase shift in a circuit, you really imply how much the current leads or lags the voltage at a given frequency. That’s a essential idea: phase shift or angle is the amount of time the current leads or lags the voltage. You can also measure phase between other things like two different voltage sources, but typically when you say “this circuit has a phase shift of 22 degrees” you imply the voltage vs the current time delay.

Keep in mind a sine wave is like a circle bent to fit a line. So if the start of the sine wave is at 0 degrees, the top of the positive top is 90 degrees. The second 0 crossing is 180 degrees, and the negative top is 270 degrees–just like the points on a circle. because the sine wave is at a fixed frequency, putting something at a particular degree mark is the same as expressing a time.

In the case of a resistor, the shift is 0 degrees. So in complex notation, a 100 ohm resistor is 100 + 0j. It can also be 100∠0. For a capacitor, the current rises before the voltage by 90 degrees so a capacitor has a phase shift of -90. but what’s the magnitude?

You probably learned that the capacitive reactance is equal to 1/(2πfC) where f is the frequency in Hz. That’s the magnitude of the polar form. Of course, because -90 degrees is straight down the number line, it is also the imaginary part of the rectangular form (and the real part is zero). If capacitive reactance (Xc) is equal to 50, for example, then you could write 0-50j or 50∠-90. Inductors work the same but the reactance (Xl) is 2πfL and the phase angle is 90 degrees. So an inductor with the same reactance would be 0 + 50j or 50∠90.

Finding the Power

Let’s look at a quick example of what these phase angles are good for: calculating power. You know that power is voltage times current. So if a capacitor has 1 V across it (peak) and draws 1 A through it (peak), is the power 1 watt? No, because it doesn’t draw 1 V at 1 A at the same time.

Consider this simulation (see figure to the right). You can see the traces to the left show the 90 degree phase shift very clearly (the green trace is voltage and the yellow one is current). The top voltage is 1.85 V and the current peaks at about 4.65 mA. The product of the voltage times the current is 8.6 mW. but that’s not the best answer. The power is actually 4.29 mW (see the graph on the right). In an ideal capacitor, power isn’t consumed. It is stored and released, which is why the power goes negative. real capacitors, of course, exhibit some loss.

Note that the power supply doesn’t offer 4.29 mW, but much less. That’s because the resistor is the only thing consuming power. The voltage and current are in phase for it and some of the power it dissipates is coming from the capacitor’s stored charge.

Circuits

The magnitude of the vector is usable in Ohm’s law. For example, at 40 Hz, the Xc of the example circuit is just under 400 ohms. So the total complex impedance for the RC circuit is 1000 – 400j.

If you are adept with vectors you could do polar by writing 1000∠0 + 400∠-90. However, it is typically simpler to write the rectangular version and convert to polar (Wolfram Alpha is good at that; just remember to use i instead of j). The magnitude is just the Pythagorean theorem and the angle is easy trig. I am not going to go into it, but here’s the formula where R and J are the real and imaginary parts, respectively.

mag=SQRT(R^ 2 + j ^ 2)
Fase = Arctan (J / R)

Vårt eksempel er da 1077∠-21,8.

Så hva kommer kraften ut av spenningskilden? Power er E ^ 2 / R (eller faktisk E ^ 2 / Z i dette tilfellet). Så 25/1077 = 23 MW Peak. Simuleringen viser 22,29 og fordi jeg avruffet noen få verdier, det er nært nok.

Det er det?

Det er ikke det, selvfølgelig, men det er alt du trenger å vite for mange formål. Tallrike elektroniske tekster på hobby-nivå på detaljene og bare arbeid med størrelser. For enkle kretser kan dette fungere, men for noe kompleks (ingen ordspill ment), blir det håret raskt.

Forresten, dette eksempelet viste elementer i serie. Du kan imidlertid legge til reaktanser i parallell akkurat som du gjør motstander parallelt.

De grunnleggende konseptene du trenger å huske er:

Analysen av en vekselstrømkrets forekommer hovedsakelig ved en enkelt frekvens med en sinusbølgeinngang.

Imaginære tall er ikke imaginære.

Magnitudene til komplekse tall i polare former kan behandles som en motstand.

Fasevinkel er tidsforsinkelsen mellom spenningen og den nåværende bølgeformen.

Det er mange detaljer jeg glanset over. Du trenger sannsynligvis ikke å vite hvordan jeg egentlig er kvadratroten av negativ. Eller hvordan Eulers nummer spiller inn i dette og enkelheten til å integrere og skille med sinusbølger skrevet med en amplitude og en fasevinkel. Hvis du er interessert i matematikkhistorie, har imaginære tall ganske en historie bak dem. Hvis du vil ha noe mye mer praktisk, har Khan Academy noen nyttige videoer. Men det som er dekket her burde være alt du trenger å vite for å jobbe med vekselstrømskretser.