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Appendix E. Complex math

Calculations on a capacitor and resistor in series.

We already know how to calculate the impedance of a capacitor:

At 10kHz a 1nF capacitor has an impedance of 15.9k.

When we connect a 10k resistor in series with this capacitor, we may expect that the total impedance will be 25.9k is. But that's not the case, because a capacitor causes a -90 degrees phase shift in the current flow. The equation

doesn't show this. But what if we plot the impedance as a vector? The length will be the (absolute) impedance and the angle the phase shift:

This component has an absolute impedance of 1 ohm and causes a 45 degrees phase shift between voltage and current.

A resistor doesn't cause a phase shift; this vector will be on the x-axis. a capacitor causes a -90 degrees phase shift and will be on the (negative) y-axis. The total impedance of R and C is R + XC. However, we have to add vectors instead of plain numbers. Since we have 90 degrees angles, it's easy to calculate the (absolute) impedance: we can just use Pythagoras' theorem. Zt = √(R2+XC2). The phase shift is equal to arctan(-XC/R)

Wouldn't it be nice if we had a more simple way for saying: the impedance is x ohms and causes a y degrees fase shift? A 180 degrees phase shift is easy; in that case we could say: the impedance equals -x ohms. A 180 degrees phase shift equal to multiplying by -1. Now suppose that a phase shift of 90 degrees is equal to multiplying by j. A 180 degrees phase shift will be the same as multiplying by j2. This means that j2 = -1. A negative square is only possible in so called 'complex math'. (Mathematicians among us may be accustomed to use i instead of j. But we already use i as a symbol for current, so that's confusing.)

Every impedance can be written as: a + bj. Number a is called the real part and is plotted on the x-axis. Number b is called the imaginary part and is plotted on the y-axis.

A resistor doesn't cause a phase shift and is therefore purely real.

We know that a capacitor causes a -90 degrees phase shift; its impedance is therefore purely imaginary, and can be written as:

As an example, we'll look again at our 10k resistor in series with a 15.9k capacitor. The total (complex) impedance is 10k - 15.9kj. The absolute value (|Z|) equals √(10k2 + 15.9k2) = 18.78k. The phase shift it causes in the current (arg(Z)) is arctan(-15.9k/10k) = -57.8 degrees.

So, when we connect a resistor and capacitor in series: ZR+C = R - XCj.

The absolutie value is: |ZR+C| = √(R2+XC2).

The phase shift is: arg(ZR+C) = arctan(-XC/R).

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