Chapter 4 - AC Theory

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4.1 AC Theory - basics

In the previous section, the concept of apparent power was introduced as being a vector composed of the elements of reactive power and active power. A vector is a two-dimensional quantity composed of two single dimensional quantities called scalars at right angles to each other (P and Q in figure 3.3).

• A vector can be denoted by an arrow above it (e .g. S) or more simply by bold type, e.g. S.
• The length of a vector (its magnitude or modulus) is the square root of the sum of the squares of the two scalars that define it and is shown by bars on either side |S|. So, in figure 3.3:

Just as apparent power is a vector, so are impedances. The reactance of an inductor consists of two elements, its DC resistance, (DCR), which does not change with frequency and its impedance, which is directly proportional to frequency. As both are measured in Ohms, they can be represented on the same vector diagram.

If the frequency is zero (DC), then XL = 2πfL is also zero which makes |Z|=R, where R is the DCR of the inductor. As the frequency is increased, the scalar XL (the AC impedance) increases and the resulting AC reactance is the vector, Z, with modulus |Z|. At infinite frequency, the AC impedance is infinite and the resistance scalar R becomes zero. The angle between the scalars, φ, is dependent on the angular frequency 2πf.



The same concept applies to capacitive reactances:
If the frequency is infinite, then XC = -1/2πfC is zero which makes |Z|=R . The resistance R is just the capacitor equivalent series resistance (ESR). As the frequency is decreased, the scalar Xc (the AC impedance) increases and the resulting AC reactance is the vector, Z, with modulus |Z|. At zero frequency (DC), the impedance is infinite and the resistance scalar R is zero. The angle between the scalars, φ, is dependent on the angular frequency 2πf.

Note: The reason why φ is conventionally shown as positive with an inductor and negative with a capacitor is because voltage leads current in an inductor, but lags it in a capacitor.

A phasor is a particular type of vector. If an impedance vector is multiplied by a current, it is transformed to a voltage (this is commonly known as Ohm’s Law, |V| = I|Z|), but it does not change direction. The phase information is retained. Such constant phase vectors are called phasors.

The advantage of phasors is that we can add reactances in series (the current is therefore the same for all elements and can be represented by a reference phasor Î, which is equal to the magnitude of the current and work out the resulting voltage phasor ÎX.)




The same applies to adding reactances in parallel (then the voltage phasor is therefore the same for all elements) and work out the resulting current phasor. The same relationship for adding resistors in parallel applies to adding reactances in parallel:



This is all well and good, but we can’t solve every impedance problem by drawing vector or phasor diagrams. We need some more mathematics.

The Z phasor consists of resistive and reactive elements, R and X, so if we get Eq. 4.2 for the magnitude of Z for a parallel circuit:

This expression has the form of a quadratic binomial (ax² + bx + c = 0), which if you remember from your school maths lessons has the general solution of

The problem is, if the term 4ac turns out to be larger than b², then we have the square root of a negative number. We can’t simply ignore this. This is applied mathematics based on real components, so such results exist in real life.

Leonhard Euler (*1707 - †1783) gave the term “i” for the quantity √ (-1), but as “i” can be confused with the symbol for current, in electronics we use “j” instead.

Any relationship including √ (-1) is a complex number with a real part and an imaginary part. The word “imaginary” somehow implies that the term does not really exist, which is not true. It is maybe more helpful to think of a number line of real numbers from –infinity to + infinity with imaginary numbers placed at 90° also going from –infinity to + infinity in the imaginary plane:



If we now spin the imaginary number line around the real number axis, we get this image:



Does this look familiar? Maybe like the typical image of the field surrounding a straight wire? And lo and behold! Maxwell’s electromagnetic field equation can be written in the form of:

Where F is the combined EM field created by the combination of the E electric and H magnetic fields. In the case of magnetics, the fact that the imaginary part has both a positive and a negative solution is not relevant; we can choose just the positive or the negative part as they are symmetrical.

In other situations, the ± terms are not equivalent: when Equation 4.3 is applied to light transmission, for example, then the positive and negative terms are more commonly called right and left circularly polarized light.

In general, we can simplify the description of the rectangular form reactance vector diagram into a much neater complex number representation: Z = cosφ |Z|+sinφ |Z|, → R + jX

The beauty of this notation is that it allows us to extend the familiar Ohm’s Law relationships from DC to AC situations (from resistances to reactances) and to use traditional solutions such as Thévenin’s Theorem to analyse component networks.




Adding reactances together becomes simpler because the results are always in the form of Z = R + jX. For example, the reactance of an inductor, capacitor and resistor placed in series becomes:





To also show how useful this notation is, let us take the network shown above which is a resonant tank and work out its response. As the network is in series, the current flowing through all three components must be the same. At resonance, the L and C reactances cancel out, so the peak current, Io, flowing through the network is simply │V│/ R (assuming that R is much larger than the capacitor ESR and the inductor DCR). At other frequencies, the current I that flows through the network is:



The magnification factor, Q, determines how quickly the current decreases away from the peak at the resonant frequency. It is defined as X0/R where X0 is the reactance of the network at resonance,

If the results of Equation 4.5 are plotted with different Q values, we get the following typical curves:


Thus we can calculate in advance the response of our resonant tank circuit without actually having to build it and determine the optimum Q factor experimentally.