Vectors Show How Circuits Work
July 1966 Radio-Electronics

July 1966 Radio-Electronics [Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Radio-Electronics, published 1930-1988. All copyrights hereby acknowledged.

 Norman Crowhurst's 1966 Radio-Electronics article argues that vectors, though heavily used in power engineering, are underutilized in electronics despite their broad utility. Mr. Crowhurst demonstrates their application in modulation analysis, impedance calculations, feedback circuits, harmonic distortion, oscillator design, and filter construction. Vectors simplify complex alternating-current relationships by representing magnitudes and phases as rotating projections, translating into sine waves when plotted over time. Key diagrams illustrate their use in inductive/capacitive reactance, resonant circuits, and vector addition via parallelograms. He clarifies conventions - like vertical cosine references - and emphasizes visualizing entire vector systems rotating counterclockwise at circuit frequencies. He dismisses the need for advanced trigonometry, advocating geometric vector addition instead. The article promises future installments on phase modulation, positioning vectors as indispensable for demystifying electronics. Crowhurst's approach bridges theoretical clarity with practical design, urging engineers to adopt vectors beyond power systems. See his Making Modulation Easy to Understand follow-on in the September edition.

Vectors Show How Circuits Work

Everybody knows what a vector is - or do you? Maybe you weren't taught. They're helpful, though

By Norman H. Crowhurst

Power engineers use vectors extensively in their circuit analysis, but, in the usual treatments of electronics, vectors are hardly mentioned. Vectors can be far more useful to electronics than in the relatively simple concepts of power engineering, so limiting their use is a considerable loss.

I use vectors in my book Mathematics for Electronic Engineers and Technicians* as an aid in modulation analysis (amplitude, frequency and phase), in impedance analysis, in feedback circuit analysis and in multifrequency analysis, including harmonic distortion. I use vectors in analyzing the performance of different oscillator types, in active reactance (tube or transistor) design, and in the design of filters, particularly twin-T and twin-π types. In each of these areas, vectors can shed considerable light as well as aid in actual calculations.

What Are Vectors?

This whole vector business starts with the notion of a varying amplitude (voltage, current or any other quantity) being represented by the projection of a rotating vector. For a picture of the idea of projection, see Fig. 1.

A vector quantity has direction as well as magnitude, which is to say that it makes an angle with some reference line. A vector that represents an alternating voltage or current is constantly changing its angle as the current rises and falls with time. In other words, it is rotating. The projection of a vector rotating at a constant angular velocity (equal change in angle for equal time) is a sine wave, if we make the projection screen "move" in a straight line at constant speed.

As the vector rotates - we've shown it (in Fig. 2) stopped every 30° - the horizontal projection to each corresponding point in time along an amplitude-time graph indicates the correct point on the sine wave.

In dealing with just one simple quantity, this seems a laborious way to arrive at a sine wave. But in dealing with the many alternating currents and voltages we have to cope with in electrical or electronic circuits, a simple vector diagram can tell as much as the detailed plotting of a great many sine waves-and with far greater clarity, once you understand the vectors.

Take the relationship of voltage and current in a pure inductance (see Fig. 3) or capacitance. (Note that in all standard vector diagrams, an open arrowhead stands for voltage, while a solid one stands for a current vector. A double arrowhead represents magnetic flux.) From the principles of reactance, we know that maximum voltage occurs at the same time as zero current, and vice versa. When we put vectors at right angles and project them, we portray sine waves with the correct 90° phase displacement for inductive or capacitive reactance. We have numbered 12 corresponding points at 30° intervals of rotation (or phase), in each case.

Now we should begin to think what the vectors really mean. If we think of them as rotating counterclockwise (as is usual) their projection on a vertical axis represents the instantaneous variation of the quantities they represent. We don't have to plot complete curves.

In power engineering, polyphase diagrams take off from just that point. This is as far as most engineers have gone in their application of vectors. They may also apply them to impedances, in particular to transformer operation. A transformer contains rather a lot of circuit elements to visualize any other way (invisible ones like winding capacitance and leakage inductance). But this use is about the limit for the majority. Yet a transformer vector diagram is more complicated than some more common things in electronics, such as modulation analysis.

Before we go on, we should clear up a small point that occasionally confuses people. The conventional reference vector is often vertical, which means it coincides with a cosine wave. A sine wave is represented by a vector pointing horizontally to the right. This means, in conventional counterclockwise rotation, that it lags the vertical by 90°. (Notice this relationship in Fig. 3) A sine wave and a cosine wave are exactly alike in shape; they differ only by 90° in phase.

If we go into the mathematical analysis that corresponds with the vector presentation, this means the reference wave (corresponding to the vertical vector) is a cosine, rather than sine, term. If you work your vectors with trigonometrical equations for solutions, this is important. But actually you don't need to know much trig to use this method of analysis.

The important part of the vector concept is to be able to think of the whole group of vectors as rotating counter-clockwise together. In a simple form this is shown in Fig. 3. A resonant circuit, considered at a frequency slightly above resonance, shows the same thing in more detail (Fig. 4).

In the resonant circuit, it is most convenient to start with a vector for the quantity that is common to all elements, which in the series circuit is current. Then voltage drop across the resistance in the circuit (V_R) is in phase with current. The drop across the capacitance (V_C) falls behind (lags) that by 90°, and the drop in the inductance (V_L) runs ahead (leads) by 90°. In this diagram, the inductive voltage is a little bigger than the capacitive, so the resultant is the difference between these two, added vectorially to the resistance component by taking the diagonal of the rectangle, or "completing the parallelogram."

Normally R is physically part of the inductor so, while the vector diagram at Fig. 4 breaks the voltage across the inductor into its resistive and reactive parts, the completed parallelogram uses components that are measurable - the voltage across the composite inductor- being V_LR. This is also used in curve development, which shows the relationship between angles on the vector diagram and the maximum points (on the curves) that represent the vectors.

If you're not sure this works, you should check out the procedure called completing the parallelogram (Fig. 5). The construction is simple enough: you just draw parallels and then fill in the diagonal. You can check that it works by projecting the sine waves from each vector. You will find that, at each point, the diagonal vector projects an instantaneous magnitude that is the sum of the instantaneous magnitudes of the two vectors that make it up.

We've plotted points at each 15° interval through the full rotation of 360°. Note that the magnitudes of the waves agree with the lengths of the vectors correspondingly lettered, and that geometrical angles A, B and C correspond with the phase angles between similar points along the curves marked A, B and C.

You must grasp the notion that a whole vector diagram, with all its vectors, rotates counterclockwise en masse. The operating frequency of the circuit elements sets the frequency of rotation. Once you learn that, plus this easy method of vectorial addition by simple geometry, you're ready to move on to more exciting things.

In the months to come, we hope to show you how you can apply your new understanding of vectors to clear up some of the more puzzling concepts of electronics. Watch for a vector analysis of phase modulation next.

* Howard W. Sams/Bobbs Merrill, 1964.