# Complex Impedance Part 1

One of the more advanced topics in amateur radio electronics, and one that is often difficult for students to fully grasp, is complex impedance and the phase angle between voltage and current in AC circuits that helps define the behavior of the circuit.  This topic is all cluttered up with inductive reactance, capacitive reactance, and resistance, in addition to some heady trigonometric concepts like the sine of angles, polar coordinates, and even those kooky imaginary numbers!

Well, let’s see if we can keep it real, ham hipster, and disentangle this testy topic with some common sense consideration and explanations.

1. Voltage and current applied to AC circuits are each represented by smoothly changing sine waveforms of equal frequency, depicting the regular reversals of direction and smoothly changing magnitudes of each.
2. The applied voltage and current sine waves often get out of step with one another so the two representations no longer oscillate together, as if one sine wave is shifted ahead or behind the other in time, or phase.
3. The amount of deviation between the voltage and current sine wave signals in a circuit is described by a phase angle between the two signals, in units of degrees.
4. Phase angle shifts between voltage and current are imposed by a type of opposition to current flow called reactance in AC circuit components, specifically inductive reactance and capacitive reactance, measured in units of ohms.
5. Inductive and capacitive reactances combine in a complex way with resistance in a circuit to determine the overall impedance of the circuit.
6. Complex impedance is described with both a magnitude in ohms and a phase angle in degrees, and there are two primary shorthand methods of representing complex impedance in writing.
7. Impedance magnitude and phase angle impact the behavior of AC circuits, particularly with respect to power transfer and resonance, as in RF antenna circuits, oscillator circuits, matching networks, power supply circuits, and many others.

We will explore these enumerated facets of complex impedance, reactance, and phase angles in more detail for some in-depth understanding of these electrical principles.  We start with some simple facts about voltage and current in AC circuits and build from there.  Press on, intrepid reader!

Voltage and Current:  I’m sure you’re already familiar, but just to be safe let’s state a couple of things that are probably obvious.

Current is the motion or flow of electric charges in the circuit. You may think of this as negatively charged free electrons flowing through the wires and components. Keep in mind that in AC circuits the current reverses direction on a regular basis. Back and forth motion. And that is key to understanding phase angles.

Voltage is the difference in electric potential between two points in the circuit. Using the water analogy of a circuit from the HamRadioSchool.com books, potential difference is like water pressure differences in a plumbing circuit. The pressure may be great in advance of driving a load like a waterwheel (or an electric appliance), and since some of the potential energy is expended on the load the pressure beyond it is reduced. Measure the pressure before and after the load and you’ll get a pressure difference, or a potential difference, expressed as a voltage. Sometimes you’ll hear this difference characterized as a “voltage drop.”

In an AC circuit the electric potential or voltage also reverses on a regular basis, with the same frequency as the current reversals.  Again, a key concept for really getting the phase angle topic.

Sine Waves:  The most common alternation of current and of voltage in AC circuits may be depicted as a sine wave, that smoothly curving wave image you have probably seen many times. You may ask, “Why is the sine wave so common?” and that is a great question, but a question for another lesson altogether. I will ask for now that you simply accept as fact the lovely sine wave variation of current and voltage is common and perhaps easiest to deal with among other alternating waveforms.

The most common form for AC signals is the sine wave.

What the sine wave is representing is the regularly varying magnitude and direction of current flow or of potential difference over time in a circuit. For current, the upward portion of the waveform indicates the magnitude of current flowing in the positive direction in the circuit, smoothly increasing over time, peaking, and then decreasing in magnitude over time as measured in units of amperes. The bottom half of the current sine wave depicts a similar cycle of current magnitude, only in the other (negative) direction of flow.

For voltage, the sine waveform indicates an analogous cycle of voltage, with potential difference increasing, peaking, and decreasing in the positive direction, or more correctly stated, “with positive polarity” (the potential drop from high to low between two measured points is in the positive direction). This is followed from the zero voltage line by an identical cycle of voltage magnitude in the opposite (negative) direction, or polarity, in which the potential drop direction is reversed.

If you are not a student of trigonometry you may now inject, “But, what is sine?” That, too, is a great question that we’ll try to get at graphically for a little better understanding.

Imagine a wheel rotating about is center axis, and one rotation is a standard 360 degrees of angular measure. Put a mark on the outside edge of the wheel, and let’s say the wheel has a radius of 1 unit. (That may be one inch, one foot, or one mile. It doesn’t matter. It is one unit from the axis to the edge, or “one radius.”) Place the mark to the right as shown in the diagram and designate that location the 0 degree position of the wheel. Now rotate the wheel counter-clockwise. The mark on the wheel moves up, and let’s stop it at a 30-degree angle above the 0 degree position. If you measure the height of the mark above the 0 degree position in units of the wheel’s radius, you will find that the mark is exactly one-half of a radius, or 0.5. The sine of a 30 degree angle is 0.5. Continue rotating the wheel counter-clockwise until the mark is at the 45 degree position and again measure its height above the 0 degree position. The mark is 0.707 radiuses high.  The sine of 45 degrees is 0.707. You get the idea. Every angle around the 360 degree circle from the 0 degree rotational start point will have an associated height relative to the radius of the circle, and that is the sine of the angle. Notice that at the 90 degree point the sine is 1.0. At the 150 degree point the sine is again 0.5. At the 180 degree point the sine is zero, same as the 0 degree angle sine. Notice also that the graphic depiction of sine above for 30 degrees and 45 degrees forms a right triangle, and the value of sine for any rotational angle of the wheel is computed as the opposite side of the triangle (side opposite the angle in question) divided by the radius “mark” side of the triangle. In our case, the radius “mark” side is always a value of 1. But for any right triangle the sine may be computed as the value of the triangle side opposite the angle divided by the side across from the right angle, the longest side called the hypotenuse. Sine is “opposite over hypotenuse” for any angle in a right triangle.

Sine is calculated as the angle’s opposite side divided by the hypotenuse.

As you continue to rotate the wheel and increase the angle beyond 180 degrees the sine values become negative values, as indicated by a distance below the 0 degree line.  At 225 degrees the sine is -0.707. At 270 degrees sine is -1.0. At 315 degrees sine is again -0.707, and so on. With a scientific calculator set to use degrees for angular values (in lieu of radians), simply enter the angle value in degrees and press the sine button to see the exact sine value of the angle. Try it!

You can see now that a sine wave is a continuous plot of the mark’s vertical distance from the 0 degree position (the sine of the angle) as the wheel rotates at a constant rate over time. In fact, the positions along the horizontal time axis of the sine wave plot are identified by the circular angles. For instance, the first upper peak of a standard up/down cycle of a sine wave (one full rotation) is the 90 degree point of the sine wave. The point at which the sine wave decreases and crosses the zero line again is the 180 degree position. The downward pointing peak position is 270 degrees, and so on. There is a one-to-one relationship between the angle of rotation of the wheel and positions across the sine waveform. A sine wave is a continuous repeating plot of the value of sine as the value of the angle cycles from 0 to 360 degrees.

Consider that if you spin the wheel faster you obtain a higher frequency sine wave, with more cycles per unit of time. Slower rotation yields a lower frequency wave.

Once again, the most common AC signals you will encounter, whether oscillating electrons in a circuit or oscillating electromagnetic fields in a traveling RF wave, are accurately characterized by the sine wave variations of magnitude and direction. Further, in common AC electric circuits both the voltage and the current follow the sine form with equivalent frequency of oscillation.

Phase Relationships:  So, we have two sine waveforms to work with in an AC circuit, one for voltage and one for current, and they are of equal frequency. Imagine that each of these two waveforms is derived from its own rotating wheel, as described above. The voltage wheel and the current wheel must rotate at the same rate for the frequencies to be equal between the two waveforms.

Imagine the two wheels begin rotation at exactly the same instant in time. The two sine waves that result will be perfectly in-step with one another. Each wave will peak at the same point on the time axis, each will cross the zero line at the same position, and so on. Nice.  (Although there is no required equivalency of amplitude, as shown.) But what if one of the wheels is delayed in starting as compared to the other wheel? What if one wheel begins rotating before the other? Then the two sine waves will no longer be perfectly in-step with one another. Instead, the waveforms will reach peaks and cross the zero line at different times, at different positions along the time axis. The size of the disparity between the two out-of-step waveforms is determined by the difference in the start time of wheel rotation.

Remember, each position on a sine wave has an angle analog from the wheel rotation in units of degrees. So, you may describe the extent to which the two waveforms are out-of-step with one another in units of degrees. For example, in the graph below evaluate the degree position of the earlier waveform (black dashed) that is matched up with the zero degree starting point of the later waveform (blue solid). In this graphic the later waveform begins its cycling when the earlier waveform is already at its 90 degree position (the upper peak position of any full cycle). We may state that these two waveforms are 90 degrees out of phase. Further, we can state that the earlier (black-dashed) wave leads the later (blue-solid) wave by 90 degrees, or that the later wave (blue-solid) lags by 90 degrees. Any two sine waves of identical frequency have a phase relationship. The phase relationship is a description of how much the two waves are out-of-step with one another, as described above, in units of degrees (or radians).

The phase angle describes the time disparity between the voltage and current sine waves in a circuit.

However, notice that when a waveform position is specified with an angle, such as the 90 degree position, that also specifies a time or duration as measured from the time of the zero start point of rotation. As the time axis implies, time and angle of the waveform depiction are rigidly related. That is, as the wheel turns at a constant rate the angle of the mark changes constantly with it, and every angle around the wheel is associated with a unique amount of time required for the mark to rotate around to it from the 0 degree starting point.

The difference between the two waveform times or phases, as stated in units of degrees, is called the phase angle.

The voltage and current phase angle is important because it affects the behavior of the AC circuit, particularly with respect to characteristics of resonance and power transfer.

We will get around to the topics of resonance and power transfer, including the notion of how phase angle impacts circuits in these arenas.  But for now, let’s consider how AC circuit components produce phase variations between voltage and current.

Go to Part 2, Reacting Nicely
Go to Part 3, Putting It All Together