One of the most fundamental concepts in our profession is impedance. Due to the nuances of the mathematics related to this quantity, the minutia of what is taking place and the associated nomenclature on occasion is sometimes confusing. Although a more rigorous discussion of impedance probably belongs in a textbook, our discussion here will demystify some of the elusive language consulting engineers tend to use.
Nearly everyone is familiar with Ohm''s Law, which states that the current through a conductor between a pair of points is directly proportional to the voltage across the two points. When two quantities are directly proportional to each other, we are discussing a division problem. In this case current is equal to voltage divided by a constant of proportionality, or resistance. Rearranging the terms gives three different equivalent forms.
Resistance is, however, really a special case of impedance. Not only is impedance similarly the ratio between voltage and current, but it is also the sum of resistance and reactance. In other words, resistance can be thought of impedance when the reactance portion is zero. Impedance is denoted by Z, and reactance uses X giving the following two equations, in which the bold-faced variables are intentional. More on that shortly.
Reactance is somewhat similar to resistance, although it does differ in certain aspects. Resistance can be considered as the opposition to the flow of current. Reactance differs from resistance in that it is the opposition of a change in the current or voltage depending on the type of circuit element. While resistors dissipate energy in the form of light and/or heat, inductors and capacitors, our reactive elements, store energy in the form of a field. Inductors store energy in magnetic fields, while capacitors utilize an electric field as their method. The energy is stored up during half of the cycle, and returned in the other half. Reactance therefore depends on a signal that varies with time, it is frequency dependent, and is zero in direct current circuits. Reactance in ohms is calculated by the following to equations in which L is inductance, C is capacitance, and f is frequency.
The storage of the field results in a shift or phase change in the voltage and current relative to each other. In the inductor, a change in the circuit current induces a voltage that opposes the change in the current. This causes the current in an inductor to lag the voltage. Conversely, the capacitor opposes a change in voltage through the inducement of a current. Therefore, the current in a capacitor leads the voltage. A good moniker for remembering this is ELI the ICE man, where E represents voltage.
Since the reactive elements come in two different flavors, the quantity of reactance comes in two different flavors. Positive reactance is created by an inductor, or combination of components that is inductive, while negative reactance arises from a capacitor or capacitive circuit. So when we refer to a circuit that is “fifty j zero” we are saying 50 ohms of resistance, and 0 ohms of reactance. Similarly, an impedance of “thirty-five plus j five” would be 35 ohms of resistance and 5 ohms of inductive reactance.
In the above impedance equations, the j operator is equivalent to the square root of -1. This makes the reactance portion of the impedance the imaginary portion of the complex quantity, while the resistance portion is the real part. This really is somewhat of a misnomer as reactance is a very real quantity as anyone who has gotten across a charged capacitor realizes. What happens, without going into the really gross mathematics, is to give rise to the phase shift that occurs between the voltage and the current.
To visualize how this all relates, consider two perpendicular lines. These two perpendicular lines form a plane. On this plane, we will assign the x or horizontal axis as resistance, and the y or vertical axis as reactance. The location where the two lines intersect is the zero point where both resistance and reactance are zero ohms. If we move to the right of the vertical axis, then our x or resistance value increases, while a move to the left is a decrease. Similarly, a move up from the horizontal axis results in positive reactance, or inductive characteristics, while a downward move increases the capacitance.
Note that a move to the left of the vertical axis results in a negative value for resistance. This is of course not a realizable quantity in a passive circuit, but is perfectly acceptable in an active circuit. Incidentally, towers in a directional array where the resistance portion of the drive point impedance is less than zero ohms are negative towers. Negative towers are not always the specter they are made out to be in popular engineering circles, but that is a topic for another day.
Now let us consider a run-of-the-mill impedance value of 35+j50Ω. To plot this impedance value on our plane, we would move 35 units to the right of the zero point, and then move up 50 units. If our impedance were 35-j50Ω, we would still make that 35-unit move to the right, but then would move below the x-axis 50 units. From our point, we can then draw a line back to the zero point, as well as a second line downward to the x-axis. The resulting shape is a right triangle.
On this right triangle, we will assign two additional quantities. The first is Z, which lies along the hypotenuse of the triangle, and is the magnitude of the impedance. The other quantity is denoted by the Greek letter theta (Θ), and is the phase angle between the voltage and current in the circuit.
From high school math, we can find the value of Z, which is determined by the Pythagorean Theorem. The magnitude of the impedance is the square root of the sum of the squares of the resistance and reactance.
The tangent of the phase angle is equal to the quotient of the reactance and the resistance.
From this, we can see that the impedance can be written in two different notations. While the R+jX or Cartesian notation allows for greater ease in visualizing the magnitude of the two components, the polar or phasor notation consisting of the magnitude and phase angle allows for greater visualization of the phase angle.
In the end, the typical station engineer may only limited contact with a consulting engineer, and perhaps even less contact with antenna designers. That being said, the tip of the iceberg covered here should go a long way in demystifying impedance numbers we sometimes tend to spout. That way when the occasion arises you can confidently say, “I got this.”
Ruck is the principal engineer of Jeremy Ruck and Associates, Canton, IL.