Estimated reading time: 12 minutes

Resistance, Reactance and Impedance, Part 1

This is Part 1 of a three-part primer on resistance, reactance and impedance. Most of us are familiar with resistance - at least we think we are. But few of us really understand reactance and its relationship to resistance. Yes, Part 2 will get a lot more interesting! And while most of us sort of understand what impedance is, few really understand the relationship between it and the other two properties. This three-part series will tie them all together.

Overview

In the world of electronics, there are only three types of passive components: Resistors, capacitors, and inductors. That's all. (Purists would add things like fuses and piezoelectric crystals, but these are either trivial or too special to include here. Others might add transformers and related items, but they are, fundamentally, inductors and rely on the principles of inductance for their operation.) Generally, the difference between a passive and an active component is that a passive component can't amplify or fundamentally change the characteristic of a signal. And passive components are symmetrical in that they have no polarity. (A capacitor may have a polarity for fabrication reasons, and an inductor for possible EMI field polarity reasons, but these do not change the fundamental nature of the component.)

These three passive components "impede" current in some way. When a resistor impedes current, we call it resistance. When a capacitor or inductor impedes current we call it reactance. When we have resistance and reactance in combination we have impedance. Impedance is the general term that encompasses both resistance and reactance, but it is not wrong to use the term impedance when referring to pure resistance or pure reactance.

The resistance of a resistor is not frequency-dependent, but the reactance of a capacitor or an inductor is (Note 1). Reactance results in a phase shift between voltage and current (see Part 2), but resistance does not. We will see that the phase shift caused by reactance is either +90 or -90 degrees, nothing else. The only way to get a phase shift other than 0, +90, or -90 degrees is to use resistance and reactance in combination (coming up in Part 3).

Finally, since the phase shift associated with resistance is exactly zero (which is exactly halfway between that caused by a capacitor and that caused by an inductor), and since resistance is not frequency-dependent while the reactance of a capacitor or an inductor is, it is not hard to imagine that resistance might be a very special case of the more general property "impedance."

In Parts 1 and 2 of this series, we will be treating the components as if they are pure. Of course, in the practical world, components are not pure. For example, a capacitor has series resistance (ESR) and some series inductance contributed by the device itself and its mounting to a PCB. Similarly, an inductor may have series resistance. But we can treat these complexities as if there were a series connection of pure components. For example, a real-world capacitor can be treated as a series connection of a pure capacitance in series with a pure resistance and a pure inductor. In that case we have - impedance (Part 3)!

Part 1: RESISTANCE

Resistivity

All materials have a property called resistance (more properly called resistivity.) Low resistivity means low resistance to current flow. High resistivity means high resistance to current flow. Therefore, metals with low resistivity make good conductors; materials with high resistivity make good insulators. Gold, silver, and copper are very desirable conductors because they are solids at normal temperatures and have very low resistivity. See Table 1.

Table 1. Resistivity of selected materials.

Resistors are made from a variety of materials using a variety of techniques. For a good discussion of various types of resistors see: http://en.wikipedia.org/wiki/Resistor

Units of Resistivity

Units of resistivity are ohms-length, typically ohm-m.  If we divide the resistivity of a conductor by the cross-sectional area of the conductor, the units become:

ohms-length/area = ohms/length

or ohms per unit length. Now if we multiply this by the length of the conductor we get: (ohms/length) x length = ohms.

So resistivity is a property, when divided by the cross-sectional area, gives us a measure of ohms per unit length, and when that is multiplied by the length of the conductor we get the resistance, in ohms, of the conductor. In short, resistance is inversely proportional to cross-sectional area and directly proportional to the length of a conductor.

Resistivity is temperature-dependent and typically increases with temperature (there are a couple of exceptions). The increase depends greatly on the material, but numbers on the order of 3% for each 10 degree C rise in temperature are typical. This is why resistivity values are given at a specific temperature - 20 degree C in the case of Table 1. This is why the current/temperature curves found in sources like IPC-2152 (see Note 2) are non-linear. As current increases, the temperature of the trace increases. But as the temperature of the trace increases, so does the resistance. The increased resistance causes additional thermal heating because of the power dissipated in the higher resistance.

Nature of Current

Before we can discuss the electrical properties of resistance, we need to have a basic understanding of current. Current is THE fundamental aspect of electricity. All other aspects of electricity relate to the basic concept of current. Current is the flow of electrons (charge) (Note 3). One coulomb of charge is 6.25 x 1018 electrons. The fundamental measure of current is coulomb/time. One amp of current is defined as one coulomb of charge passing across a plane (think cross sectional area of a conductor) in one second.

There are three fundamental laws of current that all electrical engineers learn very early in their education. The first is that current flows in a closed loop and is constant everywhere in that loop. One basic way that engineers can analyze a circuit is to identify the individual loops of current and then work with those loops individually. The total performance, then, is the sum of the parts. Conceptually this sounds pretty easy. But since a given component might be a part of two or more loops, this isn't as easy as it first appears on the surface.

The other two fundamental laws are Kirchhoff's First and Second Laws. Kirchhoff's First Law states that all current that flows into a node (think net in PCB terms) must flow out of the same node. If that were not true, then how would we account for the flowing charge? For example, if more current flowed out of a node than flowed into it, the node must somehow be generating charge (electrons). Intuitively we understand that that doesn't happen.

Kirchhoff's Second Law states that the voltage around a loop sums to zero. If, for example, we have a battery connected across a circuit, the voltage added up around the circuit loop must equal the voltage of the battery. There can be no unaccounted-for charge.

If current is the flow of electrons, resistance impedes that flow. Consider two metal spheres separated by a distance. Let the charge on one sphere be one coulomb and the charge on the other be two coulombs.

Since the charge difference between them is one coulomb, we say that the voltage difference between them is one volt. If we connected the two spheres with an ideal (zero resistance) conductor, the charge would instantly equalize to 1.5 coulombs on each sphere (0.5 coulombs moved from one sphere to the other) and the voltage difference between them would be zero.

Instead, place a resistor between the two spheres. Charge would flow between the two spheres, but the length of time this takes would depend on the resistance. If the resistance were very low, the charge would move fairly quickly. But it the resistance were very large, it would take much longer for the charge to transfer between the spheres. (See note 4)

The definition of resistance follows from this illustration. One ohm of resistance is equal to the resistance that causes one volt of voltage to appear across it when one amp of current flows through it.

Ohm's Law: Perhaps the most famous and well-known law in electronics is Ohm's Law. It is the law that states the relationship between voltage, current, and resistance: Voltage equals current times resistance:

v = i * r   and its variations    i = v/r   and   r = v/i

For example, if we have a resistance of 1,000 ohms (1.0 Kohm) with a current flowing through it measuring 5.0 ma (5 x 10-3 amp), the voltage across the resistor would be

v = i  x  r = 5 x (10-3)  x  1.0 x (103) = 5.0 volts.

Note that this relationship does not depend on frequency (something that will not be true for inductance and capacitance). If we graph the relationship (Figure 1) it is a straight line, i.e., it is a linear relationship.

Figure 1. Graph of Ohm's Law for a resistance of 1,000 ohms, including the point in the illustration above.

Series and Parallel Resistor Combinations

Before you write off this section as trivial, something you already understand, wait! That is not the point. The point is that serial and parallel combinations of capacitors are not like those of resistors. So how do we determine, in a general sense, how various combinations of impedances combine together? With Kirchhoff's Laws.

Consider the case shown in Figure 2 (a) and 2 (b). The first case shows a parallel combination of two resistors, the second case shows a single resistor (Req) carrying the same current at the same voltage. Req is the resistor that is equivalent to the two other resistors combined in parallel. So how do we solve for Req?

Figure 2. We determine how impedances combine in parallel using Kirchhoff's First Law

In Figure 2(a) we apply Kirchhoff's First Law to note that the current into the node (i) must equal the current out of the node (i1 + i2), or

i = i1 + i2

Then, using Ohm's Law, we note that since i = V/R

i = V/R1 + V/R2 = V/Req

Therefore:

V/R1 + V/R2 = V/Req

1/R1 + 1/R2 = 1/Req

Or, finally

Equation 1

This expression can be generalized for any number, n, of resistors connected in parallel:

Equation 2

For a series combination of resistors, refer to Figure 3. The first case shows two resistors in series. The second shows a single resistor (Req) carrying the same current at the same voltage. Req is the resistor that is equivalent to the two other resistors combined in series. So how do we solve for Req?

Figure 3. We determine how impedances combine in series using Kirchhoff's Second Law.

In a manner similar to the above, we apply Kirchhoff's (this time) Second Law, the one that states that voltages around a loop sum to zero (See note 6). Again, from Ohm's Law, we have:

V = V1 + V2 = i*R1 + i*R2 = i*Req

which reduces to the expected

Req = R1 + R2                                                                                   Equation 3

This can be generalized to any number of resistors, n, in series:

Req = R1 + R2 + R3 + R4 + ... + Rn                                                Equation 4

Before you consider this to be a trivial result, wait until we apply this concept to series capacitors!

Power

Power is defined as voltage times current, or Power = V*I. Alternative expressions of this, based on Ohm's Law, are:

Power = V*I = V2/R = I2 *R                                                    Equation 5

We frequently talk about the power losses in a wire or a conductor as the i2R loss (pronounced i-squared-R). This is the predominant heating effect in traces on a PCB. Even though the overall resistance of the conductor on a board is very small, if we push enough current down the trace then the i2R loss can be significant and cause undesirable heating of the trace (See again Brooks, Note 2).

It is important to notice the square operator in the formulas. Power goes up with the square of the voltage or the current. Thus a 100 ohm half-watt resistor can carry 0.07 amps (at 7 volts) while a one-watt resistor (twice as large) can carry only 0.1 amp at 10 volts (1.414 times as much)!

Of the three passive components in electronics, only resistance causes the generation of power, and therefore power losses. Note the expression includes "R" in the formula. Pure reactance does not dissipate power, as we shall see in Part 2.

Summary

This has been a review for many of us. But it provides the background for looking at very similar relationships for reactance and then impedance in Parts 2 and 3 of this series. I think you will be surprised at some of the similarities and differences that are coming.

Notes:

• 1. Skin effect is not an exception here. Skin effect is a function of the physical structure of the conductor, not of the general property of resistance. See Brooks' article, "Skin Effect" available at www.ultracad.com.
• 2. Refer to the IPC's latest standard, IPC-2152 "Standard for Determining Current Carrying Capacity in Printed Board Designs." Also Brooks' articles on current/temperature effects at www.ultracad.com/article_temperature.htm.
• 3. See Brooks' article on current, "What is This Thing Called ‘Current:' Electrons, Displacement, Light, or What?" available at www.ultracad.com/article_other.htm.  See also http://en.wikipedia.org/wiki/Electric_current.
• 4. For an analogy of this effect, consider a capacitor with a large charge on it. The charge slowly dissipates from one plate to the other through the leakage resistance inherent in the capacitor.
• 5. Some people prefer the form (R1*R2)/(R1+R2). I prefer the form in the body because (1) it is easier to remember and (2) it is easily generalizable to any number of resistors in parallel.
• 6. Students sometimes have a problem with this application of Kirchhoff's Second Law. Consider Figure 3(a). Starting at the top and adding voltages in a clockwise fashion we have V1 + V2 + (-V) = 0. We have (-V) (with a minus sign) because we are approaching V from the bottom side (the (-) side) as we go around the loop. This results in the expected V1 + V2 = V.

Douglas Brooks has an MS/EE from Stanford University and a PhD from the University of Washington. He has spent most of his career in the electronics industry in positions of engineering, marketing, general management, and as CEO of several companies. He has owned UltraCAD Design Inc. since 1992. He is the author of numerous articles in several disciplines, and has written articles and given seminars all over the world on Signal Integrity issues since founding UltraCAD. His book, Printed Circuit Board Design and Signal Integrity Issues was published by Prentice Hall in 2003. Visit his Web site at www.ultracad.com.