Last month, we looked at conductivity and electricity, the lifeblood behind our industry. Without electricity, and conductors to direct it, there would be no professional audio. We discussed what defines electrons, electric and magnetic fields, and how electrons moving inside of conducting materials give rise to resistance and the famous Ohm’s law equation, V = IR. That article was focused on the internal effects of electrons inside the crystal lattice of a conductor. This month’s article now turns to the realm of how the electric and magnetic fields characteristic of electrons influence several other aspects of the conductors we use in professional audio.
Now, let’s examine how fields store energy, and walk through the physical behavior of electrons in a capacitor to make this more tangible. Then we will talk about how fields affect the behavior of oscillating signals (e.g., music) and introduce the concept of impedance. In addition to this written description, we are including a sidebar that walks through some of the math that can be used to describe the physical effects discussed in this article.
Fields Store Energy
Electrons contained in specific materials are the transit roadway for electrical effects. As we discussed last month, the path the electrons take through the conductor, bumping into atoms as they go, gives rise to V = IR. But in addition to resistance (the “R” in V = IR), the two fields — electric and magnetic — of electrons traveling in a conductor give rise to other important effects, namely capacitance and inductance.
The concepts of capacitance and inductance are really about how fields store and release energy. Capacitance involves the electric field, and inductance the magnetic field. Unlike resistance, inductance and capacitance are not “lossy” phenomena. Resistance is lossy, in that as the electrons bump their way through the conductor, they give up some of their energy to the conductor’s atoms as heat. By contrast, the electric and magnetic fields don’t dissipate energy, but rather store it by the way in which their force acts to arrange the electrons. This effect is easier to understand physically for capacitance, so we will use that example.
What is Capacitance?
The fundamental properties of capacitance are revealed by a device called a capacitor. The simplest capacitor is two conducting plates parallel to each other with empty space in between. The electric field can pass between the two plates, as the electric field can move through empty space. The physical electrons, however, cannot transition from one plate to the other. Thus an electron can “feel” the electrons on the other plate, but only due to the electric field crossing the gap. The electron cannot physically travel to the other plate, except by moving through an external conducting circuit.
Imagine that we use an external voltage attached to the conducting circuit to push electrons onto one of the plates of our capacitor. As the electrons build up on the plate, the greater the electric field that exists between the plates. Because electrons naturally repel each other, the electrons on the plate push back as we try to add more. Therefore it is more difficult, in terms of energy expended, to move each additional electron onto the plate than it was the last one. In other words, the energy required to add the next electron to the plate is proportional to how many that are on the plate already.
Now voltage is merely the electric field intensity divided by distance. Since the electric field varies by the distance from an electron, voltage is a more convenient quantity than using electric field intensity directly. You can think of voltage as the electric field intensity normalized for distance.
As mentioned above, the voltage between the plates of the capacitor is a function of the number of charges on the plate. An external force (i.e., voltage) pushes electrons on to the capacitor plate, and that therefore raises the value of the voltage (or electric field) between the plates. If you remove the external voltage, the electrons will rapidly leave the plate, because the electric field between the plates is trying to push the electrons away from each other. The electric field between the plates is a caged beast, waiting to drive electron movement, and therefore stores energy.
Anyone who has ever worked inside of amplifiers (especially tube amps) has experienced the great potential for danger with “charged” capacitors. The power supply capacitors are “holding” a charge between their plates, and if a conductive path is given, those electrons move rapidly! We typically use a “bleeder” resistor, which slows down how quickly the current leaves the capacitor, to “drain” its charge safely. The bleeder resistor provides a valve to regulate how fast the electric field between the plates can push the electrons apart.
Fields Affect Oscillating Signals
Now that we have provided a physical explanation for capacitance, it should be mentioned that there is an analogous behavior for inductance, where energy is stored by a component called an inductor because of the magnetic field. In a perfect inductor or capacitor, there is no dissipation of energy, only storage. Real-world inductors and capacitors have some resistance. This is because electrons are still bumping into the atoms that make up these elements. In electric circuit diagrams, it’s common to show the capacitor or inductor along with a small resistor that represents this real-world resistance.
Despite the fact that electric and magnetic fields are purely storing energy, they have interesting effects on electrical signals that oscillate. An oscillating signal, for pro audio, can be thought of as a repeating voltage that is positive part of the time, and negative part of the time. Positive voltages move electrons one direction, and negative voltages move electrons the opposite direction. Whether playback of music from an MP3 player, or the moving in and out of a microphone’s diaphragm, repeating signals with both positive and negative voltages are all over pro audio.
Returning again to the capacitor, we can think about how an oscillating signal is going to be influenced by the building up of electrons on the plate. The first thing to realize is that, because there is no internal electron path between the two capacitor plates, capacitors can be used to block the continued flow of electrons. We call the rate electrons flow due to voltage “current,” and say that capacitors block direct current (DC), or the continued current flow in one direction. Capacitors are routinely utilized inside electronics to separate DC current flow from oscillating signals, like audio.
Thinking about the capacitor a little more, it becomes apparent they will have another effect on oscillating signals. Specifically, the rate of oscillation will influence how strongly a capacitor opposes the flow of electrons. To pick this apart, let’s first imagine a signal that changes from positive to negative voltage very rapidly. We already established above that adding each new electron to the plate of a capacitor is slightly more difficult than the last. For a signal that changes rapidly from positive to negative voltage, electrons are being added to the capacitor plate for a brief period of time, and the sucked right back off again by the oscillating voltage. Because the time period involved is short, not that many electrons pile up, and the electric field built up on the capacitor therefore provides little opposition back towards the external oscillating voltage.
Reactance and Impedance
By contrast, if the oscillating signal changes from positive to negative slowly, there exists a long time for electrons to pile up on the capacitor plate, and therefore the electric field between the plates provides a very strong opposition to the external voltage that is trying to bring in more electrons. A rapidly oscillating signal has a high frequency, and slowly oscillating signal a low frequency. Therefore, capacitors provide increasing opposition to low frequency signals. We call this frequency-dependent opposition reactance, and reactance combined with resistance are the commonly used term “impedance.” Impedance is the sum of the electrical storage effects (capacitance and inductance) and the heat producing effect (resistance).
The key take-away from the capacitor’s increasing opposition to low frequency signals is that it results from the electric field that builds up between the capacitor plates opposing the external voltage. We already established that the electric field does not dissipate energy, so this means that the opposition (i.e., reactance) is not dissipating energy. Only resistance dissipates energy. Thus, the low frequency signal sent into the capacitor is not being converted into more heat than the high frequency signal. Instead, the low frequency signal builds up a greater field internal to the capacitor, and that field produces ever-increasing opposition to the outside voltage.
There is an analogous reactance effect for inductors, but with a slightly different twist. While capacitors provide increased opposition to low frequencies, inductors provide increased opposition to high frequencies. With these two reactive components having similar, but opposite, effects on frequency we now have the starting palette to fill the pro audio world with all of the filters that we use regularly. Parametrics, crossovers, tone controls, etc. all trace their genesis to reactance, impedance and the field effects we’ve discussed here. Even the digital implementations are often designed to behave like simple analog circuits composed of inductors, capacitors, and resistors. The shaping of music really starts with electricity, and the fields electrons create!
Capacitors In Depth: The Math Behind the Words
Underlying our description of the behavior of capacitors that we see in the main article is some mathematics. Involved in this math is some basic calculus. Calculus is merely the math of movement and change. It was developed by Newton and Leibniz to help them describe how physical objects moved and changed, usually with time.
In electricity, we deal with moving electrons, so this math of movement finds another interesting application. Even though some of the symbols below may seem to be new, the underlying equations are quite simple. They use multiplication, division, and basic algebra. The only difference is that sometimes we are using descriptions of movement, or changes in quantity, rather than fixed values.
Equation 1: The voltage (V) between the plates is a function of the total number of charges and the number of charges on the plate. We’ll call that “Q”. As algebra, this sentence is written:
V = m x Q
There’s nothing magical about the letters “Q” or “m,” they are just convenient placeholders, with Q for the number of charges, and m to tell us how many charges are added based on V.
Equation 2: Next we have to define current (I). Current is equal to the rate of electron movement. That is to say that current tells how many electrons pass a point every second. This is electrons/second. Because this a rate, we can write it in calculus terms:
I = dQ/dt
Or in English as: “Current (I) equals the rate of flow of charge (dQ) per time (dt).”
Equation 3: We can also write Equation 1 from the rate of change standpoint, again using calculus notation:
dV/dt = m x dQ/dt
In English: “The change in voltage (dV) versus time (dt) equals m times the change in charge (dQ) versus time (dt).”
Equation 4: Now look at Equation 2 and see that it is similar to the right side of Equation 3. We can therefore substitute in using algebra and get:
dV/dt = m x I
In English: “The rate of change of voltage (dV) versus time (dt) equals m times current (I).”
Equation 4 is a math way of saying: “the rate that the voltage is being built up between the capacitor plates is proportional to the rate that charge is being added to the plate.” This is exactly one of the written conclusions from the article.
Equation 5: Now the magic of calculus can come into play. Let’s imagine we have a simple oscillating signal defined by a function:
f(X, t) = A x sin(X x t)
In English: “The function of the signal in time (t) equals the sine of (t) times the frequency (X), and then times the signal’s
amplitude (A)”
Equation 6: If we plug in f(t) for the voltage (V) in Equation 4, we get:
d(f(X, t))/dt = m x I
In English: “The rate of change of a periodic voltage (V) function of frequency (X) is proportional to the current.”
Equation 7: Using some calculus details that we won’t go into here, we can solve Equation 6 for I. The solution of I ends up being:
I(X, t) = A x X x (cos(X x t))
There are two key takeaways from Equation 7. First is that the capacitor changes the phase. This is demonstrated by the cosine term. Sine and cosine both step through the same range of values, but they have a phase shift between them.
The second takeaway is that the current is small when frequency is low, or A x X is small. Alternatively, the current is large when frequency is high (i.e., A x X is large). This is a mathematical representation of the filter effect we see from capacitors! They block current at low frequencies (i.e. when A x X is small), but allow current at high frequencies (i.e., when A x X is large).
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Capacitance in the Real World
Capacitors are among the most-used components in any electronic design, but have particular significance in pro audio applications. The most basic is a simple filter, such as a passive speaker crossover, where a single non-polar capacitor in-line with a tweeter can create a first-order, -6 dB/octave slope high-pass filter (as shown here) that blocks low frequencies from reaching the tweeter and the impedance of the speaker and value of the capacitor determines the roll-off frequency. For example, in-line with an 8-ohm tweeter, a 10-microFarad capacitor creates a roll-off that’s approximately of 2 kHz; a 20-microFarad cap creates a filter in the 1 kHz range. Conversely, adding a single inductor (coil) in-line with the woofer results in a first-order low-pass filter. More complex (second/third/fourth-order) passive crossovers with slopes up to -24 dB/octave can be created with various combinations of multiple capacitors and inductors. —George Petersen
