Sound is fundamentally about change. Changes in pressure on the eardrum lead to bone movement in the inner ear that triggers changes in neurons that cause the brain to perceive sound. And for pro audio, the initial change in air pressure results from a loudspeaker whose physical movement caused the change in pressure.
This month, let’s dive a little deeper into the idea of change and some of the machinery used to describe and track change. Understanding change is critical to pro audio, because the way in which we can — or cannot — change audio signals defines much of the performance of audio equipment.
To have a simpler way to investigate change, we will use an electronic component called a capacitor as our reference system to talk about change. Capacitors are devices that store energy using electric charge and are contained in practically every piece of pro audio gear, ranging from loudspeakers to power amplifiers to power supplies. We’ll start with the definition of a capacitor.
What is a Capacitor?
Technically, capacitors are devices that store energy in the form of an electric field by separating electrical charges from each other. From the real-world perspective, “electrical charges” mean “electrons,” and the electric field is our way of describing how electrons separated by a distance physically repel each other. Capacitors have been known for a long time, as illustrated by the apocryphal story of Benjamin Franklin flying kites in a rainstorm.
A simple capacitor is two conducting plates parallel to each other separated by an empty space in between. Because of how the universe is built, an electric field can pass between the two plates. The physical electrons, however, cannot cross the empty space from one plate to the other, but instead must move through an external conductor between the plates. For readers who have worked on tube guitar amplifiers, a “bleed resistor” is an example of an external conducting path. Thus, electrons on one plate can “feel” the electrons on the other plate, but only due to the electric field crossing the empty space.
Now imagine that we use an external voltage attached to the capacitor to “push” electrons onto one of the plates of our capacitor. The electric field emitted from that plate is a result of adding up the individual electric field of all of the electrons on that plate. As the electrons build up the first plate, the electric field grows between the two separated plates of the capacitor.
Because electrons naturally repel each other, the electrons already on the first plate push back as we try to add more by pushing them with voltage. It is more difficult, in terms of the “push,” to move each additional electron onto the plate than the one before. To increase (i.e., change) the charge on the capacitor becomes ever more difficult. In other words, the energy required to add the next electron to the plate is proportional to how many electrons are already on the plate.
All These Words!
Before we continue further with the capacitor, let us take a breather from the written descriptions. Depending on your wiring, the above paragraphs about a parallel plate capacitor may be helpful, confusing or somewhere in between. If you are confused, the key takeaway should be that voltage can move electrons onto one plate or another inside a capacitor, and the electric field can travel between the plates.
This is a case where mathematics can be helpful, because math acts as a shorter and often clearer language to describe what is happening. Hundreds of years ago, two gentlemen named Isaac Newton and Gottfried Leibniz developed some mathematics to help describe circumstances that involve change, and we will apply a little bit of that math to help clear up the description of our parallel plate capacitor.
Equations Simplify (Really!)
The underlying equations are quite simple. They use multiplication, division, and basic algebra and provide a short way to describe what is happening with the capacitor. The key difference in the equations below to what may be familiar is that sometimes there are new symbols that we will use to track quantities that change. We begin this bit of math by defining voltage.
Voltage (V) is defined as the strength of the electric field divided by distance from the charge. Voltage is a more convenient quantity than using the electric field directly. Because the distance between the plates of our capacitor is fixed, it is easy to think of the field between the plates in terms of voltage. The voltage between the plates is proportional to the number of charges. We call the number of charges “Q”. As algebra, this is written as Equation 1:
Equation #1:
V = m × Q
There is nothing magical about the letters “Q” or “m”, they are just place holders. Q is used for the number of charges, and m tells us how many charges are added for a given value of V.
Next, we have to define current (I). Current is equal to the rate of electron movement. A rate tracks something that changes with time. That is to say that current defines how many electrons pass a point every second, like counting electrons crossing an imaginary finish line. Current is the rate electrons per second. We could call current “I,” but we can also write it using a new symbol (dQ/dt) that clearly indicates that it is a rate. Q is the same as in Equation 1; it represents the number of electrons:
Equation #2:
I = dQ/dt
In English: “Current (I) equals the rate of flow of charge (dQ) per time (dt).” We can also re-write Equation 1 from the standpoint of rates, using dQ/dt in place of Q, and dV/dt in place of V:
Equation #3:
dV/dt = m × dQ/dt
In English: “The change in voltage (dV) versus time (dt) equals m times the change in charge (dQ) versus time (dt).” Equation 3 is what is known as a differential equation. It expresses the rate of change for voltage (dV/dt) in terms of the rate of change of charge (dQ/dt).
Now look up at Equation 2 and see that it is similar to the right side of Equation 3. We can therefore substitute for dQ/dt using algebra and get:
Equation #4:
dV/dt = m × 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 increasing on the capacitor plate is proportional to the rate that charge is being added to the plate.” This is a different way of wording our previous conclusion about capacitors, which stated “The energy required to add the next electron to the plate is proportional to how many electrons are already on the plate.”
About That Change
We have shown, using words and an equation, that the voltage between the plates of the capacitor is a function of the number of charges on the plate, or the rate at which we add charges to the plate. Now if we remove the external voltage pushing electrons onto the plate, 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, storing energy waiting to drive electron movement.
The energy storing electric field has an interesting effect on electrical signals that oscillate. An oscillating signal 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. As an example, the moving in and out of a microphone’s diaphragm is a repeating signal with both positive and negative voltages.
Returning to the electric field in the capacitor, we can think about how an oscillating signal is going to be influenced by the building up of electrons on the plate. Specifically, the rate of oscillation will influence how strongly a capacitor opposes the flow of electrons onto the plate.
First, imagine a signal that changes from positive to negative voltage very rapidly. We 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 then sucked right back off again by the oscillating voltage. Because the time period involved is short, there is little time for electrons to build up and the electric field on the capacitor therefore provides less opposing voltage.
By contrast, if the oscillating signal changes from positive to negative slowly, electrons pile up for a long time on the capacitor plate and therefore the electric field on the capacitor provides strong opposition to the external voltage that is trying to cram in more electrons. A rapidly oscillating signal has a high frequency, and slowly oscillating signal a low frequency. Capacitors therefore provide increasing opposition to low frequency signals. Capacitors let high frequencies through while progressively blocking low frequencies. The reader may recognize this as a high-pass filter, which is a very useful tool in professional audio. Equalizers, crossovers, tone controls, etc. that contain high-pass filters can trace their start to this electric field effect.
Look Mom, More Math
Returning to the equation approach we can also show the blocking effect on low frequencies using the clever math of Newton and Leibniz. First, we need to define a simple oscillating signal. One convenient way to do this is with the sine function. Functions are equations that return exactly one number back from other number inputs:
Equation 5:
f(x, t) = A × sin(x × t)
In English: “The function of the oscillating signal in time (t) equals the sine (sin) of t times the frequency (x), and then times the signal’s amplitude (A).”
Now we will define the voltage (V) in equation 4 in terms of equation 5. All we have done is replace dV/dt (i.e. “voltage is somehow changing with time”) with the idea that voltage is changing as a sine wave (i.e. f(x, t) from equation 5):
Equation 6:
d( V(x, t) )/dt = m × I
In English: “The rate of change of a voltage (V) that changes as a sine function at frequency (x) is proportional to the current (I).”
Now introducing the clever math, also called calculus. Using details that we won’t go into here, we can solve equation 6 for the current (I) given that voltage changes as a sine ( V(x, t) ). The solution of I ends up being:
Equation 7:
I(x, t) = A × x × cos(x × t)
In English: “The rate of change of current (I), as a function of time (t) and frequency (x), is equal to the amplitude (A) times frequency (x) times the cosine (cos(x × t) ).” The cosine is an oscillating function similar to the sine function.
There are two takeaways from Equation 7. Firstly, the current I(x, t) is small when frequency is low because A × x is small. Alternatively the current is large when frequency is high (i.e. A × x is large). Equation 7 is a mathematical representation of the high pass filter effect of capacitors that we unpacked above. Capacitors block current at low frequencies (i.e., when A × x is small), but allow current at high frequencies (i.e., when
A × x is large).
The second takeaway is one that our word picture above did not describe. The capacitor has a specific phase
relationship between voltage (V(x, t)) and current ( I(x, t) ). This is a consequence of the cosine term in Equation 7. Sine and cosine both step through the same range of values, between -1 and 1 but they have a phase shift between them. This is illustrated in Fig. 1, which shows how sine and cosine have the same shape, but are displaced by a specific amount (i.e., the phase shift).
Conclusion
Capacitors store up energy in the form of the electric field between the plates. That field grows as more electrons pile up on the plate. At lower frequencies, there is more time for electrons to pile up, so the opposition to adding each additional electron is greater at low frequencies than at high frequencies. The net result is that capacitors act as a high-pass filter. We can mathematically describe the same effect using a differential equation and a bit of math from the 17th century. Additionally, we learned about a capacitor’s phase relationship between voltage and current.
The capacitor makes a good model system for unpacking the mathematics of change, as it behaves in a clear manner with a comparatively simple solution. The world is full of many other physical systems that can be described by differential equations, but many of those differential equations are very hard, or impossible, to solve directly in the manner above. When people talk about “finite element,” “FEA” or “numerical analysis,” they are usually talking about techniques to calculate approximate solutions to differential equations that don’t produce a result like Equation 7 above.
While it is doubtful this article will cause most FRONT of HOUSE readers running out to purchase a calculus textbook, it is my hope that we have opened a small window on calculus and what it is used for. Newton and Leibniz were interested in taking word-based descriptions of physical systems and replacing intuition with clear mathematical description. Their cleverness then developed the mathematical machinery to solve equations based on those mathematical descriptions. In the modern world, we use computers to further attack problems that could be described by the language of differential equations, but weren’t readily solvable in the days of pen and paper.
