This page is a section of Ultimate Electronics: Practical Circuit Design and Analysis, a free, online, interactive electronics textbook by Michael F. Robbins.

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Thermodynamics, Energy & Equilibrium

The laws of thermodynamics are some of the most important and wide-reaching physical laws known to science.

In the context of electronics, the most important is the First Law of Thermodynamics: energy is neither created nor destroyed, but it can be transformed between forms of energy. This is sometimes known as conservation of energy.

As we'll discuss further in Chapter 2, conservation of energy is embedded into Kirchhoff's Voltage Law and Kirchhoff's Current Law, where the structure of the equations essentially guarantees conservation of energy (see Power). We'll also pick up this topic again when we discuss the potential energy interpretation of voltage, relating to the electric field and electrical forces.


Forms of Energy

Various forms of energy include:

  • Electric field energy (like in a capacitor)
  • Magnetic field energy (like in an inductor)
  • Chemical potential energy (like in a battery)
  • Linear and/or rotational kinetic energy (like in a moving motor)
  • Gravitational potential energy (like in a pendulum or swing)
  • Elastic potential energy (like in a spring)
  • Vibrational kinetic energy (like in sound waves in the air)
  • Photon or electromagnetic wave energy (like in light and radio waves)
  • Heat (like that dissipated by a resistor)

All of these forms of energy can be measured in the same units of Joules (J). In mechanical systems, energy is a force times a distance. A Joule is equal to one Newton-meter:

$$E \ \text{[J]} = F \ \text{[N]} \cdot x \ \text{[m]}$$

In electronics, as we'll discuss later in the Power section, a Joule is equal to one Volt-Amp-second, or one Volt-Coulomb:

$$ \begin{align} E \ \text{[J]} & = v \ \text{[V]} \cdot q \ \text{[C]} \\ E \ \text{[J]} & = v \ \text{[V]} \cdot i \ \text{[A]} \cdot t \ \text{[s]} \\ 1 \ \text{Joule} & = 1 \ \text {Volt} \cdot \text{Amp} \cdot \text{second} \\ 1 \ \text{J} & = 1 \ \text {V} \cdot \text{A} \cdot \text{s} \end{align} $$

The last of these forms of energy, heat, is a particularly special case.


Heat and the Second Law

Heat is a special kind of energy because it is a low quality form of energy, because it is not readily transformed into other forms. In particular, due to the Second Law of Thermodynamics, total entropy (a way of measuring unusable waste heat) is always increasing.

Whenever we convert energy between any two different forms, such as from chemical potential energy to kinetic energy, we never do so at 100% efficiency. There are always losses. Those losses result in heat generation.

$$ \begin{align} \eta & = \frac {E_{out}} {E_{in}} \\ \eta & \le 1 \\ E_{in} & = E_{out} + Q_{waste} \end{align} $$

Caution: these rules, when applied to heat engines (such as combustion engines, refrigerators, air conditioners, heat pumps, and others), where heat itself makes up a part of $E_{in} \ \text{or} \ E_{out}$, require some special attention and a slightly different approach. These cases demand a more detailed study of thermodynamics, which is beyond the scope of this book.


The First Law of Thermodynamics in Electronics Design

The First Law of Thermodynamics is a useful tool in practical electronics and higher level system design.

Very often we are interested in using an electronic system as a tool to transform one form of energy to another:

  • Flashlight: chemical (to electrical) to light
  • Electric vehicle: chemical (to electrical) to kinetic
  • Radio receiver: electromagnetic wave (to electrical) to kinetic sound
  • Battery charger: electrical to chemical potential
  • Diesel generator: chemical to heat to kinetic to electrical

First of all, we care about how efficiently we do the energy conversion:

$$\eta = \frac {\text{Useful work done}} {\text{Energy from source}}$$

Second, we care about the amount of waste heat we generate. As electronics designers, we have to deal with removing that waste heat somehow so our components don't overheat and fail!

For example, if we're designing an LED flashlight and we know our LED circuit has an electrical-to-optical efficiency of 30% and we've designed it to consume 1 Watt of electrical power, we know two things:

  1. The flashlight will emit 0.3 Watts of light.
  2. The flashlight will dissipate 0.7 Watts of heat.

Both of these values may or may not be within the bounds of our desired design, but should guide us in terms of how to approach our design:

  • Do we need to find a more efficient LED circuit?
  • How long will the battery last?
  • How hot will the case get?

Even before we approach the design details, this high-level overview of the system from the thermodynamic perspective helps us quickly approximate various system design choices.

Note that for purely computational work, such as using digital logic gates to do arithmetic inside a CPU or GPU, we often don't have a measurement of "useful work done" as Joules, even if our computation is useful. Work here is specified in the thermodynamic sense.


Equilibrium

Intuitively, lots of systems try to reach their lowest energy state. For example, consider a ball that rolls downhill until it finds a minima to stay in:

  1. First, its gravitational potential energy is converted into kinetic energy.
  2. Then, that kinetic energy is gradually dissipated away to the environment as heat through friction.
  3. Eventually, the ball is left motionless in a lower potential energy state -- the remainder was dissipated as heat via friction.

The ball will only start rolling down the hill if it isn't already in a local minima, such as a little divot or a sticky or frictional patch. Sometimes, a small addition of energy is needed to overcome this local energy minima and literally get the ball rolling so that it can eventually reach an even lower energy minima state.

Similarly, two higher-energy chemical reactants will (sometimes) react to produce one or more lower-total-energy chemical products, usually with waste heat generated. The "sometimes" is due to activation energy.

For electrical systems, the analogy may be a little harder to see, but the same inutition applies:

  • A battery left alone will tend to self-discharge.
  • A capacitor will try to discharge itself to a lower-energy state, either through internal or external resistance.
  • The excited electrons within an LED's p-n junction will try to fall back down to their resting state, emitting a visible photon in the process.

The rate at which these actions happen depends the balance of many things, but energy balance can be useful problem solving tool on its own.


Energy Balance for Problem Solving

For example, we'll later study RC circuits and see how a capacitor discharging through a resistor produces an exponentially decaying voltage signal.

But, without even considering the differential equation of the RC circuit, we can instead take an energy-based approach. In this simple system with just one resistor and one capacitor, we can observe that the rate of potential energy loss in the capacitor is equal to the power dissipation in the resistor.

$$\frac {d} {dt} \big( E_{\text{capacitor}} \big) + P_{\text{resistor}} = 0$$

We'll later learn that the energy stored in a capacitor is:

$$E_{\text{capacitor}} = \frac {1} {2} C V^2 = \frac {1} {2 C} Q^2$$

and that the power dissipation in a resistor is:

$$P_{\text{resistor}} = V I = R I^2$$

In a small time increment $dt$ the capacitor will lose some stored energy as it loses charge $dQ$. That same charge must flow through the resistor with instantaneous current $I = \frac {dQ} {dt}$:

$$ \begin{align} \frac {d} {dt} \big( E_{\text{capacitor}} \big) + P_{\text{resistor}} & = 0 \\ \frac {d} {dt} \big( \frac {1} {2 C} Q^2 \big) + R I^2 & = 0 \\ \frac {1} {C} Q \frac {dQ} {dt} + R I^2 & = 0 \\ \frac {1} {C} Q \frac {dQ} {dt} + R \big( \frac {dQ} {dt} \big)^2 & = 0 \\ R \frac {dQ} {dt} & = - \frac {1} {C} Q \\ \frac {dQ} {dt} & = - \frac {1} {R C} Q \\ \end{align} $$

Or since $Q = C V$, we can substitute:

$$ \begin{align} C \frac {dV} {dt} & = - \frac {1} {R C} (C V) \\ \frac {dV} {dt} & = - \frac {1} {R C} V \end{align} $$

That's the first-order differential equation of exponential decay for an RC circuit. Instead of deriving it from Kirchhoff's Voltage Law and Kirchhoff's Current Law, here we've derived it from thermodynamic energy balance considerations. We have another angle to approach design and analysis problems by recognizing that energy is conserved.


Energy Balance as Systems-Level Design

Thinking about energy conservation and transferring energy between forms is often a great high-level abstraction at a systems design level design strategy.

In fact, since energy is conserved (much like charge is conserved), we can use electric circuits to model and simulate heat flows.

In the Lumped Element Model section we demonstrated how to use circuit elements to model mechanical systems such as a spring-mass-dashpot. Similarly, here's a quick example that shows how to use circuit elements to model thermal systems:

Click the circuit, click "Simulate," and "Run Time-Domain Simulation."

This demonstrates a simplified model of a 1500W electric hotplate heating up an empty skillet for cooking. Thermal resistances (in units of K/W) for conduction, convection, and radiation are represented by electrical resistances (in units of Ohms). Heat capacities (in units of J/K) are represented as capacitors. Heat flows (in Watts) are represented by electrical current (in Amps), and temperatures (in degrees K relative to ambient) are represented by electrical voltages (in Volts relative to ground).

The simulation shows the hotplate element heating up, and then heating the skillet itself. You can plot the hotplate temperature too: it has a much shorter time constant than the skillet. Eventually, the radiation and convection from the skillet to the surroundings balance out the heat addition, and after a few minutes, the pan reaches a steady state temperature.

Similar thermal equivalent circuit modeling can also be used for analyzing the heatsinks and heat dissipation of components like transistors, processors, and resistors within an electronic system. Many manufacturer datasheets include some thermal specifications for maximum operating temperature and for thermal resistance from the semiconductor junction to ambient. Be careful: these numbers have various assumptions included, such as whether there's free airflow. If the component is encased in epoxy, or is in a case that restricts airflow and could get hot inside, then the thermal properties may not match the datasheet!


What's Next

Return to the Table of Contents to continue with chapter 2, where we'll start exploring the simplest electronic systems.

↑ Up: Table of Contents
← Prev: Lumped Element Model
→ Next: Chapter 1 Example Circuits


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