12.2: Electrical Energy Conversion
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Electrical can be described either in circuits language or electromagnetics language. Using circuits language, electrical systems are described by four fundamental parameters: charge in coulombs \(Q\), voltage in volts \(v\), magnetic flux in webers \(\Psi\), and current in amperes \(i\). For circuits described in this language, resistors, capacitors, and other electrical energy storage and conversion devices are treated as point-like with no length or extent, and forces and fields outside the path of the circuit are ignored. An alternative approach is to use electromagnetics language the electrical properties of materials are studied as a function of position and forces and fields outside of the path of a circuit are studied.
We can use circuits language to describe a number of energy conversion devices. Resistors convert electrical energy to thermal energy, and thermoelectric devices convert thermal energy to or from electrical energy. A charging capacitor converts electrical energy to energy stored in a material polarization, and a discharging capacitor converts the energy of the material polarization back to electrical energy. In an inductor, electrical energy is converted to and from energy of a magnetic field.
In Sec. 11.5, energy storage in a capacitor was studied in detail and described in the language of calculus of variations. Table 11.5.2 summarized the use of calculus of variations language to describe the energy conversion process, and it is repeated in the second column of the Table \(\PageIndex{1}\). In that example, charge built up on the capacitor plates, \(Q\), was the generalized path. The generalized potential was \(v\), the voltage across the capacitor. From these choices, other parameters were selected.
Instead of choosing charge \(Q\) as the generalized path, we could have chosen the generalized path to be one of the other fundamental variables of circuit analysis, voltage \(v\), magnetic flux \(\Psi\), or current \(i\). Table \(\PageIndex{1}\) summarizes parameters that result when we describe energy conversion processes occurring in a capacitor or inductor in the language of calculus of variations with these choices of generalized path. More specifically, the third column shows parameters when voltage is chosen as the generalized path. The fourth column shows parameters when magnetic flux is chosen as the generalized path, and the fifth column shows parameters when current is chosen as the generalized path. By reading down a column of the table, we see how to describe a process with this choice of generalized path. By reading across the rows of the table, we can draw analogies between parameters of energy conversion processes.
To describe the energy conversion processes occurring in a capacitor, we can choose either the charge or voltage to be the generalized path then use the language of calculus of variations. Notice that if charge is chosen as the generalized path as seen in column two of Table \(\PageIndex{1}\), voltage becomes the generalized potential. However, when voltage is chosen as the generalized path as seen in column three, charge becomes the generalized potential. The path found in nature minimizes the action, and we saw in Sec. 11.5 that we could use the Euler-Lagrange equation to set up an equation of motion for the system. Each term of the equation of motion has the same units as the generalized potential. The equation of motion found when using \(Q\) as generalized path is Kirchoff's Voltage Law (KVL), which says the sum of all voltage drops around a closed loop in a circuit is zero. The equation of motion found when using \(v\) as the generalized path is the law of conservation of charge. Both of these concepts are fundamental ideas in circuit theory, and they are shown in the second to last row of the table.
| Energy storage device | Capacitor | Capacitor | Inductor | Inductor |
|---|---|---|---|---|
| Generalized Path | Charge \(Q\) in C | Voltage \(v\) in V | Mag. Flux \(\Psi\) in Wb | Current \(i\) in A |
| Generalized Potential | Voltage \(v\) in \(\frac{J}{C}\) = V | Charge \(Q\) in C | Current \(i\) in A = \frac{J}{Wb}\) | Mag. Flux \(\Psi\) in Wb |
| Generalized Capacity | Capacitance \(C\) in F = \(\frac{C^2}{J}\) | \(\frac{1}{C}\) | Inductance \(L\) in H = \(\frac{Wb^2}{J}\) | \(\frac{1}{L}\) |
| Constitutive relationship | \(Q=Cv\) | \(v=\frac{Q}{C}\) | \(\Psi = Li\) | \(i = \frac{\Psi}{L}\) |
| Energy | \(\frac{1}{2}Cv^2\) | \(\frac{1}{2}\frac{Q^2}{C}\) | \(\frac{1}{2}Li^2\) | \(\frac{1}{2}\frac{\Psi^2}{L}\) |
| Law for potential | KVL | Conservation of Charge | KCL | Conservation of Mag. Flux |
| This column assumes | AC current and voltage | AC current and voltage | AC current and voltage | AC current and voltage |
Similarly, to describe the energy conversion processes occurring in an inductor, we may choose either magnetic flux or current as the generalized path. If we choose magnetic flux as the generalized path, the generalized potential is current. If we choose current as the generalized path, the generalized potential is magnetic flux. From the first choice, the equation of motion found is Kirchoff's Current Law (KCL). From the second choice, the equation of motion found is conservation of magnetic flux.
The relationship between the generalized path and the generalized potential is known as the constitutive relationship [168, p. 30]. For a capacitor, it is given as
\[Q = Cv. \nonumber \]
The constant \(C\) that shows up in this equation is the capacitance in farads. Analogously for an inductor, the constitutive relationship is
\[\Psi = Li \nonumber \]
where \(L\) is the inductance in henries. We will see that we can identify constitutive relationships for other energy conversion processes, and we similarly can come up with a parameter describing the ability to store energy in the device. In analogy to the capacitor, we will call this parameter the generalized capacity. Capacitance \(C\) represents the ability to store energy in the device, so generalized capacity represents the ability to store energy in other devices. Overloading of the term capacity is discussed in Appendix C.
| Circuit Quantity | Electromagnetic Field |
|---|---|
| \(Q\): Charge in C | \(\overrightarrow{D}\): Displacement flux density in \(\frac{C}{m^2}\) |
| \(v\): Voltage in V | \(\overrightarrow{E}\): Electric field intensity in \(\frac{V}{m}\) |
| \(\Psi\): Magnetic flux in Wb | \(\overrightarrow{B}\): magnetic flux density in \(\frac{Wb}{m^2}\) |
| \(i\): Current in A | \(\overrightarrow{H}\): Magnetic field intensity in \(\frac{A}{m}\) |
Using electromagnetics language, four vector fields describe systems: \(\overrightarrow{D}\) displacement flux density in \(\frac{C}{m^2}\), \(\overrightarrow{E}\) electric field intensity in \(\frac{V}{m}\), \(\overrightarrow{B}\) magnetic flux density in \(\frac{Wb}{m^2}\), and \(\overrightarrow{H}\) magnetic field intensity in \(\frac{A}{m}\). These electromagnetic fields are generalizations of the circuit parameters charge \(Q\), voltage \(v\), magnetic flux \(\Psi\), and current \(i\) respectively as shown in Table \(\PageIndex{2}\). However, the electromagnetic fields are functions of position \(x\), \(y\), and \(z\) in addition to time, and they are vector instead of scalar quantities. More specifically, displacement flux density is the charge built up on a surface per unit area, and magnetic flux density is the magnetic flux through a surface. Similarly, electric field intensity is the negative gradient of the voltage, and magnetic field intensity is the gradient of the current. We encountered these electromagnetic fields when discussing antennas in Chapter 4.
A capacitor can store energy in the charge built up between the capacitor plates. Analogously, an insulating material with permittivity greater than the permittivity of free space, \(\epsilon > \epsilon_0\), can store energy in the distributed charge separation throughout the material. We can describe the energy conversion processes occurring in a capacitor using the language of calculus of variations by choosing either charge \(Q\) or voltage \(v\) as the generalized path. Parameters resulting from these choices are shown in the second and third column of Table \(\PageIndex{1}\). Analogously, we can describe the energy conversion processes occurring in an insulating material with \(\epsilon > \epsilon_0\) using the language of calculus of variations by choosing either \(\overrightarrow{D}\) or \(\overrightarrow{E}\) as the generalized path. Parameters resulting from these choices are shown in the second and third column of Table \(\PageIndex{3}\). The equation of motion that results in either case is Gauss's law for the electric field,
\[\overrightarrow{\nabla} \cdot \overrightarrow{D} = \rho_{ch} \nonumber \]
where \(\rho_{ch}\) is charge density. The derivation is beyond the scope of this text, however, because it involves applying calculus of variations to quantities with multiple independent and dependent variables. Gauss's law is one of Maxwell's equations, and it was introduced in Section 1.6.1. In Chapter 2, piezoelectric energy conversion devices were discussed, and in Chapter 3, pyroelectric and electro-optic energy conversion devices were discussed. All of these devices involved converting electrical energy to and from energy stored in a material polarization of an insulating material with \(\epsilon > \epsilon_0\). Calculus of variations can be used to describe energy conversion in all of these devices with either displacement flux density or electric field intensity as the generalized path. For a device made from a material of permittivity \(\epsilon\) with an external electric field intensity across it given by \(\overrightarrow{E}\), the energy density stored is \(\frac{1}{2}\epsilon|\overrightarrow{E}|^2\) in \(\frac{J}{m^3}\). The energy stored in a volume \(\mathbb{V}\) is found by integrating this energy density with respect to volume, and this energy stored in a volume is listed in the second to last row of Table \(\PageIndex{3}\). Notice the similarity of the equation for the energy stored in a capacitor (second column, second to last box of \(\PageIndex{1}\)) and this equation for the energy density stored in a material with \(\epsilon > \epsilon_0\) (second column second to last box of the Table \(\PageIndex{3}\)).
| Energy storage device | Dielectric Material, \(\epsilon > \epsilon_0\) | Dielectric Material, \(\epsilon > \epsilon_0\) | Magnetic Material, \(\mu > \mu_0\) | Magnetic Material, \(\mu > \mu_0\) |
|---|---|---|---|---|
| Generalized Path | Displacement Flux Density \(\overrightarrow{D}\) in \(\frac{C}{m^2}\) | Electric Field Intensity \(\overrightarrow{E}\) in \(\frac{V}{m} = \frac{J}{C \cdot m}\) | Magnetic Flux Density \(\overrightarrow{B}\) in \(\frac{Wb}{m^2}\) | Magnetic Field Intensity \(\overrightarrow{H}\) in \(\frac{A}{m} = \frac{J}{Wb \cdot m}\) |
| Generalized Potential | Electric Field Intensity \(\overrightarrow{E}\) in \(\frac{V}{m} = \frac{J}{C \cdot m}\) | Displacement Flux Density \(\overrightarrow{D}\) in \(\frac{C}{m^2}\) | Magnetic Field Intensity \(\overrightarrow{H}\) in \(\frac{A}{m} = \frac{J}{Wb \cdot m}\) | Magnetic Flux Density \(\overrightarrow{B}\) in \(\frac{Wb}{m^2}\) |
| Generalized Capacity | Permittivity \(\epsilon\) in \(\frac{F}{m} = \frac{C^2}{J \cdot m}\) | \(\frac{1}{\epsilon}\) | Permittivity \(\mu\) in \(\frac{H}{m} = \frac{Wb^2}{J \cdot m}\) | \(\frac{1}{\mu}\) |
| Constitutive relationship | \(\overrightarrow{D} = \epsilon \overrightarrow{E}\) | \(\overrightarrow{E} = \frac{1}{\epsilon}\overrightarrow{D}\) | \(\overrightarrow{B} = \mu \overrightarrow{H}\) | \(\overrightarrow{H} = \frac{1}{\mu}\overrightarrow{B}\) |
| Energy | \(\int_{\mathbb{V}} \frac{1}{2} \epsilon|\overrightarrow{E}|^{2} d \mathbb{V}\) | \(\int_{\mathbb{V}} \frac{1}{2} \frac{1}{\epsilon}|\overrightarrow{D}|^{2} d \mathbb{V}\) | \(\int_{\mathbb{V}} \frac{1}{2} \mu|\overrightarrow{H}|^{2} d \mathbb{V}\) | \(\int_{\mathbb{V}} \frac{1}{2} \frac{1}{\mu}|\overrightarrow{B}|^{2} d \mathbb{V}\) |
| Law for potential | Gauss's Law for Elec. \(\overrightarrow{\nabla} \cdot \overrightarrow{D} = \rho_{ch}\) | Gauss's Law for Elec. \(\overrightarrow{\nabla} \cdot \overrightarrow{E} = \epsilon \rho_{ch}\) | Gauss's Law for Mag. \(\overrightarrow{\nabla} \cdot \overrightarrow{B} = 0\) | Gauss's Law for Mag. \(\overrightarrow{\nabla} \cdot \overrightarrow{H} = 0\) |
Energy can also be stored in materials with permeability greater than the permeability of free space, \(\mu > \mu_0\). Hall effect devices and magnetohydrodynamic devices were discussed in Chapter 5. These devices are all inductor-like, and the parameters used to describe inductive energy conversion processes in the language of calculus of variations are summarized in the last two columns of the Table \(\PageIndex{3}\). Calculus of variations can be used to describe energy conversion processes in these devices with either magnetic flux density or magnetic field intensity as the generalized path and the other choice as the generalized potential. The equation of motion resulting from using calculus of variations to describe inductive systems corresponds to Gauss's law for the magnetic field,
\[\overrightarrow{\nabla} \cdot \overrightarrow{B} =0 \nonumber \]
The physics of antennas is described by electric and magnetic fields, and any of the columns of Table \(\PageIndex{3}\) can be used to describe energy conversion between electricity and electromagnetic waves in antennas using the language of calculus of variations.