7 Circuits, Transmission Lines, and Waveguides Electric and magnetic fields contain energy, which can propagate. These are the ingre- dients needed for communications; in this chapter we will look at how electromagnetic energy can be guided. We will start with low-frequency circuits, then progress through transmission lines to high-frequency waveguides. 7.1 CIRCUITS The elements of an electrical circuit must satisfy Maxwell’s equations. In the low- frequency limit this provides a fundamental explanation for the familiar circuit equations. These simple relationships will hold as long as the frequencies are low enough for the size of the circuit to be much smaller than the electromagnetic wavelength. Above this there is a tricky regime in which the entire circuit acts like a distributed antenna, and then when the wavelength becomes small compared to the size of the circuit things become simpler again (this is the subject of Chapter 9 on optics). 7.1.1 Current and Voltage The voltage or potential difference between two parts of a circuit is defined by the line integral of the electric field V = E~ d~l . (7.1) − · Z As long as dB/dt~ = 0 then E~ = 0, which implies that the electric field is the gradient of a potential and the value of∇× its line integral is independent of the path; it can go through wires or free space as needed and will always give the same answer. Conversely, if there are time-varying magnetic fields then the potential difference does depend on path and can no longer be defined as a function of position alone. The electric field is defined to point from positive to negative charge so that the potential increases along a path from negative to positive charge. A charge q such as an electron in a wire feels a force F~ = qE~ , and so according to these definitions electrons flow from low to high potentials (Figure 7.1). The current I~, in amperes, at a point in a wire is equal to the number of coulombs of charge passing that point per second. It is defined to be in the same direction as the electric field and hence opposite to the direction in which electrons travel. The current density J~ is equal to the current divided by its cross-sectional area J~ = I/A~ . 7.1 Circuits 81 V x dx L s A E,J F Figure 7.1. A resistive element, showing the relations among the voltage, electric field, current, and charge motion. 7.1.2 Kirchhoff’s Laws There are two Kirchhoff Laws that can be used to analyze the current flow in a circuit: The sum of currents into and out of a circuit node must be zero. • If multiple wires meet at a point, the sum of all their currents must be equal to zero. This is just a statement of the conservation of charge. The sum of voltages around a circuit must vanish. • This follows because the line integral of the electric field around a closed path ∂ V = E~ d~l = E~ dA~ = B~ dA~ (7.2) − · − ∇× · ∂t · I Z Z will vanish if there is no time-varying magnetic flux linking the circuit. 7.1.3 Resistance In an isotropic conductor the current and electric field are related by J~ = σE~ , (7.3) where σ is the material’s conductivity. For very large fields there may be nonlinear deviations from this linear relationship, and in a complex material the conductivity may be a tensor that depends on direction. The voltage drop across the resistor in Figure 7.1 with length L, cross-sectional area A, conductivity σ, and carrying a current I is therefore + + J~ + I IL V = E~ d~x = d~x = dx = IR . (7.4) − · − σ · σA σA ≡ Z− Z− Z− Remember that the integral goes from low to high potential, but that current flows from high to low potentials, so J~ d~x = J dx = I dx/A. This is just Ohm’s Law, and it defines the resistance − · L ρL R = = (7.5) σA A 82 Circuits, Transmission Lines, and Waveguides V = IR1+IR2 V = I1R1=I2R2 I1 I2 I R1 V V = IR2 V R1 R2 R2 V =0 V =0 Figure 7.2. Series and parallel circuits. in terms of the conductivity σ, which has units of siemens per meter (S/m), or the resis- tivity ρ, which has units of ohm-meters (Ω m). For a two-dimensional film of thickness · T , the resistance of a region of length L and width W is L R = ρ A L = ρ T W ρ L = T W L R . (7.6) ≡ ⊔⊓ W This defines the sheet resistivity R (“R square”). Since L/w is dimensionless, R has units of resistance without any other⊔⊓ length. ⊔⊓ Kirchhoff’s Laws can be used to simplify the circuits in Figure 7.2. For the series circuit on the left, V = IR1 + IR2 (7.7) or V I = , (7.8) R1 + R2 therefore the total resistance is Rtotal = R1 + R2 . (7.9) Series resistances simply add. For the parallel circuit on the right, the voltage drop across both legs must be equal since potential is independent of path, V = I1R1 = I2R2 , (7.10) and the current in both legs must add up to the total current V I1 + I2 = . (7.11) Rtotal 7.1 Circuits 83 Therefore V V V + = (7.12) R1 R2 Rtotal or 1 1 1 = + Rtotal R1 R2 R1R2 Rtotal = . (7.13) R1 + R2 Parallel resistances add inversely. More complex networks of resistances can always be simplified to a single effective resistance by repeated application of these rules. 7.1.4 Power Now let’s now consider a slab of charge of cross-sectional area A and thickness dx moving through the resistor in Figure 7.1. If the charge density is ρq, the total charge in this slab is Q = ρqdxA and it feels a net force F~ = QE~ . Because a current is flowing, charge is moving relative to this force and so work is being done. The work associated with the slab moving from one end of the resistor to the other is equal to the integral of the force times the displacement: + + dW = F~ d~x = Q E~ d~x = QV = ρ dxAV (7.14) · − · − − q Z− Z− for a negative charge. This decrease in energy is dissipated in the resistor; the power is equal to the rate at which work is being done dW dx P = = ρ AV = JAV = IV = I2R . (7.15) − dt q dt The power dissipated in a resistor is equal to the current flowing through it times the voltage drop across it, which by Ohm’s Law is also equal to the square of the current times the resistance. This appears as heat in the resistor. 7.1.5 Capacitance There will be an electric field between an electrode that has a charge of +Q on it and one that has a charge of Q, and hence a potential difference between the electrodes. Capacitance is defined to− be the ratio of the charge to the potential difference: Q C = . (7.16) V The MKS unit is the farad, F. Capacitances range from picofarads in circuit components up to many farads in supercapacitors based on electrochemical effects [Gonzalez´ et al., 2016]. The current across a capacitor is given by dV dQ C = = I . (7.17) dt dt A capacitor is a device that stores energy in an electric field by storing charge on its plates; 84 Circuits, Transmission Lines, and Waveguides in Problem 6.2 we saw that this stored energy is equal to CV 2/2. The current flowing across a capacitor is a displacement current: from the point of view of the overall circuit it is a real current, but it arises from the time-varying electric field associated with the capacitor plates storing or releasing charge rather than from real charge passing through the capacitor. If the applied voltage is V = eiωt, then the current is dV I = C = iωCeiωt . (7.18) dt The impedance (complex resistance) is defined to be the ratio of the voltage and current at a fixed frequency, V eiωt 1 Z = = = . (7.19) I iωCeiωt iωC The current leads the voltage by a phase shift of i = 90◦. When ω = 0 the impedance is infinite (no current flows at DC), and when ω = the impedance is 0 (the capacitor acts like a wire). ∞ 7.1.6 Inductance An inductor stores energy in a magnetic field arising from current flowing through a coil. The inductance is defined to be the ratio of the magnetic flux Φ = B~ dA~ (7.20) · Z linking a circuit to the current that produces it: Φ L = . (7.21) I The MKS unit of inductance is the henry, H. In Figure 7.3 the electric field vanishes along the dotted line for an ideal solenoid, therefore the line intergral of the electric field along the dotted line and around the solenoid is equal to the voltage drop across the solenoid. And the magnetic field vanishes outside the solenoid if it is approximated to be a section of an infinite solenoud, therefore the integral of the magnetic field across the surface bounded by the path is equal to the flux linking the solenoid times the number of turns of the coil. This lets us relate the flux to the potential. If this ideal solenoid is taken to have just a single turn then ∂ ∂Φ ∂ dI V = E~ d~l = B~ dA~ = = (LI) = L (7.22) − · −∂t · − ∂t −∂t − dt I ZS and so dI V = L .