## Notes on phasors in electrical engineering

• Phasors are complex representations of sinusoidal (usually cosine) functions with constant angular frequency, amplitude, and initial phase. In electrical engineering, they are used to simplify circuit analysis differential equations to algebraic equations. A cosine function and its equivalent phasor are given below.  • The phasor domain is constructed using the complex plane. The amplitude corresponds to the magnitude of a complex number, while the phase shift corresponds to the angle of the complex number with respect the real axis.
• Using Euler’s formula, cosine functions are converted to exponential form. VA∠φ is shorthand for VAe. • When adding sinusoids that possess the same frequency, their phasor representation is a sum of the component phasors. • Using the 90° phase shift that comes from taking the derivative of a sinusoid, the following expression is obtained. As such, multiplying a phasor by jω is equivalent to differentiating the corresponding sinusoid. • Phasors are converted into complex numbers using their geometry. For instance, consider a phasor 5∠30°. The corresponding complex number is 5cos(30°) + 5sin(30°)j = 4.33 + 2.5j.
• Phasors can be applied to circuit analysis when the voltage or current sources take on sinusoidal waveforms.
• Kirchhoff’s voltage law and current law applies to phasor voltages and phasor currents.
• Impedance is a generalization of resistance. As such, all passive circuit elements are treated as resistors (except that the given element’s impedance is used rather than resistance). The impedances of resistors, capacitors, and inductors are given below. Impedance is a complex quantity with units of ohms. • An analogous version of Ohm’s law is true for impedance.  • As a result of the equations for impedance, frequency influences the passive circuit elements (resistors, capacitors, and inductors) in distinct ways. The relationship between the magnitude of the impedance and the frequency are plotted at right.
• To carry out phasor analysis on a circuit, the following steps are performed.
1. Convert sinusoidal voltage and current sources into the phasor domain and characterize passive current elements via their impedances.
2. Use standard algebraic circuit analysis techniques (i.e. KCL, KVL, source transformations, etc.) to solve the circuit in terms of its phasor domain quantities.
3. Transform the phasor domain responses back into their corresponding time domain responses to obtain the waveforms across each circuit element.
• Impedances possess series equivalence. The real part of Z is called the resistance and the imaginary part is called the reactance. If the imaginary part X is negative, it is called a capacitive reactance. If the imaginary part X is positive, it is called an inductive reactance. • Voltage division is carried out in the phasor domain using the following equation. • Impedances possess parallel equivalence and can be written in terms of the admittance Y = 1/Z. Admittance has units of siemens (S). The real part of Y is called the conductance G and the imaginary part is called the susceptance B. • Current division is carried out in the phasor domain using the following equation. Reference and source of images: Thomas, R. E., Rosa, A. J., & Toussaint, G. J. (2016). The Analysis and Design of Linear Circuits (8th ed.). Wiley.

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## Notes on wave optics

The wave equation

• Because light exhibits wave-particle duality, wave-based descriptions of light are often appropriate in optical physics, allowing the establishment of an electromagnetic theory of light.
• As electric fields can be generated by time-varying magnetic fields and magnetic fields can be generated time-varying electric fields, electromagnetic waves are perpendicular oscillating waves of electric and magnetic fields that propagate through space. For lossless media, the E and B field waves are in phase.
• By manipulating Maxwell’s equations of electromagnetism, two relatively concise vector expressions that describe the propagation of electric and magnetic fields in free space are found. Recall that the constants ε0 and μ0 are the permittivity and permeability of free space respectively. • Since an electromagnetic wave consists of perpendicular electric and magnetic waves that are in phase, light can be described using the wave equation (which is equivalent to the expressions above). Note that the speed of light c = (ε0μ0)-1/2. Electromagnetic waves represent solutions to the wave equation. • Either the electric or the magnetic field can used to represent the electromagnetic wave since they propagate with the same phase and direction. With the exception of the wave equation above, the electric field E will instead be used to represent both waves. Note that either the electric or magnetic field can be employed to compute amplitudes.

Solutions to the wave equation

• Plane waves represent an important class of solutions to the wave equation. The parameter k is the wavevector (which points in the direction of the wave’s propagation) with a magnitude equal to the wavenumber 2π/λ. In a 1-dimensional system, the dot product k•r is replaced by kx. The parameter ω is the angular frequency 2πf and φ is a phase shift. • To simplify calculations, Euler’s formula can be used to convert the equation above into complex exponential form. Only the real part describes the wave as the real part corresponds to the cosine term. • Spherical waves are another useful solution to the wave equation (though they are an approximation and truly spherical waves cannot exist). Because of their geometry, the electric field of a spherical wave is only dependent on distance from the origin. As such, the equation for a spherical wave can be written as seen below with origin r0. • Gaussian beams are a solution to the wave equation that can be used to model light from lasers or light propagating through lenses. If a Gaussian beam propagates in the z direction, then from the perspective of the xy plane, it shows a Gaussian intensity distribution. For a Gaussian beam, the amplitude decays over the direction of propagation according to some function A(z), R(z) represents the radius of curvature of the wavefront, and w(z) is the radius of the wave on the xy plane at distance z from the emitter. Often these functions can be approximated as constants. Intensity and energy of electromagnetic waves

• The Poynting vector S is oriented in the direction of a wave’s propagation (assuming that the wave’s energy flows in the direction of its propagation). • The magnitude of the Poynting vector represents the power per unit area (W/m2) or intensity crossing a surface with a normal parallel to S. Note that this is an approximation since, according to a quantum mechanical description of electromagnetic waves, the energy should be quantized. • Power per unit area (intensity) of plane waves, spherical waves, and Gaussian beams can also be calculated using the equations below. The formula for the Gaussian beam’s power represents the power at a plane perpendicular to the direction of light propagation z. • For electromagnetic waves, instantaneous energy per unit area is difficult to measure, so the average energy per unit area over a period of time Δt is often worked with instead. Since waves are continuous functions, taking their time-average requires an integral. • After using the above integral on the function eiωt and then taking the real and imaginary parts of the result, the time-averages of the functions cos(ωt) and sin(ωt) are found. Superposition of waves

• Let two waves E1 and E2 of the same frequency traveling in the same direction undergo superposition. E1 and E2 may or may not possess the same amplitude or phase. The substitution α = –(kx+φ) will be carried out. • If the phases of the waves are different, some special equations are necessary to find the amplitude E0 and the phase α of the resulting wave. • For the superposition of any number of waves, the equations above can be extended. Polarization of light

• The waves comprising linearly polarized light are all oriented at the same angle which is defined by the direction of the electric field of the light waves. For linearly polarized plane waves with electric fields oriented along the x or y axes that propagate in the z direction, the following equations describe their electric fields. • The superposition of two linearly polarized plane waves that are orthogonal to each other (and out of phase) is the vector sum of each electric field. • The superposition of two linearly polarized plane waves that are orthogonal to each other (and in phase) is computed via the following equation and has a tilt angle θ determined by the ratio of amplitudes of the original waves. This process can also be performed in reverse with a superposed polarized wave undergoing decomposition into two orthogonal waves.  • When two constituent waves possess equal amplitudes and a phase shift of nπ/2, the superposed wave is circularly polarized (as it can be expressed using a sine and a cosine term). Equations for the constituent waves and the superposed wave are given below. • When circularly polarized light propagates, it takes a helical path and so rotates. As such, a full rotation occurs after one wavelength. If a circularly polarized wave rotates clockwise, it is called right-circularly polarized and has a positive sine term. If a circularly polarized wave rotates counterclockwise, it is called left-circularly polarized and has a negative sine term. • If a right-circularly polarized light wave and a left-circularly polarized light wave of equal amplitude are superposed, then they create a linearly polarized light wave with twice the amplitude of the individual waves. • Linearly polarized and circularly polarized light are special cases of elliptically polarized light. For elliptically polarized light, the amplitudes of the superposed waves may differ and the relative phase shift does not need to be nπ/2. As such, the electric field traces an elliptical helix as it propagates along the z direction. • For elliptically polarized light with a positive phase shift φ, it is called right-elliptically polarized if E0x > E0y and left-elliptically polarized if E0x < E0y.
• Most light is unpolarized (or more appropriately, a mixture of randomly polarized waves). To obtain polarized light, polarizing filters are often used.

References

Boudoux, C. (2017). Fundamentals of Biomedical Optics. Blurb, Incorporated.

Degiorgio, V., & Cristiani, I. (2015). Photonics: A Short Course. Springer International Publishing.

Hecht, E. (2017). Optics. Pearson Education, Incorporated.

## Notes on fiber biomechanics

Elastic fiber models

• For an elastic fiber in which a linear relationship between force and change in length is assumed, the force is given by F = k(L – L0).
• To normalize for other elastic fibers with different starting lengths, this equation is divided by L0 to give F = k(L/L0 – 1). It is common practice to represent L/L0 as a parameter λ (called the stretch ratio).
• As such, F is found using the formula below. Note that the quantity λ – 1 is referred to as the strain. • While linear models are often useful, many real fibers exhibit finite extensibility (a nonlinear phenomenon) after exceeding a certain critical strain value λc. That is, the force necessary to extend the fiber farther after exceeding λc increases rapidly. Finite extensibility can be modeled using the following equation which divides k by a term dependent on λ and λc. • To model a muscle, let L0 represent the muscle’s length in its inactive state and Lcontracted represent the muscle’s length in its contracted state. Unlike the spring, the contracted state is used as the reference length. The contraction stretch is described by the ratio λcontracted = Lcontracted/L0 while the stretch ratio remains as λ = L/L0. • If this muscle is contracted without carrying a load such that F = 0, then λ = λcontracted. If the muscle acquires a load and so must maintain a constant length equal to its original length L0 (to “hold the load steady”), then the force in the muscle is F = k(1/λcontracted – 1).
• To generalize this model for 3-dimensional space, the locations of the fiber’s endpoints A and B are used. The fiber’s length and orientation are given below. • The following force vectors can act on point B and on point A. The stretch ratio is still λ = L/L0. Viscous fiber models

• Purely viscous behavior (as with liquids) can be described 1-dimensionally using the equation below where cη is a damping coefficient. • The normalized rate of deformation is equivalent to the above formula without the damping coefficient. In addition, the rate of deformation can be written in terms of the stretch ratio λ = L/L0. • If one endpoint of a filament of fluid is moved with constant velocity, its position is given by xB = L0 + vt. This means that the rate of deformation is v/xB. • Solving the above equation gives the following result. For a constant rate of elongation, the point xB must be displaced exponentially over time. • Given endpoint displacements uA and uB, the total displacement is ΔL = uB – uA. Using this quantity, the stretch λ and the strain ε can be written using the equations below. • For the small strains (i.e. |ε| is much less than 1) that result from small stretches, some approximations can be made which reduce the force equation to the following form. Viscoelastic fiber models

• Many biological materials exhibit viscoelastic behavior rather than elastic behavior. In viscoelastic systems, the force on a fiber with a constant length decreases over time and applying constant force causes the length to increase.
• The strain response of a fiber to force is given as ε(t). The creep function J(t) describes the fiber’s tendency to permanently deform as strain is applied. Many possible creep functions can be devised depending on the system. The creep function is related to the strain by a factor of force increase F0. • Strain responses follow the principle of superposition. That is, if a force is applied at τ1 and then another force is applied at τ2, a total strain response can be expressed as a sum of the two individual strain responses. • For an arbitrary history of applied forces, an integral formulation of strain response is used. In this equation, the change in force at each infinitesimal time interval is multiplied by the creep function. • Similarly, the force resulting from an imposed strain history can be expressed as an integral where G is a function that describes the relaxation of the fiber with time (analogous to the creep function, but the opposite concept).

Reference: Oomens, C., Brekelmans, M., & Baaijens, F. (2009). Biomechanics: Concepts and Computation. Cambridge University Press.