Used when modelling physical quantities such as magnetic and electrical fields. Maxwell's equations describe fields and are used to model electrical phenomena.
Vectors are points and directions in 3d space. \[\mathbf{v}=\overrightarrow{OQ}-\overrightarrow{OP}=v_x\hat{x}+v_y\hat{y}+v_z\hat{z}=\begin{pmatrix}v_x\\v_y\\v_z\end{pmatrix}\] The magnitude is: \[\lvert \mathbf{v} \rvert =\sqrt{v_x^2+v_y^2+v_z^2}\] Its unit vector is: \[\mathbf{\hat{v}}=\frac{\mathbf{v}}{\lvert \mathbf{v}\rvert}\] Addition of vectors is done by adding the corresponding elements of each vector. Scalar multiplication is a multiplication of each element by the scalar. The scalar (dot) product is the sum of the pairwise product of the elements. \[\mathbf{v\cdot w}=v_xw_x+v_yw_y+v_zw_z\] The cross product produces a vector result \[\mathbf{v\times w}=\begin{vmatrix}\mathbf{\hat{x}}&\mathbf{\hat{y}}&\mathbf{\hat{z}}\\v_x&v_y&v_z\\w_x&w_y&w_z\end{vmatrix}\]
Uses \(\hat{x},\hat{y},\hat{z}\). Each vector represents an orthogonal direction in 3d space.
Uses \(\hat{r},\hat{\phi},\hat{z}\) \(\hat{r}\) represents the radius of the point from the centre to the point, and \(\hat{\phi}\) represents the angle to that point and thus is tangential to the surface of the cylinder. \(\hat{\phi}\) and \(\hat{r}\) are in the x-y plane. \(\hat{z}\) represents the height of the point. This can be useful in representing wires.
Uses \(\hat{r},\hat{\phi},\hat{\theta}\). The representation is from the surface of a sphere. \(\hat{r}\) is the distance from the origin, and \(\hat{\phi}\) and \(\hat{\theta}\) represent the angles to the point. The three unit vectors are orthogonal.
To convert between coordinate systems involves projecting the unit vectors onto the new vector space in order to find scaling factors. These scaling factors are then multiplied by the elements of the original vector.
A vector valued function which assigns a vector to every point within a plane. \[\mathbf{F}(x,y)=y\hat{x}+x\hat{y}\] We can redefine the vector field to be a function of a single variable for providing it with a vector argument \[\mathbf{G}(s)=a_1s\hat{x}+a_2s\hat{y}\] \[\mathbf{F}(\mathbf{G}(s))=a_2y\hat{x}+a_1x\hat{y}\]
We can also take the gradient of a vector (its divergence), which produces a vector field. The gradient of a vector field produces a scalar vector function perpendicular to the field. This is roughly analogous to a measurement of the field's spread. \[\nabla \mathbf{F}=\frac{\partial \mathbf{F}}{\partial x}\hat{x}+\frac{\partial \mathbf{F}}{\partial y}\hat{y}+\frac{\partial \mathbf{F}}{\partial z}\hat{z}\] There is also the curl, where 0 refers to no rotation of the field. This maps a vector field to another field. \[\text{curl }\mathbf{F}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x}+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\hat{y}+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\hat{z}\]
Line integrals are integration along a path. \[W=\int_\ell F\cdot dl\] Surface integrals are the integral over a surface. \[\phi=\iint_A F\cdot dA\] Volume integrals are the integration through a volume. \[q=\iiint_v\rho dv\]
Flux is the measure of flow of a vector field through a surface. \[\Phi=\iint_AF\cdot dA\] We can find that the divergence is the incremental limit of total flux leaving a closed surface per unit volume, or: \[\nabla \mathbf{F}\equiv\lim_{v\to 0}\frac{\iint_{A(v)}\mathbf{F}\cdot d\mathbf{A}}{\iiint_vdv}\] Gauss' divergence theorem states that the net flux out of a closed surface is related to the divergence of the vector field: \[\iint_A\mathbf{F}\cdot d\mathbf{A}=\iiint_{vol(A)}\nabla \mathbf{F}dv\] The circulation is the flux through a closed path: \[C=\oint_{\ell(A)}\mathbf{F}\cdot d\mathbf{l}\] We can view curl as circulation per unit enclosed area. \[\frac{\mathbf{C}}{dA}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x}+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\hat{y}+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\hat{z}\] Stokes theorem states this: \[\oint_{\ell(A)}\mathbf{F}\cdot d\mathbf{l}=\iint_A \text{curl }\mathbf{F}\cdot d\mathbf{A}\] A vector field is conservative is there exists a scalar valued function such that: \[\mathbf{F}(x,y,z)=\nabla\phi(x,y,z)\]
Resistors can be modelled by \(v=Ri\), capacitors by \(i=C\frac{dv}{dt}\) and inductors by \(v=L\frac{di}{dt}\), but where do those models come from? The electromagnetic field model can help us to understand the derivation. Coulomb's law describes the distance between charges, their sized and the force between them: \[\mathbf{F}=q\left(\frac{kQ}{\lvert r\rvert^2}\hat{r}\right)=q\mathbf{E}\] From this, we can calculate the word done in moving a charge through the field, and from that the voltage \[W=-\int_l\mathbf{F}\cdot d\mathbf{l}=-q\int_l\mathbf{E}\cdot d\mathbf{l}=qV\] The electric field is a conservative one, where the integral along a closed loop is 0, regardless of the loop.
We have a conservation of charge, relating the charge density (\(\rho\)) and the current density (\(\mathbf{J}\)). \[\iint_A\mathbf{J}d\mathbf{A}=-\frac{\partial}{\partial t}\iiint_{vol(A)}\rho dv\] The LHS is the current out through the surface A, and the RHS is the time rate of change of charge in a volume. We can use this to give us one of Maxwell's equations, Gauss' Law of electricity, relating the flux density (\(\mathbf{D}=\epsilon\mathbf{E}\)) and charge. \[\iint_A \mathbf{D} \cdot d\mathbf{A}=q=\iiint_{vol(A)}\rho dv\]
We can also relate current and magnetic field with Ampere's law: \[\oint_\ell\mathbf{H}\cdot d\mathbf{l}=I\] We can further generalise this to \[\oint_{\ell(A)}\mathbf{H}\cdot d\mathbf{l}=\underbrace{\iint_A\mathbf{J}\cdot d\mathbf{A}}_{\text{Conventional Current}}+\underbrace{\frac{\partial}{\partial t}\iint_A\mathbf{D}\cdot d\mathbf{A}}_{\text{Displacement Current}}\] We need to do this as Ampere's law only works with non-time varying currents.
Faraday's law relates the electric field and the magnetic flux density (\(\mathbf{B}=\mu\mathbf{H}\)). \[-\oint_{\ell(A)}\mathbf{E}\cdot d\mathbf{l}=\frac{\partial}{\partial t}\iint_A\mathbf{B}\cdot d\mathbf{A}\]
We also have Gauss' law for magnetism \[\iint_A\mathbf{B}\cdot d\mathbf{A}=0\] This is always equal to zero as a source of a magnetic field has never been found.
Together these four equations form Maxwell's equations.
| Name | Differential | Integral |
|---|---|---|
| Faraday's Law | \[\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\] | \[\oint_{\ell(A)}\mathbf{E}\cdot d\mathbf{l}=-\frac{\partial}{\partial t}\iint_A\mathbf{B}\cdot d\mathbf{A}\] |
| Gauss Law (electric) | \[\nabla\cdot\mathbf{D}=\rho\] | \[\iint_A\mathbf{D}\cdot d\mathbf{A}=\iiint_{vol(A)}\rho dv\] |
| Ampere's Law | \[\nabla\times\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}\] | \[\oint_{\ell(A)}\mathbf{H}\cdot d\mathbf{l}=\iint_A \mathbf{J}\cdot d\mathbf{A}+\frac{\partial}{\partial t}\iint_A\mathbf{D}\cdot d\mathbf{A}\] |
| Gauss' Law (magnetic) | \[\nabla\cdot\mathbf{B}=0\] | \[\iint_A\mathbf{B}\cdot d\mathbf{A}=0\] |
A resistor is a two terminal, one port device characterised by:
The ideal resistor obeys Ohm's law. \[v(t)=Ri(t)\]
We can find Ohm's law by analysing what is happening in the resistor. A charge in an electric field experiences a force, which causes it to accelerate. \[F=qE\] \[x''=\frac{q}{m}E\]
We can also find the amount of charge that exits a volume in a time and use it to find the current density. \[dn=n(\lvert v\rvert dtA)\] \[dq=qdn=qnA\lvert v\rvert dt\] \[I=\frac{dq}{dt}=qnA\lvert v\rvert\] \[J=\frac{I}{A}\hat{v}=qn\lvert v\rvert\hat{v}\] \[J=qn\vec{v}\] Using this and the motion of particles in an electric field we can model the velocity as a function of time. \[v(t)=\frac{q}{m}Et+v(0)\] This suggests the current increases indefinitely, so there must be something unaccounted for. What happens is that there are collisions between the charged particles and static particles in the material. These collisions effectively stop the charges, releasing the energy as heat. We can find the average charge velocity accounting for collisions as: \[\mathbf{v}=\left(\frac{q\tau}{m}\right)\mathbf{E}\] Where \(\tau\) is the mean time between collisions. Combining this with the current density from above gives the vector form of Ohm's Law. \[\mathbf{J}=\sigma\mathbf{E}\] Where \(\sigma\) is the conductivity and \[\sigma=\frac{q^2\tau n}{m}\] We can take this form of Ohm's law and find the standard form. If we find the potential difference along a path and find the current, we can arrive at the standard form. \[V=-\int_\ell\mathbf{E}\cdot d\ell=\lvert\mathbf{E}\rvert\ell\] \[I=\iint_A\mathbf{J}\cdot d\mathbf{A}=\sigma\iint_A\mathbf{E}\cdot d\mathbf{A}=\sigma A\lvert\mathbf{E}\rvert\] \[V=RI\] \[R=\frac{\ell}{\sigma A}=\frac{\rho\ell}{A}\] This is the Drude model, and predicts resistance, thermal conductivity, hall effect, etc. There is also the Free electron model and the Nearly-free electron model. The Nearly-free electron model is useful for modelling semiconductors. The mobility of charge is given by \[\mu_q=\frac{q\tau}{m}=\frac{\sigma}{qn}\] The mobility scales with the mean time between collisions, it peaks at some temperatures then decreases for greater ones.
We can also find that there is a skin effect, where the current only flows on the “skin” or surface of the conductor. The skin depth is given by \[\delta=\sqrt{\frac{2}{\omega\mu\sigma}}\] Where \(\omega\) is the angular frequency, \(\mu\) is the permeability and \(\sigma\) is the conductivity. The resistance of a conductor becomes: \[R(\delta)=\frac{\ell}{\sigma A(\delta)}\]
Transistors can provide non-linear resistance. Thermistors are resistors which are designed to exploit the temperature dependence of resistance.
The conductivity of a material determines whether it is a conductor or insulator, with a high conductivity being the former and a low being the latter. The charge redistribution time determines how charge can be redistributed in a material. \[T_{CR}=\frac{\epsilon}{\sigma}\] This gives rise to the charge redistribution bandwidth: \[B_{CR}=\frac{1}{2\pi T_{CR}}=\frac{\sigma}{2\pi\epsilon}\] At frequencies significantly lower than the charge redistribution bandwidth a substance is a conductor, at those greater, an insulator. Charge redistribution requires current, which is limited by resistivity. Charge can not be redistributed arbitrarily quickly and the time to redistribute is material dependent. This is all modelled with Ohm's law, conservation of charge and Gauss' law of electricity. Pulling this all together gives the charge redistribution model: \[\frac{\partial\rho}{\partial t}+\frac{\sigma}{\epsilon}\rho=0\] The solution to this PDE gives: \[\rho(t)=\rho_0e^{-t/T_{CR}}\]
The skin effect is observed as high frequencies where the current flows along the skin of the conductor. This reduces the cross sectional area of the conductor. \[\delta=\sqrt{\frac{2}{\omega\mu\sigma}}\] \(\omega\) is the angular frequency, \(\mu\) is the permeability and \(\sigma\) is the conductivity. The resistance of the conductor skin is: \[R(\delta)=\frac{\ell}{\sigma A(\delta)}\] Where \(A(\delta)=\pi\delta(2r-\delta)\).
When an external electric signal intersects a conductor, its generates a current. The current is accompanied by a resistive loss in the form of heat. The magnitude of the electric field decreases as it propagates into the material. Non-conductive materials allow the signal to pass as it fails to generate a current. A consequence of the skin effect is that only a small coating of a conductor is needed to transmit high frequency signals. This allows for wires consisting of insulator cores and conductive coverings. At high frequencies, a thin metal covering can attenuate any external signals, reducing any electromagnetic interference. The telegrapher's equation for the electromagnetic field in a conductor describes the propagation of an electric field: \[\nabla^2\mathbf{E}-\mu\epsilon\frac{\partial^2\mathbf{E}}{\partial t^2}-\mu\sigma\frac{\partial\mathbf{E}}{\partial t}=0\] For a plane wave, we get: \[\mathbf{E}(t,z)=E(z)e^{j\omega t}\hat{x}\] Making this approximation: \[\frac{d^2E(z)}{dz^2}+\mu\epsilon\omega^2\left(1-\frac{j\sigma}{\omega\epsilon}\right)E(z)=0\] When the frequency is significantly less than the charge redistribution bandwidth you get the diffusion equation: \[\frac{d^2E(z)}{dz^2}-j\omega\mu\sigma E(z)=0\] When the frequency is greater you get the wave equation: \[\frac{d^2E(z)}{dz^2}+\mu\epsilon\omega^2E(z)=0\] The diffusion equation is when the material acts like a conductor and the electromagnetic field is tied to current flow. The wave equation is when the electromagnetic field is not tied to current flow and the material behaves as a dielectric. The solution to the diffusion model gives the skin depth model. \[\mathbf{E}(t,z)=E_0e^{-z/\delta}e^{j(\omega t-z/\delta)}\hat{x}\]
Charge separation gives rise to an electrostatic electric field. The field has an associated potential. If there is a charge in one conductor, another conductor nearby will mobilise its charges in order to eliminate the electric field. \[i=\frac{dQ}{dt}=\left(\frac{dQ(V)}{dV}\right)\frac{dV}{dt}=C(V)\frac{dV}{dt}\] Here the capacitance as the charge separation per volt: \[C(v)=\frac{dQ(V)}{dV}\] The capacitance is dependent on the geometry and permittivity of the material and can be linear or nonlinear.
We can model the electric field from charge separation with Gauss' law: \[\iint _A \mathbf{D}\cdot d\mathbf{A}=q\] The electric field gives rise to a potential difference \[V=-\int_\ell\mathbf{E}\cdot d\mathbf{l}\] The interaction between these gives capacitance. If we consider two infinite parallel plates, the electric field is perpendicular to the plates, giving: \[V=-\int_\ell \mathbf{E}\cdot d\mathbf{l}=Ed\]
For coaxial capacitors, the two conductors are separated by a dielectric. Here the field lines are directed radially outward from the core. \[\mathbf{E}=E(r)\hat{r}\] This is due to the radial symmetry of the cable. There are no edge effects as the cable is assumed to be of infinite length. If we apply Gauss' law with a cylindrical test surface, we find that the side face is the only one perpendicular to the electric field. If we choose a cylinder centred on the charge, the radius is constant, so the LHS of Gauss' law evaluates to: \[\iint_A\mathbf{D}\cdot d\mathbf{A}=\epsilon E(r)*2\pi rd\] So the electric field and the voltage are: \[\mathbf{E}(r)=\left(\frac{q}{2\pi\epsilon rd}\right)\hat{r}\] \[V=-\int_\ell \mathbf{E}(r)\cdot d\mathbf{l}=\frac{q}{2\pi\epsilon d}\log\left(\frac{r_2}{r_1}\right)\] Where \(r_1\) is the distance to the end of the inner wire and \(r_2\) is the distance to the inside of the outer shield.
Capacitors have dielectrics, which separates the conductors in a capacitor. The dielectric varies \(\epsilon\) and allows charge storage on either side of the capacitor. The dielectric also affects how close the plates of a capacitor can be, as smaller distances leads to larger capacitance. A good dielectric has low conductivity and high permittivity.
A real capacitor has capacitance, but also a series resistance and inductance due to the leads as well as a parallel resistance due to the dielectric's conductivity. Semiconductors can have nonlinear capacitance.
Inductors are coils of wire used to generate magnetic fields. Flowing current gives rise to a magnetic field, which corresponds to a flux over an area. \[\Phi=\iint_A\mathbf{B}\cdot d\mathbf{A}=Li\] The second relationship arises from Ampere's law. \[\oint_\ell\mathbf{H}\cdot d\mathbf{l}=\iint_A\mathbf{J}\cdot d\mathbf{A}\] We can find the induced emf from Faraday's law: \[\oint_\ell\mathbf{E}\cdot d\mathbf{l}=-\frac{\partial}{\partial t}\iint_A\mathbf{B}\cdot d\mathbf{A}\] This gives the induced emf as: \[v=\frac{d\Phi}{dt}\] From the combination of these we get self inductance: \[v=L\frac{dt}{dt}\]
We can calculate the self-inductance of an infinite length coaxial cable. By symmetry, the flux only depends on the radius from the centre and is directed along the \(\hat{\Phi}\) direction. Ampere's law gives: \[\oint_\ell\mathbf{H}\cdot d\mathbf{l}=(2\pi r)H(r)=i\] So we can find the magnetic field to be: \[\mathbf{H}=\frac{i}{2\pi r}\hat{\Phi}\] From the flux integral and the relationship between flux and current, we can find the inductance per unit length: \[\frac{L}{\ell}=\frac{\mu}{2\pi}\log\left(\frac{b}{a}\right)\] Where \(a\) is the radius of the inner wire and \(b\) is the distance to the inside of the outer wire.
We can also work out the inductance of an infinite length solenoid. Here we have a magnetic field directed along the z axis inside the core but a nonexistent field outside the core. \[\mathbf{H}=\begin{cases}0,&outside\\H(r)\hat{z},&inside\end{cases}\] If we use Ampere's law, we can find the field to be: \(H(r)=ndi\) Where \(n\) is the number of turns, \(d\) is the length and \(i\) is the current. Again finding the flux gives: \[\Phi=\mu ni\mathbf{A}\] We can use this to find the flux linkage (flux through all windings in the length) per unit length as: \[\Lambda=\mu n^2Ai\] Here \(\hat{L}=\mu n^2A\) where \(\mu\) is the permeability of the material, \(n\) is the number of turns per unit length and \(A\) is the cross sectional area. We can approximate the inductance of a finite length inductor as \[L\approx\hat{L}\ell=\frac{\mu N^2A}{\ell}\] Where \(N\) is the number of turns, \(A\) is the cross sectional area and \(\ell\) is the solenoid length.
For parallel wires flowing in opposite directions, the inductance per unit length is: \[\frac{L}{\ell}=\frac{\mu}{\pi}\log\left(\frac{d-a}{a}\right)\] Here, \(\mu\) depends on the permeability of the material between the two wires, \(a\) is the radius of the wire and \(d\) is the wire separation from the axes of the wires.
Permiability is the ability of the magmetic moments of a substance's atoms to be aligned with an external magnetic field. The internal alignment is the magnetisation \(\mathbf{M}\). We can add to it the external magnetic field to find the total flux density: \[\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})\] Where \(\mu_0\) is the permeability of free space.
For some materials the magnetisation can be linear with respect to the applied field. \[\mathbf{M}=\chi(\mathbf{H})=\chi_m\mathbf{H}\] \[\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})=\mu_0(1+\chi_m)\mathbf{H}=\mu\mathbf{H}\] Here \(\mu_r=(1+\chi_m)\) is the relative permeability.
Wires denote any pair of conductors connecting different circuit parts. In the ideal case, a wire is transparent to electrical signals, however in reality they have resistance, inductance and capacitance. These properties allow the construction of filters with only wires, exploiting their capacitance and inductance. Wires tend to behave badly at high frequencies, distorting the input signal.
For digital signals, there is an assumed equal rise and fall time between the high and low logic levels. The transitions between levels are assumed to be linear. When we take the Fourier or Laplace transform of this, we get noisy transform decreasing with frequency. There is the knee frequency which characterises the approximate bandwidth of digital signals. \[f_{KNEE}=\frac{1}{\pi t_{SW}}\approx\frac{0.5}{t_{SW}}\] Up to the knee frequency the frequency response is flat, so will not distort the digital signal. We can create the lumped circuit models for a wire as \[R_W=\hat{R}\ell\] \[G_W=\hat{G}\ell\] \[L_W=\hat{L}\ell\] \[C_W=\hat{C}\ell\] Where \(R_W\) is the resistance (series), \(G_W\) is the conductance (leakage to ground) (parallel), \(L_W\) is the inductance (series) and \(C_W\) is the capacitance (parallel). All these create a RLC circuit, which has a resonant frequency and can cause unwanted behaviour around and above the resonant frequency causing distortion of the input signal. When \(f_{ring}<<f_{knee}\), there will likely be distortion, but not when \(f_{ring} >> f_{knee}\). The ringing frequency is a property of the wire, as a property of the capacitance and inductance. \[f_{ring}=\frac{1}{2\pi}\sqrt{\frac{1}{LC}-\frac{R^2}{2L^2}}\approx\frac{1}{2\pi\sqrt{LC}}\] Using a lumped circuit model for wires allows for rough calculations, but does not give insight to the behaviour.
Regardless of the geometry of the wire, the product of the inductance and capacitance of the wire is always the same, \(\mu\epsilon\). We can find that the velocity of propagation through the material is equal to: \[\nu=\frac{1}{\sqrt{\mu\epsilon}}\] This can be found with the telegrapher's equation, playing around with wave equations. We can use this to find the propagation delay per unit length: \[D=\frac{1}{\nu}=\sqrt{\hat{L}\hat{C}}=\sqrt{\mu\epsilon}\]
We can find the length of the rising edge from the switching time and the propogation delay. \[l_{SW}=\frac{t_{SW}}{D}\] Systems with physical dimensions similar to or longer than \(l_{SW}\) can suffer from distortion.
If we use a distributed model utilising multiple smaller versions of the lumped model, we can see that propagation delay begins to arise. Using a distributed model provides higher fidelity than the lumped circuit model. This is because the resistance, capacitance, inductance and conductance are distributed along the wire length, not lumped together. The distributed model is constructed from multiple smaller lumped models in series. The model has a series of components consisting of the amount per unit length over a length \(dx\). We can take the resistance of the end of the wire as \(Z(x)\) and add it as a load across another unit of the wire \(dx\). This gives resistance \(\hat{R}dx\) and inductance \({\hat{L}dx}\) in series with conductance \(\hat{G}dx\), capacitance \(\hat{C}dx\) and \(Z(x)\) in parallel. With a bit of working we get: \[\frac{Z(x+dx)-Z(x)}{dx}=\hat{Y}_{shunt}[Z(x+dx)]^2-\hat{Z}_{series}\] Where \(\hat{Z}_{series}=\hat{R}+s\hat{L}\) and \(\hat{Y}_{shunt}=\hat{G}+s\hat{C}\). This is a derivative in the limit so we get \[\frac{dZ(xz)}{dx}=\hat{Y}_{shunt}[Z(x)^2]-\hat{Z}_{series}\] From this we can get that \(Z(x+dx)=Z(x)=C, \forall x\), where the constant is the characteristic impedance of an infinite length of wire. \[Z_o=\sqrt{\frac{\hat{Z}_{series}}{\hat{Y}_{shunt}}}=\sqrt{\frac{\hat{R}+s\hat{L}}{\hat{G}+s\hat{C}}}\] We can substitute \(s=j\omega\) to express the impedance as a function of frequency. \[Z_o(j\omega)=\sqrt{\frac{\hat{R}\hat{G}+\omega^2\hat{L}\hat{C}+j\omega(\hat{L}\hat{G}-\hat{R}\hat{C})}{\hat{G}^2+\omega^2\hat{C}^2}}\] We can approximate the impedance at low frequencies as: \[Z_o(j\omega)=\sqrt{\frac{\hat{R}}{\hat{G}}}\] And at high frequencies as: \[Z_o(j\omega)=\sqrt{\frac{\hat{L}}{\hat{C}}}\]
When working with a finite length cable, there overall equation modelling the wire and load is the same as the infinite case. The load provides a boundary condition: \[Z(\ell)=Z_{load}\] We can use this to find the impedance of the wire as \(Z(0)=Z_{wire}\). We can find the impedance of a finite length of wire as: \[Z_{wire}(s)=Z_o(s)\left(\frac{1+\rho e^{-2\ell\Gamma(s)}}{1-\rho e^{-2\ell\Gamma(s)}}\right)\] \(\ell\) is the length of the wire and \(\rho\) is the reflection coefficient: \(\rho(s)=\frac{Z_{load}-Z_o}{Z_{load}+Z_s}\). \(Z_o\) is the characteristic impedance and \(\Gamma(s)=\sqrt{Z_{series}Y_{shunt}}=D_{lossless}(s)=s/v,D_{lossless}=\sqrt{\hat{L}\hat{C}}\). In the lossless case, \(Z_o=\sqrt{\frac{L}{C}}\). In the lossy case: \[\Gamma(s)=\sqrt{Z_{series}Y_{shunt}}=D_{lossless}\sqrt{s^2+s\left(\frac{\hat{R}}{\hat{L}}+\frac{\hat{G}}{\hat{L}}\right)+\frac{\hat{R}\hat{G}}{\hat{L}\hat{C}}}\] The wire can reflect the input signals at the end, causing resonance. To mitigate this, we can decrease the input frequency or reduce the length of the wire, however these are necessitated by the application. Alternately we can set the load resistance to avoid reflection. This happens when the load impedance is equal to the characteristic impedance of the wire. This causes the load seen from the wire to be equal to the characteristic impedance of the wire, causing the load to seem like an infinite length of the cable, where reflection isn't possible. This sets \(\rho=0\) in the calculation of the impedance of a finite length wire.
A semiconductor is composed of a p and n type material connected together. Where they meet, they form a junction. Semiconductors can be used to implement nonlinear functions, like in diodes and transistors. When grouped together, they can form complex components like logic gates and op amps. Semiconductors have intermediate values of conductivity. The elemental form of a semiconductor is its intrinsic form, when mixed with another element, it becomes an extrinsic form. The band gap of a material is the region of energies where there no electrons are located. Surrounding the band gap is the valence band of immobile electrons and conductive band of mobile electrons. An insulator has a large gap between its valence and conduction bands, a metal has a conduction band that partially overlaps with its valence band. Semiconductors have a small difference between the valence and conduction bands. A dopant is an element that is added to the intrinsic semiconductor to reduce the band gap. A group V dopant adds a donor level just below the conduction band, resulting in mobile electrons in the conduction band. A group III dopant adds an acceptor level just above the valence band, resulting in mobile holes in the valence band. Both of these increase the conductivity of the resultant extrinsic semiconductor. We can model the mobile electron concentration with: \[\frac{dn(t)}{dt}=\alpha[\underbrace{n_i(T)^2}_{\text{generation rate}}-\underbrace{n(t)p(t)}_{\text{recombination rate}}]+r_{in}(t)\] Here \(n\) is the mobile electron concentration, \(p\) is the hole concentration, \(n_i\) is the intrinsic mobile electron concentration, \(r_{in}\) is the electron injection rate, \(T\) is the temperature and \(t\) is the time. The number of holes is: \[p(t)=n(t)+N_A-N_D\] At equilibrium, \[\frac{dn}{dt}=\frac{dp}{dt}=0\] \[0=\alpha[n_i(T)^2-\overline{n}\overline{p}]+\overline{r}_{in}\] If there is no injection, \(\overline{n}\overline{p}=n_i(T)^2\). From this we get the simultaneous equations for the equilibrium generation rates and the space charge neutrality. \[\overline{n}\overline{p}=n_i^2\] \[\overline{n}-\overline{p}=N_D-N_A\] Solving these allows us to fund the concentrations of holes and mobile electrons. The concentration of mobile electrons is approximately the concentration of the group V dopant. A group V dopant decreases the concentration of holes.
In the presence of an external electric field, electrons will be promoted to the conduction band, where it can move. After a bit it will return to the valence band. The moving of electrons can be seen as the movement of holes also. This is drift current. This current can be characterised by Ohm's law. \[\mathbf{J}=\sigma\mathbf{E}\] Where \(\sigma=q\mu n\) as holes and electrons will have differing conductivities.
There can also be diffusion currents, due to the random motion of particles. This is the migration of charges from high charge density regions to low charge density regions. We can characterise this current from the charge gradient. \[\mathbf{J}_n(\text{diff})=qD_n\nabla n\] \[\mathbf{J}_p(\text{diff})=-qD_p\nabla p\] Where \(D=\left(\frac{kT}{q}\right)\mu\).
The mobility of electrons is always greater than holes as for a hole to move, an electron must become freed, move and reenter the valence band. This cause \(\sigma_p<\sigma_n\), which can affect circuit designs.
Overall we can describe the current density due to electrons and holes as: \[\mathbf{J}_n=\overbrace{qn\mu_n}^{\sigma_n}\mathbf{E}+qD_n\nabla n\] \[\mathbf{J}_p=qp\mu_n\mathbf{E}-qD_p\nabla p\] The total current density is the sum of each currents; \[\mathbf{J}=\mathbf{J}_n+\mathbf{J}_p\]
We can also express the current density from conservation of charge. In doing so, we need to account for the EHP generation/recombination. \[\operatorname{div }\mathbf{J}=\frac{d\rho}{dt}+(\text{EHP term})\] Using the charge densities \(\rho_p=q(p+N_D)\) and \(\rho_n=q(n+N_A)\), we can get \[\frac{1}{q}\operatorname{div }\mathbf{J}_p=-\frac{\partial p}{\partial t}+G_p-R_p\] \[-\frac{1}{q}\operatorname{div }\mathbf{J}_n=-\frac{\partial n}{\partial t}+G_n-R_n\] Here the EHP term is the net generation. We can express the rate as; \[G_n-R_n=\alpha(n_i(T^2)-np)\approx-\alpha(\overline{n}+\overline{p})\delta n=-\frac{\delta n}{\tau_r}\] Where \(\overline{n}\) and \(\overline{p}\) are the linearised electron and holes, \(\delta n\) is the perturbation of \(n\) and \(\tau_r=\frac{1}{\alpha(\overline{n}+\overline{p})}\) is the recombination lifetime. Depending on the material, we can ignore the other term in the denominator due to the difference in magnitude, i.e. for an n type, \(\tau_r\approx\tau_n=\frac{1}{\alpha\overline{n}}\) This gives the following continuity equations: \[\frac{1}{q}\operatorname{div }\mathbf{J}_p=-\frac{\partial\delta p}{\partial t}-\frac{\delta p}{\tau_p}\] \[-\frac{1}{q}\operatorname{div }\mathbf{J}_n=-\frac{\partial\delta n}{\partial t}-\frac{\delta n}{\tau_n}\] With these we can characterise the diffusion equation due to the motion of random carriers; \[\mathbf{J}_n=qD_N\nabla \delta n\] And find the diffusion equation for mobile electrons \[\frac{\partial\delta n}{\partial t}=D_n\nabla^2(\delta n)-\frac{\delta n}{\tau_n}\]
To analyse at steady state, we get: \[0=D_n\frac{d^2\delta p}{dx^2}-\frac{\delta p}{\tau_p}\] This is subject to the boundary condition \(\begin{cases}\delta p(0)=\Delta p\\\delta p(\infty)=0\end{cases}\) and assumes the electric field is 0. Solving this with a characteristic equation gives \(s=\pm\frac{1}{L_p}\), where \(L_p=\sqrt{D_p\tau_p}\), being the diffusion length. This gives the overall solution to the equation as: \[\delta p(x)=\Delta pe^{-x/L_p}\] The diffusion length is the where \(e^{-1}\) of the holes are recombined. This gives the current density as steady state as: \[\mathbf{J}_p=-qD_p\nabla p=\frac{qD_p\Delta p}{L_p}e^{-x/L_p}\hat{x}\] Which results in a current of: \[I_p(x)=\iint_A\mathbf{J}_p\dot d\mathbf{A}=\left(\frac{qAD_p}{L_p}\right)\Delta pe^{-x/L_p}\] And a total current of: \[I=I_p(0)=\left(\frac{qAD_p}{L_p}\right)\Delta p\]
A junction is a contact surface between two materials. Common types are p-n junctions, between a p-type and n-type semiconductor and Schottky junctions between semiconductors and metals. Junctions implement rectification, meaning that they only allow current in one direction. Diodes can be modelled by the Shockley Diode Equation: \[I=I_0e^{\frac{qV}{kT}-1}\] At the a junction there is a high concentration of electrons in the n-type conductor and a high concentration of holes in the p-type. As a result, there is a flow of holes into the n-type and electrons into the p-type. This reduces the carrier gradient on either side of the junction. This leaves an immobile depletion region with excess positive charge in the n-type and excess negative in the p-type. After a distance from the junction, there is an electrically neutral region. The depletion region has a dipole, creating an electric field from the n-type to the p-type. As a result it takes energy to move a hole from the p-type to the n-type. At equilibrium there is no current, so we can characterise the electric field with: \[0=qn\mu_n\mathbf{E}+qD_n\nabla n\] \[0=qn\mu_p\mathbf{E}-qD_p\nabla n\] This gives (for holes): \[\mathbf{E}=\frac{D_p}{\mu_p}\left(\frac{1}{p}\nabla p\right)\]
We can model the contact potential as the potential along the depletion region. \[V_o=-\int_\ell\mathbf{E}\cdot d\ell=V(x_n)-V(x_p)\] \[\ell(s)=[-x_p+s(x_n+x_p)]\hat{x}\] Where \(\ell\) is any path from the p-side of the depletion region to the n-side of the depletion region. \[V_o=\frac{D_p}{\mu_p}\log\left(\frac{p(\ell(0))}{p(\ell(1))}\right)\] This gives: Knowing that \(p(\ell(0))=p_p\) and \(p(\ell(1))=p_n\) and that in a p-type material \(p_p\approx N_A\) and \(n_n\approx N_D\) gives the contact potential as: \[V_o=\frac{D_p}{\mu_p}\log\left(\frac{p_p}{p_n}\right)=\frac{kT}{q}\log\left(\frac{p_p}{p_n}\right)\approx\frac{kT}{q}\log\left(\frac{N_AN_D}{n_i^2}\right)\] Using some standard values for silicon at 300 K gives the contact potential at \(V_o\approx0.754 V\).
Applying an external DC source disrupts the equilibrium between the drift and diffusion currents. Forward bias applies an external field oriented from p to n, reducing the internal electric field and letting diffusion current dominate. Reverse bias applies an external field oriented from n to p, increasing the internal electric field and letting drift current dominate. The reverse current requires electrons arriving from p-type material and holes from n-type, which is limited, limiting the overall amount of current. This is relatively independent of applied voltage.
Applying a bias voltage gives the potential over the junction as: \[V_o-V=\frac{kT}{q}\log\left(\frac{p_p^V}{n^V}\right)\] This and the equation for the contact potential gives: \[\frac{p_p}{p_n}=\exp\left(\frac{qV_o}{kT}\right)\] \[\frac{p_p^V}{p_n^V}=\exp\left(\frac{q(V_o-V)}{kT}\right)\] We can assume that the bias does not overly affect the mobile hole concentration, i.e. \(p_p^V=p_p\), and similarly for the electrons. We can find the injected carrier concentrations as: \[\Delta p_n^V=p_n\left[\exp\left(\frac{qV}{kT}\right)-1\right]\] \[\Delta n_p^V=n_p\left[\exp\left(\frac{qV}{kT}\right)-1\right]\] Forward biasing gives rise to minority carrier injection across the junction. This causes a minority carrier concentration gradient either side of the depletion region, where holes are injected in to the n-type region and electrons into the p-type. These charge gradients give rise to diffusion currents, from a diffusion of minority holes in n-type and electrons in p-type. The total diode current is the sum of the two diffusion current components. The diffusion current from injected holes is: \[I_p(x)=\left(\frac{qAD_p}{L_p}\right)\Delta p_n^V\exp\left(-\frac{x-x_n}{L_p}\right)\] And the diffusion current due to the injected electrons: \[I_n(x)=\left(\frac{qAD_n}{L_n}\right)\Delta n_p^V\exp\left(-\frac{x-x_p}{L_n}\right)\] So the sum of the two is: \[I=I_o\left[\exp\left(\frac{qV}{kT}\right)-1\right]\] Where \(I_o=qA\left(\frac{D_pp_n}{L_p}+\frac{D_nn_p}{L_n}\right)\).