diff --git a/_psets/5.md b/_psets/5.md new file mode 100644 index 0000000000000000000000000000000000000000..40f3eb62afb1852ffd56beb11bd1a9d9320cfd78 --- /dev/null +++ b/_psets/5.md @@ -0,0 +1,250 @@ +--- +title: Problem Set 5 +--- + +## (7.1) + +{:.question} +Cables designed to carry signals with minimum pickup of interference often consist of a twisted pair +of conductors surrounded by a grounded shield. Why the twist? Why the shield? + +Let's answer this from the ground up. The simplest way to transfer a signal is to use a single wire, +where a voltage is applied at one end and measured at the other. In practice this requires at least +some current flow, so there must be a return path which for now I'll assume is literal ground. The +voltage we measure will be different than the voltage we apply if and only if the path integral of +the electric field along our wire is nonzero. + +A stationary electric field won't have any effect, since it will quickly induce a charge +distribution that cancels itself (i.e. the field inside a conductor is zero). But a quickly changing +one can induce a voltage, since the rate of charge transfer within the conductor is finite. This is +capacitive coupling. + +Similarly, a stationary magnetic field will only apply a net force on electrons perpendicular to the +current. This induces a [Hall effect](https://en.wikipedia.org/wiki/Hall_effect) but won't impact +our measurement. But a varying magnetic field (or moving the wire through a fixed one) can induce a +voltage because it causes curl in the electric field. This is inductive coupling. + +How can we mitigate these effects? If we surround our signal wire with another conductor, the +capacitively induced voltages will mostly appear there. This is a shield. The effectiveness of the +shield depends on its conductivity, its thickness, and the frequency of the interference (recall +that oscillating electric fields penetrate conductors up to a skin depth). It doesn't necessarily +need to be grounded to do its job, but doing so ensures it can source enough charge to counteract +even the strongest external fields, and dissipate static buildup. + +The inductive coupling is minimized by minimizing the magnetic flux between our signal wire and the +current return path. So if we use a second wire for our return path, and keep it next to the first, +there's much less area for flux to penetrate. By twisting the pair of wires, we create alternating +regions where the flux will have opposite sign. This causes inductively coupled signals to cancel +themselves out. If we want to take this even further, we could use +[star quad](https://en.wikipedia.org/wiki/Star_quad_cable) wire. + +Continuing with the theme of noise cancellation, if the drive, line, and load impedances are matched +in the two wires, then any noise signals that do couple will tend to cause equal voltage +fluctuations. This is why differential signaling is common. Note that this works even if we don't +put equal and opposite voltages on the wires: we just need matching electrical characteristics (and +the assumption of equal noise patterns). But applying equal and opposite voltages to the wires does +help in another way: it causes the fields generated by our wires to cancel each other out except at +small distances, so it helps our wires not induce noise in any other signal lines. + + +## (7.2) + +{:.question} +Salt water has a conductivity ∼4 S/m. What is the skin depth at $$10^4$$ Hz? + +$$ +\begin{align*} +\delta &= \frac{1}{\sqrt{\pi \nu \mu \sigma}} \\ +&= \frac{1}{\sqrt{\pi \cdot 10^4 \si{Hz} \cdot \num{1.3e-6} \si{H/m} \cdot 4 \si{S/m}}} \\ +&= 2.5 \si{m} +\end{align*} +$$ + + +## (7.3) + +{:.question} +Integrate Poynting’s vector $$P = E \times H$$ to find the power flowing across a cross-sectional +slice of a coaxial cable, and relate the answer to the current and voltage in the cable. + +We saw that the magnetic field has a magnitude of $$I/(2 \pi r)$$ and wraps around the inner +conductor according to the right hand rule. The electric field has magnitude $$Q/(2 \pi \epsilon +r)$$ and points radially from the inner conductor toward the outer. So $$P = E \times H$$ points +along the direction of current in the inner conductor, and has magnitude + +$$ +| P | = \frac{I Q}{(2 \pi r)^2 \epsilon} +$$ + +Integrating over the area between the conductors, we find that the cross-sectional power transfer is + +$$ +\begin{align*} +\int_0^{2 \pi} \int_{r_i}^{r_o} \frac{I Q}{(2 \pi r)^2 \epsilon} r \mathrm{d} r \mathrm{d} \theta +&= \frac{I Q}{2 \pi \epsilon} \int_{r_i}^{r_o} r^{-1} \mathrm{d} r \mathrm{d} \theta \\ +&= \frac{I Q}{2 \pi \epsilon} \ln \left( \frac{r_o}{r_i} \right) +\end{align*} +$$ + +But the charge is + +$$ +Q = CV = \frac{2 \pi \epsilon}{\ln (r_0 / r_i)} +$$ + +so we can write the total power as $$IV$$. + + +## (7.4) + +{:.question} +Find the characteristic impedance and signal velocity for a transmission line consisting of two +parallel strips with a width $$w$$ and a separation $$h$$. You can ignore fringing fields by +assuming that they are sections of conductors infinitely wide. + +Take the plates to have length one. Suppose the bottom plate has a charge $$Q$$, and the top plate a +charge of $$-Q$$. Then as we found last week the field between the plates will have magnitude +$$Q/(\epsilon w)$$ (since the charge density is the total charge divided by area $$1 \cdot w$$), and +point from the bottom plate toward the top plate. Integrating from the bottom plate to the top one, +we find that the voltage difference is $$Qh/(\epsilon w)$$. Thus the capacitance (charge per +voltage) per length is + +$$ +C = \frac{\epsilon w}{h} +$$ + +By symmetry the magnetic field must point perpendicular to the direction of current travel and +parallel to the surfaces of the plates. Consider a square perpendicular to the current, with a width +of $$w$$, bisected by the lower plate. Ampère's Law tells us that the integral of the +magnetic field along the boundary of this square is $$I$$. So we deduce that the magnetic field has +magnitude $$I/w$$. Integrating over the surface perpendicular to the field between the plates, we +find that the flux is $$\mu_0 I h / w$$. Thus the inductance (flux per current) per length is + +$$ +L = \frac{\mu_0 h}{w} +$$ + +From here we can use the results derived in the chapter. + +$$ +v = \frac{1}{\sqrt{L C}} = \frac{1}{\sqrt{\mu_0 \epsilon}} +$$ + +$$ +Z = \sqrt{\frac{L}{C}} = \frac{\mu_0 h^2}{\epsilon w^2} +$$ + + +## (7.5) + +{:.question} +The most common coaxial cable, RG58/U, has a dielectric with a relative permittivity of 2.26, an +inner radius of 0.406 mm, and an outer radius of 1.48 mm. + +### (a) + +{:.question} +What is the characteristic impedance? + +$$ +\begin{align*} +Z &= \sqrt{\frac{L}{C}} \\ +&= \sqrt{\frac{\mu_0}{2 \pi} \ln \left( \frac{r_o}{r_i} \right) + \frac{1}{2 \pi \epsilon_0 \epsilon_r} \ln \left( \frac{r_o}{r_i} \right)} \\ +&= \frac{1}{2 \pi} \sqrt{\frac{\mu_0}{\epsilon_0 \epsilon_r}} \ln \left( \frac{r_o}{r_i} \right) \\ +&= 51.6 \si{\ohm} +\end{align*} +$$ + +### (b) + +{:.question} +What is the velocity? + +$$ +\begin{align*} +v &= \frac{1}{\sqrt{L C}} \\ +&= \frac{1}{\sqrt{\frac{\mu_0 \ln(r_o / r_i)}{2 \pi} + \frac{2 \pi \epsilon_0 \epsilon_r}{\ln( r_o / r_i )}}} \\ +&= \frac{1}{\sqrt{\mu_0 \epsilon_0 \epsilon}} \\ +&= \num{2e8} \si{m/s} +\end{align*} +$$ + +This is two thirds the speed of light. + +### (c) + +{:.question} +If a computer has a clock speed of 1 ns, how long can a length of RG58/U be and still deliver a +pulse within one clock cycle? + +$$ +\num{1e-9} \si{s} \cdot \num{2e8} \si{m/s} = 0.2 \si{m} +$$ + +### (d) + +{:.question} +It is often desirable to use thinner coaxial cable to minimize size or weight but still match the +impedance of RG58/U (to minimize reflections). If such a cable has an outer diameter of 30 mils (a +mil is a thousandth of an inch), what is the inner diameter? + +Solving the formula above for $$r_i$$, + +$$ +\begin{align*} +r_i &= r_0 \cdot e^{-2 \pi \cdot 51.6 \sqrt{\epsilon_0 \epsilon_r / \mu_0}} \\ +&= 8.2 \si{mils} +\end{align*} +$$ + +### (e) + +{:.question} +For RG58/U, at what frequency does the wavelength become comparable to +the diameter? + +$$ +\begin{align*} +\nu &= \frac{c}{\lambda} \\ +&= \frac{\num{2e8} \si{m/s}}{2 \cdot \num{1.48e-3} \si{m}} \\ +&= \num{6.76e10} \si{Hz} +\end{align*} +$$ + + +## (7.6) + +{:.question} +CAT6 twisted pair cable used in ethernet networks has a propagation delay of 4.6 ns/m, and an +impedance of 100 ohms. + +### (a) + +{:.question} +What is the physical length of a minimum size 64 byte frame? + +The inverse of the propagation delay is the velocity, so the signal travels at $$0.22 \si{m/ns}$$. +Assuming we need one nanosecond clock cycle per bit, the length of a 64 byte frame is + +$$ +64 \si{bytes} \cdot \frac{8 \si{bits}}{1 \si{byte}} \cdot \frac{1 \si{ns}}{1 \si{bit}} +\cdot \frac{1 \si{m}}{4.6 \si{ns}} = 111 \si{m} +$$ + +### (b) + +{:.question} +Now consider what would happen if a “T” connector was used to connect one CAT6 cable to two other +ones. Estimate the reflection coefficient for a signal arriving at the T. + +The two connected CAT6 cables will look like two parallel impedances. So we can take $$Z_0$$ to be +100 $$\si{\ohm}$$, and $$Z_L$$ to be 50 $$\si{\ohm}$$. Then + +$$ +\begin{align*} +R &= \frac{Z_L - Z_0}{Z_L + Z_0} \\ +&= -\frac{1}{3} +\end{align*} +$$