From b3730b7b3d2330b4af45d45421ac06a9bbef9a11 Mon Sep 17 00:00:00 2001 From: Erik Strand <erik.strand@cba.mit.edu> Date: Thu, 2 May 2019 00:32:51 -0400 Subject: [PATCH] Answer 14.4 --- _psets/11.md | 71 +++++++++++++++++++++++++++++++ _sass/main.scss | 4 ++ assets/img/pset11_im_full.jpg | Bin 0 -> 17732 bytes assets/img/pset11_im_zoom.jpg | Bin 0 -> 18966 bytes assets/img/pset11_phase_full.jpg | Bin 0 -> 16893 bytes assets/img/pset11_phase_zoom.jpg | Bin 0 -> 20186 bytes assets/img/pset11_re_full.jpg | Bin 0 -> 15243 bytes assets/img/pset11_re_zoom.jpg | Bin 0 -> 18686 bytes 8 files changed, 75 insertions(+) create mode 100644 assets/img/pset11_im_full.jpg create mode 100644 assets/img/pset11_im_zoom.jpg create mode 100644 assets/img/pset11_phase_full.jpg create mode 100644 assets/img/pset11_phase_zoom.jpg create mode 100644 assets/img/pset11_re_full.jpg create mode 100644 assets/img/pset11_re_zoom.jpg diff --git a/_psets/11.md b/_psets/11.md index 53432ac..7803f38 100644 --- a/_psets/11.md +++ b/_psets/11.md @@ -47,6 +47,77 @@ Typical parameters for a quartz resonator are $$C_e = 5 \si{pF}$$, $$C_m = 20 \s \si{mH}$$, $$R_m = 6 \si{\ohm}$$. Plot, and explain, the dependence of the reactance (imaginary part of the impedance), resistance (real part), and the phase angle of the impedance on the frequency. +The circuit in question is depicted in figure 14.4. There's a capacitance $$C_e$$ in parallel with +a series RLC circuit ($$R_m$$, $$L_m$$, and $$C_m$$). To solve this it's helpful to know the complex +impedances of the basic electrical components: + +$$ +\begin{align*} +Z_C &= \frac{1}{i \omega C} \\ +Z_L &= i \omega L \\ +Z_R &= R +\end{align*} +$$ + +Then we just combine these, using the sum for series connections, and the inverse of the sum of +inverses for parallel connections. So the total impedance $$Z$$ is + +$$ +\begin{align*} +\frac{1}{Z} &= \frac{1}{Z_{C_e}} + \frac{1}{Z_{C_m} + Z_{L_m} + Z_{R_m}} \\ +&= i \omega C_e + \frac{1}{1/(i \omega C_m) + i \omega L_m + R_m} \\ +&= i \omega C_e + \frac{\omega C_m}{-i + \omega C_m (R_m + i \omega L_m)} \\ +Z &= \frac{1}{i \omega C_e + \frac{\omega C_m}{\omega C_m (R_m + i \omega L_m) - i}} +\end{align*} +$$ + +Let's see what this looks like with the given values plugged in. All the x axes are in radians per +second. + +The resistance (real part) is very small except for a spike near $$\num{4e6} \si{rad/s}$$. + + + +Mathematica tells me the max occurs at + +$$ +\begin{align*} +\omega_\text{max} &= \frac{1}{L_m} \sqrt{\frac{2 C_e L_m + 2 C_m L_m -C_e C_m R_m^2}{2 C_e C_m}} \\ +&\approx \num{4.09e6} \si{rad/s} +\end{align*} +$$ + +But the first term in the numerator is a couple orders of magnitude larger than the others, so we +can approximate $$\omega_\text{max}$$ by dropping them. + +$$ +\begin{align*} +\omega_\text{max} &\approx \frac{1}{L_m} \sqrt{\frac{2 C_e L_m}{2 C_e C_m}} \\ +&= \frac{1}{\sqrt{C_m L_m}} \\ +&\approx \num{4.08e6} \si{rad/s} +\end{align*} +$$ + +The reactance decreases without bound as $$\omega$$ approaches zero, and tends toward 0 as +$$\omega$$ increases. It has a kink in about the same place the resistance has a spike. + + + +Finally the phase is almost always $$-\pi$$, except where it shoots just above one (radian), again +near $$\omega = 1/\sqrt{C_m L_m}$$. + + + +Let's zoom in on those spikes. + + + + + +$$ +\begin{align*} +\end{align*} +$$ ## (14.5) diff --git a/_sass/main.scss b/_sass/main.scss index 504e663..0dc99a8 100644 --- a/_sass/main.scss +++ b/_sass/main.scss @@ -46,3 +46,7 @@ table, th, td { th, td { padding: 5px 10px; } + +img { + width: 100%; +} diff --git a/assets/img/pset11_im_full.jpg b/assets/img/pset11_im_full.jpg new file mode 100644 index 0000000000000000000000000000000000000000..dbbb893241a92bee71ab0cedd9f8edfc850eb434 GIT binary patch literal 17732 zcmex=<NpH&0WUXCHwH#VMg|WC4+e(+4;d<4D>Bm<7(6|-7&sUh7#JB08JQSD85kI( z7$Cs8C^fMpHASI3vm`^IBqLQJJU=I`C^1*TCowO*G%-C@AuYd1A-JS8B{i?4SV31I zAhoDCKQA#yA+$JE!9OpjQo%~0yu4hm9HLJzIX{<y;s0$0X9f;7Hg+~v4t92SPEHOk z9uZz1Zf+h)Az^+ISt)rrSt%JA1r<GY1tlG285s?8O&tS66B848bxT_dBO5(q6C;oz 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