From 753a61d747db2ef2bf47e7fdd852f435701879f5 Mon Sep 17 00:00:00 2001 From: Erik Strand <erik.strand@cba.mit.edu> Date: Thu, 9 May 2019 14:58:27 -0400 Subject: [PATCH] Add some schematics and reword for clarity --- _psets/12.md | 93 +- assets/img/pset12_differential_amplifier.svg | 705 +++++++++++++++ assets/img/pset12_differentiator.svg | 556 ++++++++++++ assets/img/pset12_integrator.svg | 555 ++++++++++++ assets/img/pset12_non-inverting_amplifier.svg | 566 ++++++++++++ assets/img/pset12_summing_amplifier.svg | 813 ++++++++++++++++++ .../img/pset12_transimpedance_amplifier.svg | 527 ++++++++++++ 7 files changed, 3782 insertions(+), 33 deletions(-) create mode 100644 assets/img/pset12_differential_amplifier.svg create mode 100644 assets/img/pset12_differentiator.svg create mode 100644 assets/img/pset12_integrator.svg create mode 100644 assets/img/pset12_non-inverting_amplifier.svg create mode 100644 assets/img/pset12_summing_amplifier.svg create mode 100644 assets/img/pset12_transimpedance_amplifier.svg diff --git a/_psets/12.md b/_psets/12.md index ca92e61..c05aeb9 100644 --- a/_psets/12.md +++ b/_psets/12.md @@ -9,12 +9,17 @@ title: Problem Set 12 {:.question} Show that the circuits in Figures 15.1 and 15.2 differentiate, integrate, sum, and difference. -I'll take as given that op amps with negative feedback keep their inputs at the same voltage. I also -assume ideal op-amps that draw no current from their inputs. All the proofs follow from [Kirchhoff's -current law](https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws) (aka conservation of charge). -I write $$R_f$$ to refer to the resistor in the feedback path. +I'll take as given that op amps with negative feedback keep their two input terminals at the same +voltage. I also assume ideal op-amps that draw no current from their inputs. All the proofs follow +from [Kirchhoff's current law](https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws) (aka +conservation of charge). I write $$R_f$$ to refer to the resistor in the feedback path. All +schematics are from Wikimedia Commons. -For the integrator, +#### Integrator + + + +For the integrator, current through the resistor must equal current through the capacitor. $$ \begin{align*} @@ -24,26 +29,41 @@ V_o &= - \frac{1}{RC} \int V_i \mathrm{d}t \end{align*} $$ -For the differentiator, +#### Differentiator + + + +For the differentiator, current through the capacitor must equal current through the resistor. $$ \begin{align*} -\frac{-V_o}{R} &= C \frac{dV_i}{dt} \\ +C \frac{dV_i}{dt} &= \frac{-V_o}{R} \\ V_o &= -RC \frac{dV_i}{dt} \end{align*} $$ -For the summing amplifier, +#### Summing Amplifier + + + +For the summing amplifier, the sum of the currents through the input resistors must equal the +current through the feedback resistor. $$ \begin{align*} -\frac{V_1 + V_2 + V_3}{R_i} &= -\frac{V_o}{R_f} \\ -V_o &= -\frac{R_f}{R_i} (V_1 + V_2 + V_3) +\frac{V_1 + \cdots + V_n}{R_i} &= -\frac{V_o}{R_f} \\ +V_o &= -\frac{R_f}{R_i} (V_1 + \cdots + V_n) \end{align*} $$ -For the difference amplifier, neither input terminal is grounded. Call their voltage $$V_i$$. -Applying Kirchhoff's current law to the non-inverting input, +#### Differential Amplifier + + + +For the difference amplifier, neither input terminal is grounded, so we can't assume they're at zero +volts. Call their voltage $$V_i$$. I'll also assume both resistors $$R_1$$ and $$R_2$$ in the above +schematic are the same, which I'll call $$R_i$$. Applying Kirchhoff's current law to the +non-inverting input, $$ \begin{align*} @@ -68,10 +88,11 @@ $$ {:.question} Design a non-inverting op-amp amplifier. Why are they used less commonly than inverting ones? + + Connect the signal you want to amplify to the non-inverting input. Then create a voltage divider, connected to ground on one end, the inverting input in the middle, and the output on the other end. -I'll refer to the resistor connected to ground as $$R_g$$, and the feedback resistor $$R_f$$ as -before. +I'll call the resistor connected to ground $$R_g$$, and the feedback resistor $$R_f$$ as before. $$ \begin{align*} @@ -81,8 +102,9 @@ V_o &= V_i \left( 1 + \frac{R_f}{R_g} \right) \end{align*} $$ -A potential downside is that the gain must be at least unity. A potential upside is that the input -impedance is very high. And, of course, the output isn't inverted. +A potential downside is that the gain must be at least unity. Additionally the nodes aren't tied to +ground, so if the op-amp has imperfect common-mode rejection we'll see it in the output. A potential +upside is that the input impedance is very high. And, of course, the output isn't inverted. ### (c) @@ -90,6 +112,8 @@ impedance is very high. And, of course, the output isn't inverted. Design a transimpedance (voltage out proportional to current in) and a transconductance (current out proportional to voltage in) op-amp circuit. + + For a transimpedance amplifier, replace the input resistor in an inverting amplifier with a wire. Then the current flowing through the feedback resistor is equal to the current at the input. So $$I = V_o / R_f$$, and $$V_o = R_f I$$. @@ -336,7 +360,7 @@ D_{ij} = \end{align*} $$ - + ### (c) @@ -346,7 +370,7 @@ Plot the inverse transform of the {$$f_i$$} by multiplying them by the inverse o The original samples are recovered. - + ### (d) @@ -356,7 +380,7 @@ investigate the dependence on the sample size. Here I've selected 100 samples from the original 250. The plot is recognizable but very distorted. - + ### (e) @@ -375,12 +399,12 @@ and plot the resulting estimated coefficients. Gradient descent very quickly drives the loss function to zero. However it's not reconstructing the true DCT coefficients. - + To make sure I don't have a bug in my code, I plotted the samples we get by performing the inverse DCT on the estimated coefficients. - + Sure enough all samples in the subset are matched exactly. But the others are way off the mark. We've added a lot of high frequency content, and are obviously overfitting. @@ -405,7 +429,7 @@ and plot the resulting estimated coefficients. With L2 regularization, we remove some of the high frequency content. This makes the real peaks a little more prominent. - + However it comes at a cost: gradient descent no longer drive the loss to zero. As such the loss itself isn't a good termination condition. In its place I terminate when the squared norm of the @@ -414,7 +438,7 @@ around 50. You can easily see that the loss is nonzero from the reconstructed samples. - + ### (g) @@ -434,22 +458,25 @@ dependence of the results on M to the Nyquist sampling limit of twice the highes With L1 regularization the DCT coefficients are recovered pretty well. There is no added high frequency noise. - + It still can't drive the loss to zero. Additionally it's hard to drive the squared norm of the gradient to zero, since the gradient of the absolute values shows up as 1 or -1. (Though to help prevent oscillation I actually drop this contribution if the absolute value of the coefficient in question is less than $$\num{1e-3}$$.) So here I terminate when the relative change in the loss -falls below $$\num{1e-9}$$. +falls below $$\num{1e-9}$$. I also decay the learning rate during optimization. It starts at 0.1 and +is multiplied by 0.99 every 64 iterations. -The final loss is around 40; smaller than we were able to find with L2 regularization. However it -did take more effort: this version converged after 21,060 iterations, as opposed to 44 (for L2) or -42 (for unregularized). +The final loss is around 40, so better than we found with L2 regularization. However it did take +longer to converge: this version stopped after 21,060 iterations, as opposed to 44 (for L2) or 42 +(for unregularized). If I stop it after 50 samples it's a bit worse than the L2 version (loss of 55 +instead of 50). It's not until roughly 2500 iterations that it's unequivocally pulled ahead. I +played with learning rates and decay schedules a bit, but there might be more room for improvement. -The recovered samples are also much more recognizable. The amplitude of our waveform seems overall -a bit diminished, but unlike our previous attempts it looks similar to the original. +The recovered samples are also much more visually recognizable. The amplitude of our waveform seems +overall a bit diminished, but unlike our previous attempts it looks similar to the original. - + This technique can recover the signal substantially below the Nyquist limit. The highest frequency signal is 1209 Hz, so with traditional techniques we'd have to sample at 2418 Hz or faster to avoid @@ -458,5 +485,5 @@ So my original 250 is (not coicidentally) near here. 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