# 20.309:Homeworks

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**Homework Set 1**
Due by 12:00 noon on Friday Sept. 21, 2007

shows a resistor network known as a Wheatstone Bridge, which will be part of apparatus that you'll build shortly for the DNA melting curves module. You can read a bit more about it in the handout for that module. For now, we will look at it analytically. R_x is a resistance you are trying to measure, and R_3 is a variable resistor.

Assuming $R_3$ is set such that the bridge is balanced (i.e. $V_{ab}=0$), derive an analytical expression for $R_x$ in terms of $R_1$, $R_2$ and $R_3$. \item[(b)] Now let $R_3$ also be a fixed value, and suppose that $R_x$ varies in a way that makes $V_{ab}$ nonzero. Derive an expression for the dependence of $V_{ab}$ on $R_x$

Referring again to the Wheatstone Bridge in Figure \ref{fig:hw1fig1}, suppose that $R_x$ varies with some physical parameter (strain, temperature, etc.) in the range of 1-10k$\ohm$. You want to use the circuit to measure the physical variable by observing $V_{ab}$ and correlating it to the resistance changes. In what range should the values of $R_1$, $R_2$ and $R_3$ be to make a sensitive measurement? Explain your reasoning. (Hint: using \textsc{matlab} to plot the output as a function of the varying resistances is a very useful way to think about this problem).}

Photodiode i-v characteristics:}} Using the data that you collected in the lab for the photodiode, generate 3-4 \textit{i-v} curves for a photodiode at different light levels (including in darkness). Plot these on the same graph to see how incident light affects diode \textit{i-v} characteristics.

Give a brief (qualitative) explanation for why photodiodes are best used in reverse bias?}

Transfer functions:}} For the ``black boxes that you measured in the lab, determine what kind of circuit/filter each one is (two of them will look similar, but have an important difference - what is it?). Determine a transfer function that can model the circuit, and fit the model to the data to see whether the model makes sense.

Of the four boxes, ``D* is required, and you should choose one of*
either ``A* or ``C*. You can fit ``B* for bonus credit*

Referring to the circuit shown in Figure \ref{fig:hw1fig2}, what value
of $R_L$ (in terms of $R_1$ and $R_2$) will result in the maximum
power being dissipated in the load?

(Hint: this is much easier to do if you first remove the load, and calculate the equivalent Thevenin output resistance $R_T$ of the divider looking into the node labeled $V_{out}$. Then express $R_L$ for maximal power transfer in terms of $R_T$.)}

Consider the op-amp circuit shown in Fig. \ref{fig:hw1fig3}.

Determine an expression for the output voltage of the circuit with respect to a DC current input. Express your answer in $V_{out}/I_{in}$. \item Since this is such a high-gain circuit, it can be quite noisy, if the input current $I_{in}$ experiences high-frequency fluctuations. You can insert a capacitor across one of the resistors to reduce the noise (i.e. make a low-pass filter to eliminate high-frequency content). Where would you insert it, and how would you choose its size? \item Now write down the expression for this new circuit's output with respect to the current input for AC signals (Hint: in the expression from part (a), substitute the parallel combination $R_x \parallel C$ for the resistor $R_x$ that you chose).