3.20 Analog signal processing problems  (Page 6/6)

A circuit problem

You are given the depicted circuit .

1. Find the differential equation relating the output voltage to the source.
2. What is the impedance “seen” by the capacitor?

Analog computers

Because the differential equations arising in circuits resemble those that describemechanical motion, we can use circuit models to describe mechanical systems. An ELEC 241 student wants tounderstand the suspension system on his car. Without a suspension, the car's body moves in concert with the bumpsin the raod. A well-designed suspension system will smooth out bumpy roads, reducing the car's vertical motion. Ifthe bumps are very gradual (think of a hill as a large but very gradual bump), the car's vertical motion shouldfollow that of the road. The student wants to find a simple circuit that will model the car's motion. He istrying to decide between two circuit models ( [link] ).

Here, road and car displacements are represented by the voltages $v_{\mathrm{road}}(t)$ and $v_{\mathrm{car}}(t)$ , respectively.

1. Which circuit would you pick? Why?
2. For the circuit you picked, what will be the amplitude of the car's motion if the road has adisplacement given by $v_{\mathrm{road}}(t)=1+\sin (2t)$ ?

Transfer functions and circuits

You are given the depicted network .

1. Find the transfer function between $V_{\mathrm{in}}$ and $V_{\mathrm{out}}$ .
2. Sketch the magnitude and phase of your transferfunction. Label important frequency, amplitude and phase values.
3. Find $v_{\mathrm{out}}(t)$ when $v_{\mathrm{in}}(t)=\sin (\frac{t}{2}+\frac{\pi }{4})$ .

Fun in the lab

You are given an unopenable box that has two terminals sticking out. You assume the box contains a circuit. Youmeasure the voltage $\sin (t+\frac{\pi }{4})$ across the terminals when nothing is connected to them and the current $\sqrt{2}\cos t$ when you place a wire across the terminals.

1. Find a circuit that has these characteristics.
2. You attach a 1 H inductor across the terminals. What voltage do you measure?

Dependent sources

Find the voltage $v_{\mathrm{out}}$ in each of the depicted circuits .

Circuit a

Circuit b

Operational amplifiers

Find the transfer function between the source voltage(s) andthe indicated output voltage for the circuits shown in [link] .

Op-amp a

Op-amp b

Op-amp c

Op-amp d

Op-amp circuit

The following circuit is claimed to serve a useful purpose.

1. What is the transfer function relating the complex amplitude of the output signal, the current $I_{\mathrm{out}}$ , to the complex amplitude of the input, the voltage $V_{\mathrm{in}}$ ?
2. What equivalent circuit does the load resistor $R_L$ see?
3. Find the output current when $v_{\mathrm{in}}=V_{0}e^{-\left(\frac{t}{\tau }\right)}$ .

Why op-amps are useful

The circuit of a cascade of op-amp circuits illustrate the reason whyop-amp realizations of transfer functions are so useful.

1. Find the transfer function relating the complexamplitude of the voltage $v_{\mathrm{out}}(t)$ to the source. Show that this transfer function equals the product of each stage's transfer function.
2. What is the load impedance appearing across the first op-amp's output?
3. [link] illustrates that sometimes “designs” can go wrong. Find the transferfunction for this op-amp circuit , and then show that it can't work! Why can't it?

Operational amplifiers

Consider the depicted circuit .

1. Find the transfer function relating the voltage $v_{\mathrm{out}}(t)$ to the source.
2. In particular, $R_1=530\Omega$ , $C_1=1\mathrm{\mu F}$ , $R_2=5.3\mathrm{k\Omega }$ , $C_2=0.01\mathrm{\mu F}$ , and $R_3=R_4=5.3\mathrm{k\Omega }$ . Characterize the resulting transfer function anddetermine what use this circuit might have.

Designing a bandpass filter

We want to design a bandpass filter that has transferthe function $$H(f)=10\frac{i\times 2\pi f}{(i\frac{f}{f_l}+1)(i\frac{f}{f_h}+1)}$$ Here, $f_l$ is the cutoff frequency of the low-frequency edge of the passband and $f_h$ is the cutoff frequency of the high-frequency edge. We want $f_l=1\mathrm{kHz}$ and $f_h=10\mathrm{kHz}$ .

1. Plot the magnitude and phase of this frequency response. Label important amplitude and phase valuesand the frequencies at which they occur.
2. Design a bandpass filter that meets thesespecifications. Specify component values.

Pre-emphasis or de-emphasis?

In audio applications, prior to analog-to-digital conversion signals are passed throughwhat is known as a pre-emphasis circuit that leaves the low frequencies alone but provides increasinggain at increasingly higher frequencies beyond some frequency $f_0$ . De-emphasis circuits do the opposite and are applied after digital-to-analog conversion. Afterpre-emphasis, digitization, conversion back to analog and de-emphasis, the signal's spectrumshould be what it was.

The op-amp circuit here has been designed for pre-emphasis or de-emphasis (Samantha can'trecall which).

1. Is this a pre-emphasis or de-emphasis circuit? Find the frequency $f_0$ that defines the transition from low to highfrequencies.
2. What is the circuit's output when the input voltage is $\sin (2\pi ft)$ , with $f=4\mathrm{kHz}$ ?
3. What circuit could perform the opposite function to your answer for the first part?

Active filter

Find the transfer function of the depicted active filter .

This is a filter?

You are given a circuit .

1. What is this circuit's transfer function? Plot the magnitude and phase.
2. If the input signal is the sinusoid $\sin (2\pi f_{0}t)$ , what will the output be when $f_0$ is larger than the filter's “cutoff frequency”?

Optical receivers

In your optical telephone, the receiver circuit had the form shown .

This circuit served as a transducer, converting light energy into a voltage $v_{\mathrm{out}}$ . The photodiode acts as a current source, producing a current proportional to the light intesityfalling upon it. As is often the case in this crucial stage, the signals are small and noise can be aproblem. Thus, the op-amp stage serves to boost the signal and to filter out-of-band noise.

1. Find the transfer function relating light intensity to $v_{\mathrm{out}}$ .
2. What should the circuit realizing the feedbackimpedance $Z_f$ be so that the transducer acts as a 5 kHz lowpass filter?
3. A clever engineer suggests an alternative circuit to accomplish the same task. Determine whether the idea works or not. If itdoes, find the impedance $Z_{\mathrm{in}}$ that accomplishes the lowpass filtering task. If not, show why it does not work.

Reverse engineering

The depicted circuit has been developed by the TBBG Electronics design group. They are trying to keep itsuse secret; we, representing RU Electronics, have discovered the schematic and want to figure out theintended application. Assume the diode is ideal.

1. Assuming the diode is a short-circuit (it has been removed from the circuit), what is the circuit'stransfer function?
2. With the diode in place, what is the circuit's outputwhen the input voltage is $\sin (2\pi f_{0}t)$ ?
3. What function might this circuit have?