# 3.20 Analog signal processing problems  (Page 2/6)

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## Bridge circuits

Circuits having the form of [link] are termed bridge circuits .

1. What resistance does the current source see when nothing is connected to the output terminals?
2. What resistor values, if any, will result in a zero voltage for ${v}_{\mathrm{out}}$ ?
3. Assume ${R}_{1}=1\Omega$ , ${R}_{2}=2\Omega$ , ${R}_{3}=2\Omega$ and ${R}_{4}=4\Omega$ . Find the current $i$ when the current source ${i}_{\mathrm{in}}$ is $\Im ((4+2i)e^{i\times 2\pi \times 20t})$ . Express your answer as a sinusoid.

## Cartesian to polar conversion

Convert the following expressions into polar form. Plot their location in the complex plane .

1. $(1+\sqrt{-3})^{2}$
2. $3+i^{4}$
3. $\frac{2-i\frac{6}{\sqrt{3}}}{2+i\frac{6}{\sqrt{3}}}$
4. $(4-i^{3})(1+i\frac{1}{2})$
5. $3e^{i\pi }+4e^{i\frac{\pi }{2}}$
6. $(\sqrt{3}+i)\times 2\sqrt{2}e^{-(i\frac{\pi }{4})}$
7. $\frac{3}{1+i\times 3\pi }$

## The complex plane

The complex variable $z$ is related to the real variable $u$ according to $z=1+e^{iu}$

• Sketch the contour of values $z$ takes on in the complex plane.
• What are the maximum and minimum values attainable by $\left|z\right|$ ?
• Sketch the contour the rational function $\frac{z-1}{z+1}$ traces in the complex plane.

## Cool curves

In the following expressions, the variable $x$ runs from zero to infinity. What geometric shapes do the following trace in the complex plane?

1. $e^{ix}$
2. $1+e^{ix}$
3. $e^{-x}e^{ix}$
4. $e^{ix}+e^{i(x+\frac{\pi }{4})}$

## Trigonometric identities and complex exponentials

Show the following trigonometric identities using complex exponentials. In many cases, they were derived using this approach.

1. $\sin (2u)=2\sin u\cos u$
2. $\cos u^{2}=\frac{1+\cos (2u)}{2}$
3. $\cos u^{2}+\sin u^{2}=1$
4. $\frac{d \sin u}{d u}}=\cos u$

## Transfer functions

Find the transfer function relating the complex amplitudes of the indicated variable and thesource shown in [link] . Plot the magnitude and phase of the transferfunction.

## Using impedances

Find the differential equation relating the indicated variable to the source(s) using impedances for each circuitshown in [link] .

## Measurement chaos

The following simple circuit was constructed but the signal measurements were made haphazardly. When the source was $\sin (2\pi {f}_{0}t)$ , the current $i(t)$ equaled $\frac{\sqrt{2}}{3}\sin (2\pi {f}_{0}t+\frac{\pi }{4})$ and the voltage ${v}_{2}(t)=\frac{1}{3}\sin (2\pi {f}_{0}t)$ .

1. What is the voltage ${v}_{1}(t)$ ?
2. Find the impedances ${Z}_{1}$ and ${Z}_{2}$ .
3. Construct these impedances from elementary circuit elements.

## Transfer functions

In the following circuit , the voltage source equals ${v}_{\mathrm{in}}(t)=10\sin \left(\frac{t}{2}\right)$ .

1. Find the transfer function between the source and the indicated output voltage.
2. For the given source, find the output voltage.

## A simple circuit

You are given this simple circuit .

1. What is the transfer function between the source and the indicated output current?
2. If the output current is measured to be $\cos (2t)$ , what was the source?

## Circuit design

1. Find the transfer function between the input and the output voltages for the circuits shown in [link] .
2. At what frequency does the transfer function have a phase shift of zero? What is the circuit's gain atthis frequency?
3. Specifications demand that this circuit have an output impedance (its equivalent impedance) less than 8Ωfor frequencies above 1 kHz, the frequency at which the transfer function is maximum. Find element valuesthat satisfy this criterion.

## Equivalent circuits and power

Suppose we have an arbitrary circuit of resistors that we collapse into an equivalent resistor using the series and parallel rules. Is the power dissipated by the equivalent resistor equal to the sum of the powers dissipated by the actual resistors comprising the circuit?Let's start with simple cases and build up to a complete proof.

1. Suppose resistors ${R}_{1}$ and ${R}_{2}$ are connected in parallel. Show that the power dissipated by $\parallel ({R}_{1}, {R}_{1})$ equals the sum of the powers dissipated by the component resistors.
2. Now suppose ${R}_{1}$ and ${R}_{2}$ are connected in series. Show the sameresult for this combination.
3. Use these two results to prove the general result we seek.

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO