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

 Page 2 / 6

## 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.

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
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
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti By By By Richley Crapo By Saylor Foundation By John Gabrieli By Richley Crapo By Kimberly Nichols By OpenStax By OpenStax By Jonathan Long By Sandy Yamane By Yasser Ibrahim