Circuits having the form of
[link] are termed
bridge circuits .
What resistance does the current source see when nothing is connected to the output terminals?
What resistor values, if any, will result in a zero voltage for
${v}_{\mathrm{out}}$ ?
Assume
${R}_{1}=1\Omega $ ,
${\mathrm{R}}_{2}=2\Omega $ ,
${\mathrm{R}}_{3}=2\Omega $ and
${\mathrm{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 .
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?
$e^{ix}$
$1+e^{ix}$
$e^{-x}e^{ix}$
$e^{ix}+e^{i(x+\frac{\pi}{4})}$
Trigonometric identities and complex exponentials
Show the following trigonometric identities using complex exponentials.
In many cases, they were derived using this approach.
$\sin (2u)=2\sin u\cos u$
$\cos u^{2}=\frac{1+\cos (2u)}{2}$
$\cos u^{2}+\sin u^{2}=1$
$\frac{d \sin u}{d u}=\cos u$
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
${v}_{2}(t)=\frac{1}{3}\sin (2\pi {f}_{0}t)$ .
What is the voltage
${v}_{1}(t)$ ?
Find the impedances
${Z}_{1}$ and
${Z}_{2}$ .
Construct these impedances from elementary circuit elements.
Transfer functions
In the
following
circuit , the voltage source equals
${v}_{\mathrm{in}}(t)=10\sin \left(\frac{t}{2}\right)$ .
Find the transfer function between the source and the
indicated output voltage.
What is the transfer function between the source
and the indicated output current?
If the output current is measured to be
$\cos (2t)$ , what was the source?
Circuit design
Find the transfer function between the input and the
output voltages for the circuits shown in
[link] .
At what frequency does the transfer function have a
phase shift of zero? What is the circuit's gain atthis frequency?
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.
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.
Now suppose
${R}_{1}$ and
${R}_{2}$ are connected in series.
Show the sameresult for this combination.
Use these two results to prove the general result we seek.
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?
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