# 0.1 Lecture 2: introduction to systems  (Page 2/8)

3/ Conclusion

A system is described structurally by specifying:

• the system topology,
• the rules of interconnection of the elements,
• functional descriptions of the elements — constitutive relations.

III. DYNAMIC ANALOGIES

Physically divergent systems can have similar dynamic properties.

1/ Mechanical free-body diagram

M = mass, B = friction constant,

K = spring constant,

f(t) = external force, and

v(t) = velocity of the mass.

Summing the forces yields

$f\left(t\right)=M\frac{\text{dv}\left(t\right)}{\text{dt}}+\text{Bv}\left(t\right)+K{\int }_{-\infty }^{t}v\left(\tau \right)\mathrm{d\tau }$

2/ Electric network

Summing the currents (Kirchhoff’s current law) yields

$i\left(t\right)=C\frac{\text{dv}\left(t\right)}{\text{dt}}+\frac{v\left(t\right)}{R}+\frac{1}{L}{\int }_{-\infty }^{t}v\left(\tau \right)\mathrm{d\tau }$

3/ The mechanical and electrical systems are dynamically analogous

$\begin{array}{}f\left(t\right)=M\frac{\text{dv}\left(t\right)}{\text{dt}}+\text{Bv}\left(t\right)+K{\int }_{-\infty }^{t}v\left(\tau \right)\mathrm{d\tau }\\ i\left(t\right)=C\frac{\text{dv}\left(t\right)}{\text{dt}}+\frac{v\left(t\right)}{R}+\frac{1}{L}{\int }_{-\infty }^{t}v\left(\tau \right)\mathrm{d\tau }\end{array}$

Thus, understanding one of these systems gives insights into the other.

4/ Block diagram

A block diagram using integrators, adders, and gains

$f\left(t\right)=M\frac{\text{dv}\left(t\right)}{\text{dt}}+\text{Bv}\left(t\right)+K{\int }_{-\infty }^{t}v\left(\tau \right)\mathrm{d\tau }$

5/ Electronic synthesis of block diagram

The integrator, adder, and gain blocks are other examples of functional descriptions of systems. We can produce a structural model of each of these blocks. For example, the gain block is easily synthesized with an op-amp circuit.

The op-amp itself is a functional model of a device that we can synthesize with an electronic circuit including a number of transistors.

Conclusion: We have seen several types of descriptions of systems

$f\left(t\right)=M\frac{\text{dv}\left(t\right)}{\text{dt}}+\text{Bv}\left(t\right)+K{\int }_{-\infty }^{t}v\left(\tau \right)\mathrm{d\tau }$

All four descriptions define a system with the same dynamic properties. We will develop methods that characterize these systems efficiently and that abstract their critical dynamic properties.

IV. CLASSIFICATION OF SYSTEMS

1/ Memoryless systems

The output of a memoryless system at some time to depends only on its input at the same time to. For example, for the resistive divider network, ${v}_{o}\left(t\right)=\frac{{R}_{2}}{{R}_{1}+{R}_{2}}{v}_{i}\left(t\right)$

Therefore, ${v}_{o}\left(\text{to}\right)$ depends upon the value of ${v}_{i}\left(\text{to}\right)$ and not on ${v}_{i}\left(t\right)$ for t ≠ to.

2/ Systems with memory

$i\left(t\right)=C\frac{\text{dv}\left(t\right)}{\text{dt}}$

$v\left(t\right)=\frac{1}{C}{\int }_{-\infty }^{t}i\left(\tau \right)\mathrm{d\tau }$

Note that v(t) depends not just on i(t) at one point in time t. Therefore, the system that relates v to i exhibits memory.

3/ Causal and noncausal systems

For a causal system the output at time to depends only on the input for t ≤ to, i.e., the system cannot anticipate the input.

Physical systems with time as the independent variable are causal systems. There are examples of systems that are not causal.

• Physical systems for which time is not the independent variable, e.g., the independent variable is space (x, y, z) as in an optical system.
• Processing of signals where time is the independent variable but the signal has been recorded or generated in a computer. Processing is not in real time.

4/ Stable and unstable systems

Stability can be defined in a variety of ways.

Definition 1: a stable system is one for which an incremental input leads to an incremental output. An incremental force leads to only an

incremental displacement in the stable

system but not in the unstable system.

Definition 2: A system is BIBO stable if every bounded input leads to a bounded output. We will use this definition.

#### Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
?
Kyle
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Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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Devang Reply
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
Abhijith Reply
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
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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China
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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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