# 3.14 Formal circuit methods: node method  (Page 2/2)

The presence of a current source in the circuit does not affect the node method greatly; just include it in writing KCLequations as a current leaving the node. The circuit has three nodes, requiring us to define twonode voltages. The node equations are

Note that the node voltage corresponding to the node that we are writing KCL for enters with a positive sign,the others with a negative sign, and that the units of each term is given in amperes. Rewrite these equations in the standardset-of-linear-equations form. ${e}_{1}(\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}})-{e}_{2}\frac{1}{{R}_{2}}={i}_{\mathrm{in}}$ ${e}_{1}\frac{1}{{R}_{2}}+{e}_{2}(\frac{1}{{R}_{2}}+\frac{1}{{R}_{3}})=0$ Solving these equations gives ${e}_{1}=\frac{{R}_{2}+{R}_{3}}{{R}_{3}}{e}_{2}$ ${e}_{2}=\frac{{R}_{1}{R}_{3}}{{R}_{1}+{R}_{2}+{R}_{3}}{i}_{\mathrm{in}}$ To find the indicated current, we simply use $i=\frac{{e}_{2}}{{R}_{3}}$ .

## Node method example

In this circuit ( [link] ), we cannot use the series/parallel combination rules: The vertical resistorat node 1 keeps the two horizontal 1 Ω resistors from being in series, and the 2 Ω resistor prevents the two1 Ω resistors at node 2 from being in series. We really do need the node method to solve this circuit! Despitehaving six elements, we need only define two node voltages. The node equations are Solving these equations yields ${e}_{1}=\frac{6}{13}{v}_{\mathrm{in}}$ and ${e}_{2}=\frac{5}{13}{v}_{\mathrm{in}}$ . The output current equals $\frac{{e}_{2}}{1}=\frac{5}{13}{v}_{\mathrm{in}}$ . One unfortunate consequence of using the element's numericvalues from the outset is that it becomes impossible to check units while setting up and solving equations.

What is the equivalent resistance seen by the voltagesource?

To find the equivalent resistance, we need to find the current flowing through the voltage source. This currentequals the current we have just found plus the current flowing through the other vertical 1 Ω resistor. Thiscurrent equals $\frac{{e}_{1}}{1}=\frac{6}{13}{v}_{\mathrm{in}}$ , making the total current through the voltage source (flowingout of it) $\frac{11}{13}{v}_{\mathrm{in}}$ .Thus, the equivalent resistance is $\frac{13}{11}Ω$ . Modification of the circuit shown on the left to illustrate the node method and the effect of adding the resistor R 2 .

The node method applies to RLC circuits, without significant modification from the methods used on simple resistive circuits,if we use complex amplitudes. We rely on the fact that complex amplitudes satisfy KVL, KCL, and impedance-based v-i relations. In the example circuit, we define complex amplitudes for the input and output variables andfor the node voltages. We need only one node voltage here, and its KCL equation is $\frac{E-{V}_{\mathrm{in}}}{{R}_{1}}+Ei\times 2\pi fC+\frac{E}{{R}_{2}}=0$ with the result $E=\frac{{R}_{2}}{{R}_{1}+{R}_{2}+i\times 2\pi f{R}_{1}{R}_{2}C}{V}_{\mathrm{in}}$ To find the transfer function between input and output voltages, we compute the ratio $\frac{E}{{V}_{\mathrm{in}}}$ . The transfer function's magnitude and angle are $\left|H(f)\right|=\frac{{R}_{2}}{\sqrt{({R}_{1}+{R}_{2})^{2}+(2\pi f{R}_{1}{R}_{2}C)^{2}}}$ $\mathop{\mathrm{arg}}(H(f))=-\arctan \left(\frac{2\pi f{R}_{1}{R}_{2}C}{{R}_{1}+{R}_{2}}\right)$ This circuit differs from the one shown previously in that the resistor ${R}_{2}$ has been added across the output. What effect has it had on the transfer function, which in the original circuit was a lowpassfilter having cutoff frequency ${f}_{c}=\frac{1}{2\pi {R}_{1}C}$ ? As shown in [link] , adding the second resistor has two effects: it lowers the gainin the passband (the range of frequencies for which the filter has little effect on the input) and increases the cutofffrequency. Transfer functions of the circuits shown in [link] . Here, R 1 1 , R 2 1 , and C 1 .

When ${R}_{2}={R}_{1}$ , as shown on the plot, the passband gain becomes half of theoriginal, and the cutoff frequency increases by the same factor. Thus, adding ${R}_{2}$ provides a 'knob' by which we can trade passband gain for cutoff frequency.

We can change the cutoff frequency without affectingpassband gain by changing the resistance in the original circuit. Does the addition of the ${R}_{2}$ resistor help in circuit design?

Not necessarily, especially if we desire individual knobs for adjusting the gain and the cutoff frequency.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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

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