# 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 $\frac{13}{11}Ω$ .

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.

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.

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
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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
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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
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
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
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles