Sabado, Enero 3, 2015

AC Circuits:Nodal Analysis & Mesh Analysis



Ø Since KCL is valid for phasors, we can analyze AC circuits by NODAL analysis.
Ø Determine the number of nodes within the network.
Ø Pick a reference node and label each remaining node with a subscripted value of voltage: V1, V2 and so on.
Ø Apply Kirchhoff’s current law at each node except the reference. Assume that all unknown currents leave the node for each application of Kirhhoff’s current law.
Ø Solve the resulting equations for the nodal voltages.
Ø For dependent current sources: Treat each dependent current source like an independent source when Kirchhoff’s current law is applied to each defined node. However, once the equations are established, substitute the equation for the controlling quantity to ensure that the unknowns are limited solely to the chosen nodal voltages.



ØPractice Problem 10.1: Find v1 and v2 using nodal analysis







Mesh Analysis

Practice Problem 10.4: Calculate the current Io
Ipinaskil ni sa Walang komento:

Impedance and Admittance

Susceptance and Admittance

Admittance (symbolized Y ) is an expression of the ease with which alternating current ( AC ) flows through a complex circuit or system. Admittance is a vector quantity comprised of two independent scalar phenomena: conductance and susceptance .

Impedance, denoted Z, is an expression of the opposition that an electronic component, circuit, or system offers to alternating and/or direct electric current.Impedance is a vector (two-dimensional)quantity consisting of two independent scalar (one-dimensional) phenomena: resistance and reactance.


In the study of DC circuits, the student of electricity comes across a term meaning the opposite of resistance: conductance. It is a useful term when exploring the mathematical formula for parallel resistances: Rparallel = 1 / (1/R1 + 1/R2 + . . . 1/Rn). Unlike resistance, which diminishes as more parallel components are included in the circuit, conductance simply adds. Mathematically, conductance is the reciprocal of resistance, and each 1/R term in the “parallel resistance formula” is actually a conductance.
Whereas the term “resistance” denotes the amount of opposition to flowing electrons in a circuit, “conductance” represents the ease of which electrons may flow. Resistance is the measure of how much a circuit resists current, while conductance is the measure of how much a circuit conducts current. Conductance used to be measured in the unit of mhos, or “ohms” spelled backward. Now, the proper unit of measurement is Siemens. When symbolized in a mathematical formula, the proper letter to use for conductance is “G”.
Reactive components such as inductors and capacitors oppose the flow of electrons with respect to time, rather than with a constant, unchanging friction as resistors do. We call this time-based opposition, reactance, and like resistance we also measure it in the unit of ohms.
As conductance is the complement of resistance, there is also a complementary expression of reactance, called susceptance. Mathematically, it is equal to 1/X, the reciprocal of reactance. Like conductance, it used to be measured in the unit of mhos, but now is measured in Siemens. Its mathematical symbol is “B”, unfortunately the same symbol used to represent magnetic flux density.
The terms “reactance” and “susceptance” have a certain linguistic logic to them, just like resistance and conductance. While reactance is the measure of how much a circuit reacts against change in current over time, susceptance is the measure of how much a circuit is susceptible to conducting a changing current

Series-parallel R, L, and C

Now that we've seen how series and parallel AC circuit analysis is not fundamentally different than DC circuit analysis, it should come as no surprise that series-parallel analysis would be the same as well, just using complex numbers instead of scalar to represent voltage, current, and impedance.
Take this series-parallel circuit for example: (Figure below)







Ipinaskil ni sa Walang komento:

Sinusoids and Phasors

A Sinusoid is a signal that has the form of the sine or cosine function.


A sinusoidal current is usually referred to as alternating current (ac).
Such a current reverses at regular time intervals and has alternately pos-
itive and negative values. Circuits driven by sinusoidal current or volt-
age sources are called ac circuits.


A sinusoidal forcing function produces both a transient response
and a steady-state response, much like the step function, which we stud-
ied in Chapters 7 and 8. The transient response dies out with time so
that only the steady-state response remains. When the transient response
has become negligibly small compared with the steady-state response,
we say that the circuit is operating at sinusoidal steady state. It is this
sinusoidal steady-state response.


A sinusoid having period T and angular frequency can be represented as



A periodic function is one that satisfies f (t) f (tnT), for all t and
for all integers n.




Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

  • Polar representation:$z = \vert z\vert\;e^{j\angle z}$, or simply $z=\vert z\vert\;\angle z$, where $\vert z\vert$ and $\angle z$ are the magnitude and phase angle, respectively.
complex_number.gif
The two representations can be converted from one to the other:
  • From $(x,y)$ to $(\vert z\vert,\angle z)$:

    \begin{displaymath}\left\{ \begin{array}{ll} \vert z\vert=\sqrt{x^2+y^2} & \mbox... ...gle z=tan^{-1} (y/x) & \mbox{phase angle} \end{array} \right. \end{displaymath}

  • From $(\vert z\vert,\angle z)$ to $(x,y)$:

    \begin{displaymath}z=\vert z\vert\;e^{j\angle z}=\vert z\vert(\cos\angle z+j\;\sin\angle z)=x+jy \end{displaymath}


    due to Euler identity, i.e.,

    \begin{displaymath}\left\{ \begin{array}{ll} x=\vert z\vert\;\cos\angle z & \mbo... ...ert\;\sin\angle z & \mbox{imaginary part} \end{array} \right. \end{displaymath}

The arithmetic operations of two complex numbers $z=x+jy=\vert z\vert\;e^{j\angle z}$ and $w=u+jv=\vert w\vert\;e^{j\psi}$ are listed below:
  • Add/Subtract:

    \begin{displaymath}z+w=(x+u)+j(y+v),\;\;\;\;z-w=(x-u)+j(y-v) \end{displaymath}


  • Multiply:

    \begin{displaymath}z\;w=(x+jy)(u+jv)=(xu-yv)+j(xv+yu)=\vert z\vert\;\vert w\vert\;e^{j(\angle z+\angle w)} \end{displaymath}


  • Divide:

    \begin{displaymath}\frac{z}{w}=\frac{x+jy}{u+jv}=\frac{(x+jy)(u-jv)}{(u+jv)(u-jv... ... =\frac{\vert z\vert}{\vert w\vert}\;e^{j(\angle z-\angle w)} \end{displaymath}




Ipinaskil ni sa Walang komento:
Mag-subscribe sa: Mga Post (Atom)