Lunes, Pebrero 23, 2015

Thevenin's Theorem in AC Analysis



Ipinaskil ni sa Walang komento:

Source Transformation in AC Analysis


  • Transform a voltage source in series with an impedance to a current source in     parallel with an impedance for simplification or vice versa.






Source transformations are easy to perform as long as there is a familiarity with Ohm's law. If there is a voltage source in series with an impedance, it is possible to find the value of the equivalent current source in parallel with the impedance by dividing the value of the voltage source by the value of the impedance. The converse also applies here: if a current source in parallel with an impedance is present, multiplying the value of the current source with the value of the impedance will result in the equivalent voltage source in series with the impedance.
Ipinaskil ni sa Walang komento:

Superposition Theorem in AC Analysis

The superposition theorem states that in a linear circuit with several sources, the current and voltage for any element in the circuit is the sum of the currents and voltages produced by each source acting independently. The theorem is valid for any linear circuit. The best way to use superposition with AC circuits is to calculate the complex effective or peak value of the contribution of each source applied one at a time, and then to add the complex values. This is much easier than using superposition with time functions, where one has to add the individual time functions.
To calculate the contribution of each source independently, all the other sources must be removed and replaced without affecting the final result.
When removing a voltage source, its voltage must be set to zero, which is equivalent to replacing the voltage source with a short circuit.
When removing a current source, its current must be set to zero, which is equivalent to replacing the current source with an open circuit. 
Now let's explore an example. 
In the circuit shown below" 
Ri = 100 ohm, R1 = 20 ohm, R2 = 12 ohm, L = 10 uH, C = 0.3 nF, vS(t)=50cos(wt) V, iS(t)=1cos(wt+30°) A, f=400 kHz. 
Notice that both sources have the same frequency: we will only work in this chapter with sources all having the same frequency. Otherwise, superposition must be handled differently.
Find the currents i(t) and i1(t) using the superposition theorem.


  • REVIEW:
  • The Superposition Theorem states that a circuit can be analyzed with only one source of power at a time, the corresponding component voltages and currents algebraically added to find out what they'll do with all power sources in effect.
  • To negate all but one power source for analysis, replace any source of voltage (batteries) with a wire; replace any current source with an open (break).

Ipinaskil ni sa Walang komento:

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:

Sabado, Oktubre 4, 2014

Maximum Power Transfer

           The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than maximum.

            This is essentially what is aimed for in radio transmitter design , where the antenna or transmission line “impedance” is matched to final power amplifier “impedance” for maximum radio frequency power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance.
            The Maximum Power Transfer Theorem states that the maximum amount of power will be dissipated by a load resistance if it is equal to the Thevenin or Norton resistance of the network supplying power.
            The Maximum Power Transfer Theorem does not satisfy the goal of maximum efficiency.

Maximum Power Transfer Example No1.

maximum power transfer theorem
Where:
  RS = 25Ω
  RL is variable between 0 – 100Ω
  VS = 100v
 
Then by using the following Ohm’s Law equations:
maximum power transfer
 
We can now complete the following table to determine the current and power in the circuit for different values of load resistance.

Table of Current against Power

RL (Ω)I (amps)P (watts)
04.00
53.355
102.878
152.593
202.297
RL (Ω)I (amps)P (watts)
252.0100
301.897
401.594
601.283
1000.864
Using the data from the table above, we can plot a graph of load resistance, RL against power, P for different values of load resistance. Also notice that power is zero for an open-circuit (zero current condition) and also for a short-circuit (zero voltage condition).

Graph of Power against Load Resistance

maximum power against load
Ipinaskil ni sa Walang komento:
Mag-subscribe sa: Mga Post (Atom)