Martes, Disyembre 30, 2014

AC Circuits

AC Circuits w/ Complex Numbers

Direct current (DC) circuits involve current flowing in one direction. In alternating current (AC) circuits, instead of a constant voltage supplied by a battery, the voltage oscillates in a sine wave pattern, varying with time as:

In a household circuit, the frequency is 60 Hz. The angular frequency is related to the frequency, f, by:
Vo represents the maximum voltage, which in a household circuit in North America is about 170 volts. We talk of a household voltage of 120 volts, though; this number is a kind of average value of the voltage. The particular averaging method used is something called root mean square (square the voltage to make everything positive, find the average, take the square root), or rms. Voltages and currents for AC circuits are generally expressed as rms values. For a sine wave, the relationship between the peak and the rms average is:
rms value = 0.707 peak value

Resistance in an AC circuit

The relationship V = IR applies for resistors in an AC circuit, so
In AC circuits we'll talk a lot about the phase of the current relative to the voltage. In a circuit which only involves resistors, the current and voltage are in phase with each other, which means that the peak voltage is reached at the same instant as peak current. In circuits which have capacitors and inductors (coils) the phase relationships will be quite different.

Capacitance in an AC circuit

Consider now a circuit which has only a capacitor and an AC power source (such as a wall outlet). A capacitor is a device for storing charging. It turns out that there is a 90° phase difference between the current and voltage, with the current reaching its peak 90° (1/4 cycle) before the voltage reaches its peak. Put another way, the current leads the voltage by 90° in a purely capacitive circuit.
To understand why this is, we should review some of the relevant equations, including:
relationship between voltage and charge for a capacitor: CV = Q

RLC Circuits

Consider what happens when resistors, capacitors, and inductors are combined in one circuit. If all three components are present, the circuit is known as an RLC circuit (or LRC). If only two components are present, it's either an RC circuit, an RL circuit, or an LC circuit.
The overall resistance to the flow of current in an RLC circuit is known as the impedance, symbolized by Z. The impedance is found by combining the resistance, the capacitive reactance, and the inductive reactance. Unlike a simple series circuit with resistors, however, where the resistances are directly added, in an RLC circuit the resistance and reactances are added as vectors.
This is because of the phase relationships. In a circuit with just a resistor, voltage and current are in phase. With only a capacitor, current is 90° ahead of the voltage, and with just an inductor the reverse is true, the voltage leads the current by 90°. When all three components are combined into one circuit, there has to be some compromise.
To figure out the overall effective resistance, as well as to determine the phase between the voltage and current, the impedance is calculated like this. The resistance R is drawn along the +x-axis of an x-y coordinate system. The inductive reactance is at 90° to this, and is drawn along the +y-axis. The capacitive reactance is also at 90° to the resistance, and is 180° different from the inductive reactance, so it's drawn along the -y-axis. The impedance, Z, is the sum of these vectors, and is given by:
The current and voltage in an RLC circuit are related by V = IZ. The phase relationship between the current and voltage can be found from the vector diagram: its the angle between the impedance, Z, and the resistance, R. The angle can be found from:
If the angle is positive, the voltage leads the current by that angle. If the angle is negative, the voltage lags the currents.
The power dissipated in an RLC circuit is given by:


Overview and Insights

I learned that all of this power is lost in the resistor; the capacitor and inductor alternately store energy in electric and magnetic fields and then give that energy back to the circuit. I am also having difficulty in solving Complex Numbers because the values are not accurate when i solve.



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Impedance and Admittance

IMPEDANCE  AND ADMITTANCE 

What is Impedance ?
While Ohm's Law applies directly to resistors in DC or in AC circuits, the form of the current-voltage relationship in AC circuits in general is modified to the form:

where I and V are the rms or "effective" values. The quantity Z is called impedance. For a pure resistor, Z = R. Because the phase affects the impedance and because the contributions of capacitors and inductors differ in phase from resistive components by 90 degrees, a process like vector addition (phasors) is used to develop expressions for impedance. More general is the complex impedancemethod.

Impedance Combinations

Combining impedances has similarities to the combining of resistors, but the phaserelationships make it practically necessary to use the complex impedance method for carrying out the operations. Combining series impedances is straightforward:
Calculation
Combining parallel impedances is more difficult and shows the power of the complex impedance approach. The expressions must be rationalized and are lengthy algebraic forms.
Expressions


What is Admittance ?
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of impedance. The SI unit of admittance is the siemens (symbol S). Oliver Heaviside coined the term admittance in December 1887.[1]
Admittance is defined as
Y \equiv \frac{1}{Z} \,
where
Y is the admittance, measured in siemens
Z is the impedance, measured in ohms
The synonymous unit mho, and the symbol ℧ (an upside-down uppercase omega Ω), are also in common use.
Resistance is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as reactance). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:
Y = G + j B \,
where
  • Y is the admittance, measured in siemens.
  • G is the conductance, measured in siemens.
  • B is the susceptance, measured in siemens.
  • j^2 = -1

Conversion from impedance to admittance

The impedance, Z, is composed of real and imaginary parts,
Z = R + jX \,
where
  • R is the resistance, measured in ohms
  • X is the reactance, measured in ohms
Y = Z^{-1}= \frac{1}{R + jX} = \left( \frac{1}{R^2 + X^2} \right) \left(R - jX\right)
Admittance, just like impedance, is a complex number, made up of a real part (the conductance, G), and an imaginary part (the susceptance, B), thus:
Y = G + jB \,\!
where G (conductance) and B (susceptance) are given by:
\begin{align} G &= \Re(Y) = \frac{R}{R^2 + X^2} \\ B &= \Im(Y) = -\frac{X}{R^2 + X^2} \end{align}
The magnitude and phase of the admittance are given by:
\begin{align} \left | Y \right | &= \sqrt{G^2 + B^2} = \frac{1}{\sqrt{R^2 + X^2}} \\ \angle Y &= \arctan \left( \frac{B}{G} \right) = \arctan \left( -\frac{X}{R} \right) \end{align}
where
  • G is the conductance, measured in siemens
  • B is the susceptance, also measured in siemens
Note that (as shown above) the signs of reactances become reversed in the admittance domain; i.e. capacitive susceptance is positive and inductive susceptance is negative.

Overview and Insights

In this topic I learned about the steps you need to follow in order to use the formula. The larger C the more movement of charge the smaller resistance. The higher F the more movement of charge approaching short-circuit.





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