Inductive reactance, xl
XL = 2пfL |
where: |
XL = reactance in ohms ( ) f = frequency in hertz (Hz) L = inductance in henrys (H) |
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20. 61. Series resonance and Q-factor.
Resonance may occur in a series resistor, inductor and capacitor (RLC) circuit, as shown in Figure 10.1, when the capacitor reactance XC is equal to the inductor reactance XL.
When the magnitudes of inductive reactance XL and capacitive reactance XC are equal (XL = XC), or when reactance X is zero (X = XL - XC = 0), the equivalent or total circuit impedance Z is equal to the resistance R, i.e. Z = R + j(XL - XC)= R.
Resonant angular frequency: ω=1/(LC)1/2
As previously mentioned, when series resonance occurs, the circuit’s equivalent impedance is at the minimum (Z = R). This is illustrated in Figure 10.2, which is the response curve of the impedance Z versus frequency f in the series resonant circuit. When f = fr the impedance Z is at the lowest point on the curve.
When resonance occurs in a series RLC circuit, the impedance of the circuit is equal to the resistance (Z .=R), and the resonant current will be
Figure 10.3 illustrates the response curve of current I versus frequency f in the series resonant circuit, and the current is at the highest point on the curve when f = fr.
The resistor voltage is equal to the source voltage (VR =E) since X = 0 when series resonance occurs. Thus, the current I and source voltage E are also in phase, and the phase difference between E and I is zero A phasor diagram of the series resonant circuit is illustrated in Figure 10.4.
Quality factor
The quality factor is defined as the ratio of stored energy and consumed energy in physics and engineering, so it is the ratio of the reactive power stored by an inductor or a capacitor and average power consumed by a resistor in a resonant circuit, i.e.
The quality factor can be used to measure the energy that a circuit stores and consumes. The lower the energy consumption of a resistor (power loss) in a circuit, the higher the quality factor, and the better the quality of the resonant circuit. If substituting the equations of the reactive power and average power into the quality factor equation, the quality factor of the series resonance will
be obtained as follows:
where R is the total or equivalent resistance in the series circuit.
Voltage of series resonance
● A lower input voltage may produce a higher output voltage.
● Inductor or capacitor voltage may greatly exceed the supply voltage
VL = VC =EQ (Q > 1)
21. Power triangle and power factor.
22. Rms value, meanvalue and the form factor of a complex wave.The rms value of a periodic function is defined as the square root of the mean value of the squared function. Hence, if v = Vmcos(ωt + φ ), the rms value of v is
The quantity under the radical sign in Eq. 9.5 reduces to V2„/2.Hencethermsvalueofv is
23. Parallel AC circuit.
R–L
parallel a.c. circuit
R–C parallel a.c. circuit
24.
Frequency
of series resonance.
Resonance
may occur in a series resistor, inductor and capacitor (RLC) circuit,
as shown in Figure 10.1, when the capacitor reactance XC
is
equal to the inductor reactance XL.
When the magnitudes of inductive reactance XL and capacitive reactance XC are equal (XL = XC), or when reactance X is zero (X = XL - XC = 0), the equivalent or total circuit impedance Z is equal to the resistance R, i.e. Z = R + j(XL - XC)= R.
Resonant angular frequency: ω=1/(LC)1/2
As previously mentioned, when series resonance occurs, the circuit’s equivalent impedance is at the minimum (Z = R). This is illustrated in Figure 10.2, which is the response curve of the impedance Z versus frequency f in the series resonant circuit. When f = fr the impedance Z is at the lowest point on the curve.
Figure 10.3 illustrates the response curve of current I versus frequency f in the series resonant circuit, and the current is at the highest point on the curve when f = fr.
25. Half-power frequency.When an RLC series circuit is in resonance, its impedance will reach the minimum value and the current will reach the maximum value. The curve of the current versus frequency of the series resonant circuit is illustrated in Figure 10.8. As displayed in the diagram, the current reaches the maximum value Imax as the frequency closes in on the resonant frequency fr, which is located at the centrer of the curve.
The characteristic of the resonant circuit can be expressed in terms of its bandwidth (BW) or pass-band. The bandwidth of the resonant circuit is the difference between two frequency points f2 and f1. BW = f2 - f1, where f2 and f1 are called critical, cutoff or half-power frequencies.
The reason to define the term ‘half-power’ frequency can be derived from the following mathematical process. The power delivered by the source at the points f1 and f2 can be determined from the power formula P = I2R =0.5Pmax
26.
. Ideal voltage source. An electrical source is a device that is capable of converting nonelectric energy to electric energy and vice versa. An ideal voltage source is a circuit element that maintains a prescribed voltage across its terminals regardless of the current flowing in those terminals. Ideal voltage sources can be further described as either independent sources or dependent sources. An independent source establishes a voltage in a circuit without relying on voltages elsewhere in the circuit. The value of the voltage supplied is specified by the value of the independent source alone. In contrast, a dependent source establishes a voltage whose value depends on the value of a voltage elsewhere in the circuit.
27.
.
Inductive reactance and inductive susceptance.
An
inductor's opposition to change in current translates to an
opposition to alternating current in general, which is
by definition always
changing in instantaneous magnitude and direction. This opposition to
alternating current is similar to resistance, but different in that
it always results in a phase shift between current and voltage, and
it dissipates zero power. Because of the differences, it has a
different name: reactance. Reactance to
AC is expressed in ohms. The
exact formula for determining reactance is
as follows:
Susceptance
is the reciprocal of reactance, X and is given the symbol B.
In AC circuits susceptance is defined as the ease at which a reactance (or a set of reactances) allows current to flow when a voltage is applied. Susceptance has the opposite sign to reactance so capacitive susceptance BC is positive, +ve in value and inductive susceptance BL is negative, -ve in value.
Inductive
Susceptance, ( BL ):
