Published:2011/7/25 2:34:00 Author:Li xiao na From:SeekIC
By B. Kainka
Resonant circuits
Although resonant circuits are the most important application for inductors, resonant frequency and damping are also significant when inductors are used for other purposes, For one thing, it’s important to recognize undesired resonances, and for another thing, it’s very easy to determine the value of an unknown inductor by using frequency measurements.
If an inductor and a capacitor are connected together as shown in Figure 3, the result is a resonant circuit. Electrical energy can ’swing’ back and forth between the inductor and the capacitor, similar to the motion of a pendulum, and such a circuit has a characteristic resonant frequency. After being excited by a short current pulse, a resonant circuit will oscillate freely at a frequency given by the formula
f0 = 1÷ [2π x√(LC)] |Hz|
Resonant circuits are often used in circuits where several different frequencies are present and in frequency mixers. This allows currents and voltages to be distinguished according to their frequencies. A parallel resonant circuit has a complex impedance Z whose peak value occurs at the resonant frequency f0. At this frequency, RC= RL. and the currents through the inductor and the capacitor exactly cancel each other since they have a 180-degree phase difference. An ideal parallel resonant circuit with no damping would have infinite impedance at its resonant frequency.
However, energy losses always occur in practice, due to the.ph.mic resistance of the coil, magnetic losses in the core of the inductor and electromagnetic radiation. The resonant impedance thus remains finite. This causes the oscillation to be damped. For simplicity, the losses can be grouped into an equivalent parallel ’loss resistance’ R, as shown in Figure 4.
For every resonant circuit, it is possible to specify a quality factor, or ’Q factor’, or simply ’Q’, which is inversely proportional to the bandwidth of the circuit. Q is dimensionless and can easily be determined by taking the ratio of the parallel damping resistance R to the inductive impedance Rt = 2 n f L or capacitive impedance Rc = l/(2πr f C) at the resonant frequency:
Q = R/RL = R/RC
If a resonant circuit is excited by an alternating current I with constant amplitude and variable frequency, for example using an AC generator with a high internal impedance, the voltage across the resonant circuit will be proportional to the magnitude of the complex impedance Z. The voltage will reach its maximum value at the resonant frequency.
The amount that the voltage increases at resonance is inversely proportional to the extent to which the oscillations are damped by any sort of energy loss, and thus directly proportional to the Q factor of the resonant circuit. At either side of the resonant frequency, points can be found at which the voltage is reduced from its maximum value by a factor of l/√2 = 0.707 (-3 dB). The difference between the frequencies of these two points is defined to be the bandwidth BW of the circuit. The relationship between the bandwidth BW and the resonant frequency f0 and Q factor of the circuit is:
BW(-3dB)=f0/Q
Figure 5 shows resonance curves for several different Q factors. A circuit with Q = 50 has a greater bandwidth (BW,) than one with Q = 110 {BW2). You can also see that the peak value at resonance increases as the Q factor increases. This means that the resonant circuit oscillates more strongly at its resonant frequency. By contrast, the various circuits show nearly the same behavior in regions far away from the resonant frequency.
In practice, the circuit damping, and with it the Q factor, almost always arises from a combination of series and parallel resistances. The series resistance comes from the wire used to form the coil, and at a given frequency it is greater than the DC resistance of the inductor, due to the ’skin effect’. The parallel resistance is determined by the matching impedance in the circuit. However, iron cores and ferrite cores also have losses that can be expressed in the form of a parallel resistance. For a given inductance, an inductor with a core requires fewer turns and thus has smaller copper losses, but this comes at the price of core losses.
At high frequencies (above approximately 100 MHz), pure air-core inductors wound using thick, silver-plated wire give the best results, while at medium frequencies (around 10 MHz) the best Q factor can be obtained using closed cores, such as ring cores. However, air-core inductors can also be used down to frequencies of approximately 1 MHz. By contrast, inductors and transformers for use at low frequencies use almost always require cores.
With careful coil construction, Q factors of around 100 can be achieved. However, resonant circuits can also be damped by connected circuitry or an aerial. This damping can be counteracted by using loose coupling to the resonant circuit via a small auxiliary winding, a coil tap or a suitable capacitor. When a resonant circuit is connected directly to an amplifier, the internal impedance of the amplifier must be very high in order to minimise the damping.
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