Calculation of LC filters. Encyclopedia of radio electronics and electrical engineering

Encyclopedia of radio electronics and electrical engineering / Beginner radio amateur

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By combining inductors and capacitors, it is possible to build filters, firstly, of higher orders (the order of the filter, as a rule, is equal to the number of its reactive elements), i.e., having steeper frequency response slopes in the stopband, and secondly, introducing significantly less attenuation in the passband. Ideally, when coils and capacitors are lossless (their quality factor is infinite), LC filters introduce no losses at all.

The simplest LC filter is an oscillatory circuit. Included as shown in fig. 38 diagram, it will act as a narrow bandpass filter tuned to the frequency

f0= 1/2π√LC.

At the resonant frequency, the loop resistance is active:

R0 = pQ.

where p is the characteristic resistance, equal to the reactance of the coil and capacitor. It is more convenient to calculate it by the formula

p = √L / C.

Since the capacitor usually introduces almost no losses, the quality factor of the circuit is equal to the quality factor of the coil. It is easier to determine the resonant frequency and quality factor experimentally by assembling the cascade according to the above scheme. You will need a signal generator that creates an input voltage Uin, and some kind of output meter with high internal resistance, best of all an oscilloscope. It will serve to register the voltage Uout.

By changing the frequency of the generator, it will be possible to register the maximum Uout at the resonant frequency of the circuit f0. Resistor R1 and resonant circuit resistance r0 form a divider, and

Uout = Uin/(R1+r0).

Having measured the voltages at the input and output, it is now easy to calculate the resonant impedance, and then the quality factor of the circuit.

Another way to measure the quality factor is to measure the loop bandwidth 2Δf, where Δf is the oscillator frequency deviation at which Uout falls to 0,7 of the resonant value. The quality factor is related to the bandwidth by a simple formula

Q = f0/2Δf.

In this case, it must be borne in mind that it is not the intrinsic (constructive) quality factor of the Q0 circuit that will be measured, but a slightly smaller value - the quality factor of the circuit shunted by the resistor R1. Therefore, the resistance of the resistor in this experiment should be chosen as large as possible. Often the resistor is replaced with a small capacitor, in practice it is enough to bring the generator probe to the upper (according to the diagram) output of the circuit.

The input impedance of an oscilloscope, or other device connected to the circuit, is also not infinitely large, and, of course, it reduces its quality factor. The method for calculating the "loaded" quality factor is simple: you need to find a new resonant resistance formed by the parallel connection of R1 and R0, and then divide it by p. Then, the resistance R2 connected to the output is taken into account in the same way.

A single-loop bandpass filter is a very imperfect device. If we want to use the properties of the circuit completely, i.e., to obtain a sharp resonance curve corresponding to the constructive quality factor, then the circuit must be loaded weakly by choosing R1 and R2 much larger than R0. Then the power transfer coefficient is small, which means a large loss in the bandwidth. If the circuit is heavily loaded by selecting R1 = R2 << R0, then the transmission coefficient tends to the maximum possible (-6 dB), but the circuit almost completely loses its resonant properties. However, a single circuit is often used at the input of radio receivers or in resonant amplifiers because of its simplicity.

The voltage transfer coefficient increases if at least R2 can be made large (for example, by connecting the circuit to the gate of a field-effect transistor, which serves to further amplify the signal). It remains to coordinate the circuit from the input side (for example, with a 75-ohm antenna feeder). Use an autotransformer connection (Fig. 39) or a capacitive divider (Fig. 40).

In the first case

R1 = R0(n1/n0)²,

where n1 is the number of turns from the "ground" to the tap: n0 is the total number of turns of the coil (the connection of the parts of the coil is assumed to be strong) In the second case

R1 = R0C1²/(C1 +C2)².

If R2 is not infinite, then first you need to take it into account by calculating the new R0 (reduced by parallel connection of R2), and then calculate the input matching. The parameters of a narrow bandpass filter can be significantly improved, including two, three or more circuits. The connection between them can be inductive or external capacitive. The mutual inductance coefficient is chosen to be Q times less than the inductance of the coils, and the capacitance of the coupling capacitors is Q times less than the loop capacitances, with Q being determined from the required bandwidth of the filter. If O is much less than the constructive quality factor of the coils, the losses in the filter are small. The input and output of the filter are loaded with resistors R = pQ.

The signal to the circuit can be applied not only in parallel, as described above, but also in series, as in Fig. 41. In this case, if it is necessary to obtain a sharp resonance curve, the resistance R2, as before, must be chosen as much as possible, and R1, on the contrary, as little as possible. With a small internal resistance of the generator, such a circuit has a large voltage transfer coefficient at the resonant frequency, equal to Q in the limit. At the lowest frequencies, the transfer coefficient tends not to zero, as in the filters already considered, but to one.

A very interesting case is when in the filter according to the scheme of Fig. 41, select the resistances at the input and output equal to the characteristic, i.e. R1 \u2d RXNUMX \uXNUMXd p.

It turns out a matched low-pass filter, the transfer coefficient of which is constant and equal to 1/2 (-6 dB) at all frequencies from zero to the resonant frequency of the L1C1 circuit, and decreases with a further increase in frequency. The slope of the frequency response is 12 dB per octave, as it should be for a second-order filter.

In the filter passband 0 ... f0, the transfer coefficient is often assumed to be equal to one, considering the input voltage not the generator EMF, but the voltage between the upper output of the resistor R1 according to the circuit and the common wire. Moreover, the resistor R1 can be the internal resistance of the generator. The generator, as it were, "sees" the load resistance R2 through a filter that is transparent in the passband and gives maximum power at R1 = R2.

By the way, most measuring generators have a standard internal resistance of 50 ohms, and the output voltage scale is calibrated for the case of their load also at 50 ohms. If the output of such a generator is not loaded with anything, the output voltage will be twice as high as the scale of the output attenuator shows!

To obtain steeper slopes of the frequency response, a pair of the described L-shaped links is used, connecting them in accordance with Fig. 42 to form a T-link, or according to fig. 43 to form a U-link. In this case, a third-order low-pass filter is obtained. Usually U-shaped links are preferred, since they have less labor-intensive inductors to manufacture.

It is also possible to further "build up" the order of the filters. For example, in Fig. 44 shows how a two-link low-pass filter of the fifth order is made up of two U-shaped links.

It has a very steep frequency response in the stopband - 30 dB per octave. It can be made even cooler if additional small capacitors are connected in parallel with the coils. At the frequencies of the resulting resonant circuits, two points of "infinite damping" are obtained, lying in the stopband. In some cases, the role of additional capacitors can be performed by the interturn capacitance of the coils.

The HPF is constructed in a similar way, only the coils are replaced by capacitors, and the capacitors are replaced by coils. Broadband bandpass filters are obtained by cascading a low-pass filter and a high-pass filter, preferably with an isolating amplifier stage between them.

Question for self-test. Using the formulas in this chapter, derive the calculation formulas for the inductance and capacitance of the L-shaped link of the low-pass filter. Calculate the LPF according to fig. 44 for a radio amateur heterodyne receiver. The filter cutoff frequency is 2,7 kHz and the characteristic impedance is 1,6 kΩ.

Draw a filter circuit with the designation of the element ratings and plot its frequency response on a logarithmic scale.

Response. The parameters of the matched L-shaped link of the low-pass filter (Fig. 41, 42) are found from the relation R = p, where R is the filter load resistance; p is its characteristic impedance, equal to the reactance of its elements at the cutoff frequency:

L=R/2πf_c,C=1/2πf_cR.

Having received these formulas, it is no longer difficult to calculate the elements of a two-link low-pass filter (Fig. 44) of a heterodyne receiver, taking into account the fact that the inductances of both coils should be 2L, the capacitances of the extreme capacitors - C, the capacitance of the middle capacitor - 2C:

L= 1,6-10³/ 6,28.2,7-10³ - 0,095H = 95 mH, 2L = 190 mH;

C \u1d 6,28 / 2,7 10 XNUMX³1,6 10³ = 0,037х10^-6F \u0,037d 2 uF, 0,074C \uXNUMXd XNUMX uF.

In the practical manufacture of the filter, the number of turns of the coil is calculated using the information presented in Chapter 5. In this case, it is advisable to use ferrite rings, which provide a good quality factor of the coil and are little susceptible to interference from extraneous fields. Somewhat worse in both respects are magnetic circuits made of W-shaped steel plates, for example, from transformers previously used in portable transistor receivers.

For example, let's calculate the number of coil turns on a K16x8x4 ferrite ring made of 2000NM grade ferrite. Let's use the formula L=μμ₀N²/l. Substituting into it the values ​​μ = 2000, μ₀ = 4π-10^-7rH/M,S=16 10^-6M², l=38 10^-3M, we get L -10^-6N² or N - 10³L Substituting the value L = 0,19 H, we get N = 430 turns. It should be noted that, contrary to popular belief, such simple filters are rather uncritical to the spread of the parameters of their elements, in any case, deviations of ± 5% have practically little effect on the shape of the frequency response. Calculations can also be carried out with appropriate accuracy. The source and load resistances of the filter are even less critical, and deviations of up to ± 25% are acceptable here.

Author: V.Polyakov, Moscow

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