Parallel LC-circuit, calculation formulas

In radio engineering, a parallel resonant LC-circuit is used as the basis for frequency-selective circuits and is much more common than a series circuit. Real circuit elements exhibit losses, and circuit analysis uses a realistic model of ideal lumped elements, in which losses are accounted for using “virtual” series active resistances R_L & R_C.

The coil's parasitic self-capacitance is usually ignored, as it is simply added to the LC-circuit capacitance. The Coil64 program calculates the losses in the wire R_L without considering losses in the former, shield, core, or any other objects with which the electromagnetic field surrounding the coil interacts. However, the skin effect and proximity effect are taken into account. These losses are also accounted for by the “coil constructive quality factor” (Q_L). This is not the quality factor of the entire circuit, but the quality factor of the coil itself, which is related to its loss resistance by the following relationship:

[1]

Loss in a capacitor are an order of magnitude smaller and is characterized by the capacitor's quality factor. Since capacitor loss is concentrated primarily in the dielectric, its quality factor Q_C and loss resistance R_C can be considered related to the parameter tanδ, which accounts for dielectric loss, as follows:

[2]

When analyzing a circuit, it is often converted into an equivalent parallel RLC circuit. In this case, by replacing resistances with conductances, we simplify the analysis and obtain formulas identical to those for a series circuit. Many radio amateurs believe that series R_L and R_C circuits are simply converted into parallel R. This is not true:

As we can see, the resistances and reactances are “mixed up” with this transformation, so for clarity, we'll perform the analysis without using conductances, directly following the original circuit. The input impedance of the two-terminal network is as follows:

Active and reactive (imaginary) components:

At resonance, the currents in the reactive elements (I_L, I_C) are Q times greater than the total current of the circuit (I), therefore, for a parallel circuit, the phenomenon is called current resonance.

The resonant frequency of a parallel oscillatory circuit is the frequency at which the reactive component of the input impedance is zero, the input impedance is absolutely resistive, and, accordingly, the phase shift between the current and voltage at the circuit's input terminals is also zero. Equating X_i to zero and performing the appropriate transformations yields the following formula for the resonant frequency of a parallel LC-circuit:

[3]

One of the most important parameters of a circuit is its characteristic impedance:

The parallel LC-circuit resonant frequency formula can be represented differently:

[5]

ω₀ - resonant frequency of a series LC-circuit.

As we can see, the resonant frequency of a parallel LC-circuit is equal to the resonant frequency of a series LC-circuit composed of the same elements, with the addition of a correction factor √[(L/C - RL^2)/(L/C - RC^2)]. In practice, this coefficient is always close to one and is equal to one if R_L=R_C or R_L=R_C=0.

Example:

We have a circuit with an inductance of 3μH and a capacitance of 42pF. The coil's loss resistance is RL = 2 ohms, and the capacitor's loss resistance is RC = 0.1 ohms. According to Thompson's formula, the circuit's resonant frequency is 14.178649 MHz, which we calculate more precisely using formula [1]: 14.178253 MHz. As we can see, active loss resistance introduce additional reactance into an ideal circuit and lower its frequency, in this case by almost 400 Hz.

This very small deviation must be taken into account, but it is much smaller than the deviations introduced by unaccounted parasitic capacitances. Therefore, when the conditions R_L << ρ, R_C << ρ are met, which is usually the case in practice, we can assume that the current resonance conditions coincide with the voltage resonance conditions in a series LC-circuit composed of the same elements L and C.

ω₀ = 1/√LC or ƒ₀ = 1/(2π√LC)

The “family similarity” between series and parallel LC-circuits doesn't end there.
Under the same conditions: R_L<< ρ, R_C<< ρ
where Z_i-ser = (R_L + R_C) + j(ωL - 1 ⁄ ωC) – input impedance of a series LC-circuit composed of the same elements.

• R_Σ = R_L + R_C;

As we can see, we can assume that the loss resistances of the coil and the capacitor are summed up, so the overall quality factor of the circuit Q can be determined by the following expression:

[6]

At resonant frequency ω₀:

[7]

Since the reactances compensate each other, the LC-circuit at the resonant frequency has only active resistance equal to Re (the equivalent or effective resistance of the circuit).

From the last formula it follows that:

[8]

That is, the quality factor of a LC-circuit is equal to the ratio of its characteristic impedance to the loss resistance. In other words, at a resonant frequency, a circuit with lower capacitance and higher inductance has a higher Q-factor. How then does the quality factor of a LC-circuit relate to the constructive quality factor of a coil? To understand this, it should be borne in mind that the characteristic impedance of a circuit is numerically equal to the modulus of the reactance of inductance or capacitance at the resonant frequency. The latter, as is known, are equal in this case and differ only in sign. If we neglect loss in the capacitor, then formula [8] reduces to formula [1]. After all, at the resonant frequency ρ = |X_L|, and in the sum R_Σ = R_L + R_C, we do not take into account the last term. In other words, if we neglect loss in the capacitor, then the quality factor of the LC-circuit is equal to the constructive quality factor of the coil. As a result, we come to the conclusion that formulas [1] and [8] are equivalent in this case. If we need to take into account loss in the capacitor, then we should use formula [6].

There are two important points to note:

1. Coil64 calculates the constructive Q-factor for an “alone coil in a vacuum”. The presence of a shield increases the distributed capacitance and reduces the inductance. The characteristic impedance of the circuit drops, and the Q-factor decreases. Eddy current loss in the shield also add to this. The coil's former also reduces its Q-factor and the Q-factor of the LC-circuit, respectively.
2. The coil's Q-factor increases with frequency only at “low” frequencies, far from the coil's self-resonance frequency. As the frequency moves to self-resonance, the Q-factor reaches its maximum at frequencies of 25-75% of Fsrf and then gradually decreases. This occurs because at these frequencies, the coil's self-inductance and self-capacitance begin to show a frequency dependence.

The amplitude-frequency characteristic has the same form as the resonant curve of a series LC-circuit; the phase response is a mirror image of the phase response of a series LC-circuit.

The concept of a circuit's bandwidth is important. This is the frequency range within which the impedance Z_in is no lower than 1 ⁄ √2 (or 0.707) of the maximum at the resonant frequency. The following formula, which can be used to measure the quality factor, is valid:

In practice, the amount of attenuation of unwanted frequencies by the circuit is of interest:

[10]

For detunings greater than three passbands, the formula is simplified:

where the sign is not taken into account.

In a real circuit, the circuit is connected to an oscillation source and a load, which introduce additional losses that reduce the quality factor. The equivalent quality factor Q of a loaded parallel LC-circuit is:

Q = Q₀·R_i ⁄ (R_e + R_i)

• Q₀ : quality factor of an unloaded LC-circuit;
• Ri : source input impedance;
• Re : equivalent resistance of an unloaded LC-circuit;

This formula can be used to take into account the effect of any resistances connected to the circuit (for example, a load) on its quality factor.

To reduce the influence of external circuits, as well as to transform and match impedances, partial inclusion of the load in the LC-circuit is used.

As the figure shows, this can be accomplished in various ways: with a capacitive divider, by tapping the circuit's coil, or by using a separate magnetic coupling coil. Then the LC-circuit output impedance is:
R_out = T²R_e
where T : The effective transformation ratio (not to be confused with the magnetic coupling coefficient). It can be calculated as a ratio of resistances, similar to a resistive divider. But reactances are used instead of resistances.

• a) For a capacitive divider, after transforming the ratio of the reactances of the capacitors, we obtain:
  T = C1 ⁄ (C1 + C2)
• b) At first glance, for a coil tapped divider, the formula should be similar: T = Lc ⁄ (Lc + Lo). This would be true if both coil parts, Lc and Lo, were independent with a magnetic coupling coefficient equal to zero. Otherwise, T could be defined as the ratio of the number of turns before the tap to the total number of coil turns, as in an autotransformer, in which the magnetic coupling coefficient is known to be close to unity. However, in the case of an air core coil, the magnetic coupling coefficient is less than unity. In this case, the total inductance is calculated as L = (Lc + M) + (Lo + M), where Lc and Lo are the self-inductances of the coil parts before and after the tap, and M is their mutual inductance. Then the effective transformation ratio is:
  T = (Lc + M) / L
  where L : self-inductance of the LC-tank coil; Lс : self-inductance of the part of the coil before the tap; and M : mutual inductance between the parts of the coil before and after the tap.
• c) For inductive coupling, the transformation ratio in the case of air core coils is also not equal to the ratio of the number of turns and is determined as follows:
  T = M ⁄ L
  where L : self-inductance of the LC-tank coil, and M : mutual inductance of the LC-tank coil and the coupling coil.

Please note that when modeling a circuit with a tapped coil in LTSpice, the coil is represented as two coupled coils connected in series. This requires the actual magnetic coupling coefficient, which Coil64 can calculate. Strictly speaking, the coupling coefficient also depends on the loss in the coil, R_L. However, for a Q-factor greater than 50, this dependence can be neglected. However, by substituting the single-layer coil parameters provided by Coil64, including the loss resistance, into LTSpice, we take this dependence into account. Furthermore, the program allows calculating the coupling coefficient of both two air core coils and two parts of a tapped coil.

When connecting an external load to an LC-circuit using tap, the resulting Q-factor is determined:
Q = Q_0·R_u ⁄ (R_e + R_u)

R_u = T²R_i (R_i – external load)

It should be noted that for maximum electromagnetic energy transfer efficiency, the output impedance of the circuit must be equal to the load impedance. This also applies when the circuit is matched to the signal source.