Resonant circuits

Resonant circuits, also known as oscillating circuits, are oscillatory circuits that respond strongly when excited at a certain frequency (resonance). There are two types of resonant circuits: parallel resonant circuits and series resonant circuits.

A system is oscillatory if it can store energy in two forms and convert between the two forms. Resonant circuits therefore always have a capacitance (electrical storage) and an inductance (magnetic storage).

Resonant circuits are used in areas such as energy, measurement and communication technology. There, they are used, for example, to filter mains interference or signals.

Resonant circuits can, under certain circumstances, lead to undesirable effects such as extreme overvoltage or undervoltage.

1 Resonance phenomenon

Resonance occurs when oscillatory systems are excited at a frequency close to their resonant frequency.

This section first describes the state of a free oscillation and that of a forced oscillation. In subchapter 2, the condition for resonance is defined mathematically and calculated in the following chapter in example ?? for a series and parallel oscillating circuit.

Figure 1 shows an example of the circuit diagram of an RLC series resonant circuit. The series connection of R, L and C can be short-circuited via the switch shown. At time \(t=0\), the switch closes, the voltage across the capacitor is \(U_0>0\) (capacitor charged) and no current flows \(I=0\) (inductor discharged).

Figure 1b shows the time curve of the capacitive voltage \(u\) and the current \(i\) through the RLC series resonant circuit from \(t=0\). The resonant circuit is in a state of free oscillation. This means that currents and voltages oscillate without external excitation at the natural frequency \(f_d\) (damped) of the resonant circuit.

The amplitudes decay due to damping (\(R>0\)). They can each be described by an exponential curve, which envelops the respective time course (dashed curve). For more precise calculations of compensation processes, see Module 12: Switching Processes.

The energy \(E\) oscillates between the electrical form in the \(\vec {\underline {E}}\) field of the capacitance (\(E_{el} \sim u^2\)) and the magnetic form in the \(\vec {\underline {H}}\) field of the inductance (\(E_{mag} \sim i^2\)). The minima and maxima and the zero points of both time curves show how the energy is converted back and forth twice in each period.

Due to the real (loss) power in the ohmic resistance (\(R > 0\)), part of the energy is converted into heat in the resistance during every energy conversion. The resistance dampens the oscillating circuit and causes the current and voltage to decay in a state of free oscillation.

The natural frequency \(\omega _d = 2\pi f_d\) is independent of the amplitude. In undamped systems, it corresponds exactly to the resonant frequency \(\omega _0\). In slightly damped systems, \(\omega _d\) is slightly smaller than \(\omega _0\). Heavily damped systems do not oscillate, but decay aperiodically.

Figure 2a shows an RLC series resonant circuit excited by an ideal voltage source \(u_e\) at a capacitor (forced oscillation) for comparison. Figure 2b shows the steady-state time response of current and voltage of the capacitance under sinusoidal excitation.

In a steady state, current and voltage oscillate at the frequency of the voltage source, which in this case is also called the excitation frequency.

If the excitation frequency corresponds to the resonance frequency, resonance occurs. This can be achieved by varying the excitation frequency or the component sizes \(L\) or \(C\).

2 Definition of resonance condition, frequency, quality factor, characteristic impedance

For resonant circuits, when resonance occurs, the imaginary part of the impedance becomes zero. Respectively, the imaginary part of the admittance becomes zero and the current and voltage at the output terminals are in phase.

The resonance condition at the output terminals of the resonance circuit is:

\begin {equation} \begin {aligned}\label {eq:resonanzbedingung} \varphi _{u0} - \varphi _{i0} &\overset {!}{=} 0 &&\text {Phase shift equal to zero}\\\leftrightarrow \Im \{ \underline {Z}_0 \} &\overset {!}{=} 0 &&\text {Reactance equal to zero} \end {aligned} \end {equation}

Frequency-variable quantities are marked with index \(0\) in the following resonance case:

\begin {equation*} \underline {Z},\ \underline {U},\ \underline {I},\ \varphi _{u},\ \varphi _{i} = \underline {Z}_0,\ \underline {U}_0,\ \underline {I}_0,\ \varphi _{u0},\ \varphi _{i0} \text {für} \omega =\omega _0 \end {equation*}

The resonance frequency \(f_0\) is the frequency at which excitation produces resonance or satisfies the resonance condition. In some cases, the resonant circuit frequency \(\omega _0\) is also imprecisely referred to as the resonance frequency. In general, \(\omega _0 = 2\pi f_0\).

In simple RLC oscillating circuits, the reactance \(X_L\) of the inductance and \(X_C\) of the capacitance cancel each other out in the case of resonance. In terms of magnitude, these are equal, which is why their magnitude in the case of resonance is also referred to as the characteristic impedance \(X_k\). Similarly, the characteristic conductance \(B_k\) is defined as the magnitude of the conductance of the inductance and the capacitance at resonance:

\begin {align} X_k = |X_{L,0}| &= |X_{C,0}|\label {eq:kennwiderstand}\\ B_k = |B_{L,0}| &= |B_{C,0}| = \frac {1}{X_k}\label {eq:kennleitwert} \end {align}

Characteristic resistance and conductance have the same units as impedance [\(\Omega \)] and admittance [\(\mathrm {S}\)].

The quality factor \(Q\), also known as the resonance sharpness, is a dimensionless quantity. It is generally defined as the ratio of the reactive power of the inductance or capacitance to the (active) power of the oscillating circuit in the case of resonance:

\begin {align} \label {eq:q:leistung} Q &= \frac {|Q_{L,0}|}{P_{R,0}} = \frac {|Q_{C,0}|}{P_{R,0}}&&\text {generally} \end {align}

The quality factor is sometimes also defined specifically for series or parallel oscillating circuits.

For RLC series oscillating circuits, the quality factor corresponds to the ratio of characteristic impedance \(X_k\) to ohmic resistance \(R\), while for RLC parallel oscillating circuits, the quality factor corresponds to the ratio of characteristic conductance \(B_k\) to ohmic conductance \(G\): \begin {align} Q &= \frac {X_k}{R}&&\text {in series oscillating circuit}\label {eq:q:kennwiderstand}\\ Q &= \frac {Y_k}{G}&&\text {in the parallel oscillating circuit}\label {eq:q:kennleitwert}\\ \intertext { \index {Überspannung}\index {Überstrom}\index {Resonanz, >Spannungs- (spannungsbezogen)}\index {Resonanz, >Strom- (strombezogen)}The quality factor thus also describes the ratio of possible overvoltages to source voltages for series resonant circuits in the case of voltage resonance, or the ratio of overcurrents to source currents for parallel resonant circuits in the case of current resonance: } Q &= \frac {U_{L,0}}{U_{q}} = \frac {U_{C,0}}{U_{q}} && \text {in voltage resonance}\label {eq:q:spannung}\\ Q &= \frac {I_{L,0}}{I_{q}} = \frac {I_{C,0}}{I_{q}} && \text {in electrical resonance}\label {eq:q:strom} \end {align}

3 Resonant frequency using the example of ideal LC oscillating circuits

In example ??, the resonance circuit frequency for ideal LC series and LC parallel oscillating circuits is determined.

The examples use calculations to illustrate why resonance is caused by voltages in series resonant circuits and by currents in parallel resonant circuits. To distinguish between the two, we therefore refer to voltage-related and current-related resonance.

Resonance in LC resonant circuitsresonanz:lc Berechnung der Resonanzfrequenz \(\omega _0\) für LC-Serien- und LC-Parallelschwingkreise.

In the case of resonance, the voltages across \(L\) and \(C\) cancel each other out in an ideal series resonant circuit. Therefore, theoretically, arbitrarily high voltages can arise across \(L\) and \(C\). In real circuits, however, the voltage is limited by the ohmic resistance \(R\). [See RLC resonant circuit in section 4]

The same applies in the case of resonance for the currents through \(L\) and \(C\) in a parallel resonant circuit.

The resonance frequency of simple RLC resonant circuits is identical to that of ideal LC resonant circuits, since the resistance \(R\) does not affect the reactive component of the impedance, with \(R\), \(L\) and \(C\) all in series or all in parallel.

4 RLC series resonant circuit - resonance behaviour

This subchapter examines the resonance behaviour of an RLC series resonant circuit as an example. The resonant circuit is shown in Figure 3 and corresponds to the ideal LC series resonant circuit from Example ?? with an additional ohmic resistance \(R\) in series.

The impedance \(\underline {Z}\) of the RLC series resonant circuit shown in Figure 3 is given by:

\begin {align} \underline {U} &= \underline {I} \cdot \left (R + \mathrm {j}\omega L - \mathrm {j}\frac {1}{\omega C}\right )\\ \underline {Z} &= R + \mathrm {j}\left (\omega L - \frac {1}{\omega C}\right )\label {eq:rlcs:z} \end {align}

In the case of resonance, according to equation 1, the imaginary part of the impedance disappears. Consequently, in this arrangement, the reactances of inductance and capacitance cancel each other out. By inserting \(\underline {Z}\) into the resonance condition, the resonance frequency \(\omega _0\) can be determined:

\begin {equation} \label {eq:rlcs:w0} \begin {aligned} \Im \{\underline {Z}\} = \left (\omega L - \frac {1}{\omega C}\right ) &\overset {!}{=} 0 \\ \omega _0 L - \frac {1}{\omega _0 C} &= 0 &&\Rightarrow & \omega _0 &= \frac {1}{\sqrt {LC}} \end {aligned} \end {equation}

Since the reactance of the inductance is positive and proportional to the frequency (\(X_L \sim \omega \)), and since the reactance of the capacitance is negative and proportional to the reciprocal of the frequency (\(-X_C \sim \frac {1}{\omega }\)), the inductive component predominates for frequencies above the resonance frequency and the capacitive component for frequencies below it.

This results in the following case distinction for the RLC series resonant circuit:

\begin {equation} \Im \{\underline {Z}\} = \begin {cases} \omega L - \frac {1}{\omega C} < 0 & \qquad \omega < \omega _0 \qquad \text {ohmic-capacitive}\vphantom {\Big |}\\ \omega L - \frac {1}{\omega C} = 0 & \qquad \omega = \omega _0 \qquad \text {purely ohmich}\vphantom {\Big |}\\ \omega L - \frac {1}{\omega C} > 0 & \qquad \omega > \omega _0 \qquad \text {ohmic-inductive}\vphantom {\Big |} \end {cases} \label {eq:rlcs:fallunterscheidung} \end {equation}

The case differentiation is illustrated in Figure 4 using a pointer diagram and in Figure 5 using an impedance curve.

Digression: Free oscillation as a limiting case of resonance

The term resonance, from the Latin resonare (to echo), always refers, in accordance with the meaning of the word, to system states in the event of external excitation.

As shown in Figure 1, a charged RLC series resonant circuit oscillates at its natural frequency in a state of free oscillation when the external terminals are short-circuited.

The short circuit can be regarded as an ideal voltage source (internal resistance \(R_i=0\)) with voltage \(U=0\). The natural frequency (with damping) is obtained from the solution of the homogeneous differential equation of the oscillating circuit. More on this in Module 12: Switching Processes.

Knot and mesh equations:

\begin {align*} u(t) &= u_R + u_L + u_C \overset {!}{=} 0 \\ i(t) &= i_R = i_L = i_C \end {align*}

Differential equation:

\begin {align*} u(t)= i R + L \frac {\mathrm {d}i}{\mathrm {d}t} + \frac {1}{C} \int i \mathrm {d}t &\overset {!}{=} 0 \\ L \frac {\mathrm{d} ^2i}{\mathrm {d}t^2} + R \frac {\mathrm {d}i}{\mathrm {d}t} + \frac {1}{C} i &\overset {!}{=} 0 \end {align*}

Approach:

\begin {align*} i(t) &= k \mathrm {e}^{\lambda t}&\frac {\mathrm {d}i}{\mathrm {d}t} &= k\lambda \mathrm {e}^{\lambda t}&\frac {\mathrm{d} ^2 i}{\mathrm {d}t^2} &= k\lambda ^2 \mathrm {e}^{\lambda t} \end {align*}

Insertion:

\begin {align*} L \frac {\mathrm{d} ^2i}{\mathrm {d}t^2} + R \frac {\mathrm {d}i}{\mathrm {d}t} + \frac {1}{C} i &= 0 \\ k \cdot \bigg ( \underbrace {L \lambda ^2 + R \lambda + \frac {1}{C}}_{={\ 0}} \bigg ) \mathrm {e}^{\mathrm {j} \lambda t} &= 0 \end {align*}

Dissolve: \begin {equation*} \lambda _{1/2} = -\frac {R}{2L} \pm \mathrm {j} \sqrt {\frac {1}{LC} - \left (\frac {R}{2L}\right )^2} \end {equation*}

As explained in more detail in Module 12, if \(\frac {1}{LC} > \left (\frac {R}{2L}\right )^2\) (low damping), a complex solution results. Only in this case can the resonant circuit oscillate freely, whereby the real part of the solution defines the decay constant and the imaginary part (root term) defines the natural frequency \(\omega _d\):

\begin {equation*} \omega _{d} = \sqrt {\frac {1}{LC} - \left (\frac {R}{2L}\right )^2} \end {equation*}

4.1 Case differentiation in the pointer diagram

Figure 4 shows a pointer diagram for all three cases. The voltages across \(R\), \(L\) and \(C\) are shown, as well as the total voltage \(\underline {U}\) when excited with a constant current \(\underline {I}\). The true-to-scale length ratios \(\frac {U_L}{U_C}\) correspond to: \(\frac {1}{2}\), \(\frac {\sqrt {2}}{\sqrt {2}}\), \(\frac {2}{1}\).

Since this is a passive resonant circuit (load arrow system), the active power \(P\) of the resonant circuit is always positive with \(P=\Re \{\underline {U}\cdot \underline {I}\}\).

With a real impressed current \(\underline {I}\), the real part of the total voltage \(\underline {U}\) is therefore always positive. The pointer of the total voltage \(\underline {U}\) therefore points either to the first quadrant (resistive-inductive), to the fourth quadrant (resistive-capacitive) or to the real axis (purely resistive).

4.2 Case differentiation in impedance curve

For the series resonant circuit shown in Figure 3, the impedance is given by: \begin {equation} \begin {aligned}\underline {Z} &= R + \mathrm {j}\left (X_L + X_C\right ) \\ &= R + \mathrm {j}\left (\omega L - \frac {1}{\omega C}\right ) \end {aligned} \end {equation}

Capacitive reactance (hyperbolic) predominates for frequencies below the resonance frequency, while inductive reactance (linear) predominates for frequencies above the resonance frequency as shown in equation 3.

Figure 5 shows the impedance curve of an RLC series resonant circuit as a function of frequency relative to the resonance frequency \(1/{f}{f_0}\) equivalent to \(1/{\omega }{\omega _0}\). Shown are the magnitudes of the impedance \(|\underline {Z}|\), the reactance \(|X|\), the reactance of the inductance \(|X_L|\), the reactance of the capacitance \(|X_C|\) and the ohmic resistance \(R\).

As can be seen in the impedance curve, the impedance in the resonance case is purely ohmic and corresponds to the ohmic resistance \(R\): \begin {equation} \label {eq:rlcs:z0} \underline {Z}_0 = R \end {equation}

The reactance of the inductance \(X_L\) (positive) and the reactance of the capacitance \(X_C\) (negative) cancel each other out in the case of resonance (intersection of the curves). The magnitudes of both reactances are equal in the case of resonance and correspond to the characteristic impedance \(X_k\):

\begin {equation} \label {eq:rlcs:xk} \begin {aligned} X_k &= |X_{L,0}| = |X_{C,0}| = \sqrt {\frac {L}{C}} \\ &= \omega _0 L \ \ = \frac {1}{\omega _0 C} \end {aligned} \end {equation}

It can also be seen that the impedance value approaches the capacitive curve for very low frequencies \(\omega << \omega _0\) with \(\lim _{\omega \rightarrow 0}|Z|=|X_C|\) and for very high frequencies \(\omega >> \omega _0\) approaches the inductive curve with \(\lim _{\omega \rightarrow \infty }|Z|=|X_L|\).

4.3 Resonance curve and quality factor of an RLC series resonant circuit

The impedance of an RLC series resonant circuit can be expressed, as in equation ??, in terms of the component values \(R\), \(L\) and \(C\) as a function of frequency \(\omega \):

\begin {equation} \underline {Z} = R + \mathrm {j}\left (\omega L - \frac {1}{\omega C}\right ) \end {equation}

Using the resonant circuit frequency from equation 2, the angular frequency can be normalised to this value. To do this, the \(\omega \) of the reactance terms are extended with \(\omega _0\) (\(\omega =\omega \frac {\omega _0}{\omega _0}\)). This allows the reactances \(X_L\) and \(X_C\) to be expressed using the characteristic impedance \(X_k\) from Eq. 6:

\begin {equation} \label {eq:rlcs:zxk} \begin {aligned} \underline {Z} &= R + \mathrm {j}\left ( \omega L \frac {\omega _0}{\omega _0} - \frac {1}{\omega C}\frac {\omega _0}{\omega _0}\right )\\ &= R + \mathrm {j}\left ( X_{L,0} \frac {\omega }{\omega _0} + X_{C,0} \frac {\omega }{\omega _0}\right )\\ &= R + \mathrm {j}\left ( \frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right ) X_k \end {aligned} \end {equation}

The following applies:

\begin {align} X_L &= +X_k \frac {\omega }{\omega _0} &= +\sqrt {\frac {L}{C}}\cdot \frac {\omega \sqrt {LC}}{1} &= \omega L &&\text {(linear)}\label {eq:rlcs:xlw0}\\ X_C &= -X_k \frac {\omega _0}{\omega } &= -\sqrt {\frac {L}{C}}\cdot \frac {1}{\omega \sqrt {LC}} &= -\frac {1}{\omega C} &&\text {(hyperbolically)}\vphantom {\Big |}\label {eq:rlcs:xcw0}\\ X_k &= \sqrt {\frac {L}{C}}&\text {mit} \omega _0 &= \frac {1}{\sqrt {LC}} &&\text {(constant)}\label {eq:rlcs:xkw0} \end {align}

As stated in equation ??, the quality factor \(Q\) of a series resonant circuit is the ratio of its characteristic resistance \(X_k\) to its ohmic resistance \(R\). This allows the impedance of the resonant circuit to be expressed in terms of its characteristic resistance \(X_k\) using the quality factor \(Q\):

\begin {equation} \label {eq:rlcs:znorm} \frac {\underline {Z}}{X_k} = \frac {1}{Q} + \mathrm {j}\left (\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right ) \vphantom {\Big |} \qquad \text {mit} Q = \frac {X_k}{R} \end {equation}

The term \(\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\) is also referred to as relative detuning \(\nu _r\).[?]

Figure 6 shows several resonance curves of an RLC series resonant circuit for different quality factors \(Q\) for comparison. The y-axis shows the impedance normalised to the characteristic resistance \(X_k\) and the x-axis shows the frequency normalised to the resonance frequency \(\omega _0\).

The most significant difference between the individual impedance curves for different quality factors can be seen in the resonance frequency range. As shown in equation 5, the following applies in the case of resonance: \(Z_0 = R\).

A high quality factor means that the resonant circuit has a very low impedance relative to the characteristic resistance \(X_k\) at the resonance frequency. The quality factor is therefore also a measure of the resonant circuit’s ability to oscillate or, conversely, of its damping.

The following applies to the series resonant circuit: \begin {equation} Q = \frac {X_k}{R} = \frac {\sqrt {\frac {L}{C}}}{R} \end {equation}

For frequencies well below the cut-off frequency \(\omega << \omega _0\) and well above \(\omega >> \omega _0\), the reactive components of the impedance dominate, as shown in Chapter 4. Therefore, the influence of the quality factor on the impedance at these frequencies is small (approximation of the curves).

Figure 7 shows the phase curve of the resonant circuit for different qualities with \(\varphi = \varphi _U - \varphi _I\). For frequencies significantly below the resonance frequency \(\omega << \omega _0\), the phase shift \(\varphi \) is almost \(-90^\circ \) (purely capacitive). For frequencies significantly above the resonance frequency \(\omega >> \omega _0\), the phase shift \(\varphi \) is almost \(+90^\circ \) (purely inductive).

The higher the quality factor (resonance sharpness), the steeper the transition from capacitive to inductive in the resonance frequency range.

4.4 Voltage resonance using the example of an RLC series resonant circuit

When a series resonant circuit is excited by a constant voltage source \(\underline {U}_q\), a voltage resonance may occur, characterised by overvoltages at \(L\) and \(C\).

Consider the RLC series resonant circuit from Figure 3 with impedance \(\underline {Z}\) from Equation ??. The voltage \(\underline {U}\) across the resonant circuit corresponds to the constant source voltage \(\underline {U}_q\) at its external terminals as shown in Figure 8.

With the knot and mesh equation:

\begin {equation*} \begin {aligned} \underline {U} &= \underline {U}_R + \underline {U}_L + \underline {U}_C = \underline {U}_q\\ \underline {I} &= \underline {I}_R = \underline {I}_L = \underline {I}_C \end {aligned} \end {equation*}

and the component impedances follow the complex voltage dividers:

\begin {align*} \frac {\underline {U}_R}{\underline {U}} &= \frac {\underline {I}}{\underline {I}} \cdot \frac {R}{R+\mathrm {j}\left (\omega L-\frac {1}{\omega C}\right )}\\ \frac {\underline {U}_L}{\underline {U}} &= \frac {\underline {I}}{\underline {I}} \cdot \frac {\mathrm {j}\omega L}{R+\mathrm {j}\left (\omega L-\frac {1}{\omega C}\right )}\\ \frac {\underline {U}_C}{\underline {U}} &= \frac {\underline {I}}{\underline {I}} \cdot \frac {\frac {1}{\mathrm {j}\omega C}}{R+\mathrm {j}\left (\omega L-\frac {1}{\omega C}\right )} \end {align*}

After shortening the currents, the frequency-variable impedance terms remain. The individual voltages are obtained by multiplying them by the voltage \(\underline {U}\).

To normalise the frequency-dependent voltages, the reactances can be expressed as in equation 6 using the characteristic impedance \(X_k\) and the resonance circuit frequency \(\omega _0\) from equation 2. With the impedance normalised in this way from equation 8, the terms for the voltage dividers are obtained:

\begin {align*} \frac {\underline {U}_R}{\underline {U}} &= \frac {R}{R+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )X_k}\\ \frac {\underline {U}_L}{\underline {U}} &= \frac {\mathrm {j}X_k\frac {\omega }{\omega _0}}{R+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )X_k}\\ \frac {\underline {U}_C}{\underline {U}} &= \frac {-\mathrm {j}X_k\frac {\omega _0}{\omega }}{R+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )X_k} \end {align*}

Figure 9 shows the magnitudes of the voltages \(U_R\), \(U_L\), \(U_C\) of the RLC series resonant circuit as a function of the angular frequency normalised to the resonant angular frequency. Normalisation to \(\omega _0\) allows, for example, the comparison of the quality of series resonant circuits with different resonant frequencies.

As can be seen in the figure, in the resonance frequency range \(\omega = \omega _0\), here is a significantly higher voltage across \(L\) and \(C\) compared to the ohmic resistance \(R\).

The degree of voltage increase at the resonant frequency is expressed by the quality factor. As defined in equation ??, it corresponds to the ratio of the overvoltage of the apparent resistances (\(X_L\), \(X_C\)) \(U_{L,0}\), \(U_{C,0}\) to the voltage of the effective resistance (\(R\)) \(U_{R,0}\) in the case of resonance (\(omega = \omega _0\)).

\begin {equation} Q = \frac {U_{L,0}}{U} = \frac {U_{C,0}}{U} = \frac {X_k}{R} \end {equation}

As can be seen in the graph, the voltage values \(|U_L|\) and \(|U_C|\) do not reach their maximum at the resonance frequency!

Digression: Derivation of quality via reactive power

In general, the following applies to the apparent power of a component: \begin {align} \label {eq:ql}\underline {S} &= S \cdot \mathrm {e}^{\mathrm {j}\varphi } \\ &= U \cdot I \cos {\varphi } + \mathrm {j} U \cdot I \sin {\varphi } \\ &= P \cdot \mathrm {j} Q \end {align}

As apparent resistances, inductance and capacitance can only absorb and emit reactive power. With \(\varphi _L = +90^\circ \) and \(\varphi _C = -90^\circ \), the following applies to the resonance case for the reactive power of both components:

\begin {equation} \label {eq:blindleistung:rlcs} \begin {aligned} Q_L &= U_L I_L \cdot \sin {(+\frac {\pi }{2})} = + U_L \cdot I_L \vphantom {\big |}\\ Q_C &= U_C I_C \cdot \sin {(-\frac {\pi }{2})} = - U_L \cdot I_L \vphantom {\big |} \end {aligned} \end {equation}

When excited by a constant voltage source \(\underline {U}_q\), the same current \(\underline {I}\) flows through all components in the series resonant circuit.

Using the general definition of quality via the reactive power in equation ?? of the inductance \(Q_L\) or the capacitance \(Q_C\) and the active power \(P\) of the resonant circuit in the case of resonance, the following applies to the series resonant circuit:

\begin {equation} \label {eq:rlcs:q:guete} \begin {aligned} Q &= \frac {|Q_{L,0}|}{P_0} = \frac {U_{L,0}I_L}{U_{R,0}I_R} = \frac {U_{L,0}I}{U_{R,0}I} = \frac {U_{L,0}}{U}\\ &= \frac {|Q_{C,0}|}{P_0} = \frac {U_{C,0}I_C}{U_{R,0}I_R} = \frac {U_{C,0}I}{U_{R,0}I} = \frac {U_{C,0}}{U} \end {aligned} \end {equation}

As shown in Equation 13 and stated in Equation ??, the quality factor therefore corresponds to the ratio of the overvoltage of the apparent resistances (\(L\), \(C\)) to the voltage of the effective resistance (\(R\)) in the resonance case.

Using the voltage divider rule, the quality factor can also be expressed as the ratio of the characteristic impedance to the effective impedance by substituting the reactances for the resonance case, as stated in equation ??. With \(X_k\) from Eq. 6 and \(\underline {Z}\) from Eq. ??, the following applies:

\begin {equation} \begin {aligned} \frac {U_{L,0}}{U} &= \frac {\cancel {I}}{\cancel {I}} \cdot \frac {|\mathrm {j}X_{L,0}|}{\big |R + \mathrm {j}(X_{L,0}+X_{C,0})\big |}\vphantom {\Bigg |}\\ &= \frac {|X_{L,0}|}{ \big |R + \mathrm {j}( \underbrace {X_{L,0} + X_{C,0}}_{={\ 0}})\big |}\vphantom {\Bigg |}\\ &= \frac {|X_{L,0}|}{R} = \frac {X_k}{R} = Q \end {aligned} \end {equation}

4.5 Current curve of the RLC series resonant circuit

When excited with a constant voltage \(\underline {U}_q\), the RLC series resonant circuit shown in Figure 8 results in a frequency-dependent current curve. This is shown in terms of magnitude in Figure 10 for different quality factors \(Q\).

The current is at its maximum at the resonance frequency, as can be seen in the graph. As shown in Figure 5, the impedance is minimal in the case of resonance and corresponds to the ohmic resistance \(R\).

In general, the following relationship applies to the current \(\underline {I}\) with the impedance \(\underline {Z}\) from equation 8 and the resonance frequency \(\omega _0\) from equation 2:

\begin {equation} \begin {aligned} \underline {I} &= \frac {\underline {U}}{\underline {Z}} \\ &= \frac {\underline {U}_q}{R + \mathrm {j}\left (\omega L - \frac {1}{\omega C}\right )} \\ &= \frac {\underline {U}_q}{R + \mathrm {j}\left (\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right )X_k} \end {aligned} \end {equation}

The maximum current \(I_0\) is obtained for a real source voltage \(\underline {U}_q = U_q\) from the quotient of the source voltage divided by the resistance \(R\):

\begin {equation} I_0 = \frac {U_q}{R} \qquad \text {mit} \underline {Z}_0 = R \end {equation}

This means that both the total current \(\underline {I}\) and its active component \(\Re \{\underline {I}\}\) are limited by the resistance \(R\).

For the current \(\underline {I}\) relative to the maximum current \(I_0\), the quality factor from equation 13 applies:

\begin {equation} \label {eq:rlcs:inorm} \begin {aligned} \frac {\underline {I}}{I_0} &= \frac {U_q}{\underline {Z}} \cdot \frac {R}{U_q} = \frac {R}{\underline {Z}}\\ &= \frac {1}{1 + \mathrm {j} \left (\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right ) Q} \end {aligned} \end {equation}

4.6 Voltage curve of the RLC series resonant circuit

If an RLC series resonant circuit is excited with a constant current \(\underline {I}_q\) as shown in Figure 11, component voltages result.

The component voltages are directly proportional to the respective impedances. The following applies to the total voltage:

\begin {equation*} \underline {U} = \underline {I}_q \cdot \underline {Z} \end {equation*}

The voltage curve therefore corresponds exactly to the impedance curve in Figure 6.

5 RLC parallel resonant circuit - resonance behavior

This subchapter examines the resonance behaviour of an RLC parallel resonant circuit. The resonant circuit is shown in Figure 12 and corresponds to the ideal LC parallel resonant circuit from Example ?? with an additional parallel ohmic resistance \(R\).

The investigation is analogous to that for the RLC series resonant circuit in Chapter 4. In the parallel resonant circuit, the admittance \(\underline {Y}\) is primarily considered. This results in similar case distinctions and calculation methods as in the series resonant circuit, whereby the roles of current and voltage, inductance and capacitance, as well as resistance and conductance are reversed:

\begin {equation} \underline {Y} = \underline {Z}^{-1} = G + \mathrm {j}B = \frac {\underline {I}}{\underline {U}} \end {equation}

The admittance \(\underline {Y}\) of the RLC parallel resonant circuit shown in Figure 12 is given by:

\begin {align} \underline {I} &= \underline {U} \cdot \left (G + \mathrm {j}B_C + \mathrm {j}B_L\right )\\ \underline {Y} &= G + \mathrm {j}\left (\omega C - \frac {1}{\omega L}\right )\label {eq:rlcp:y} \end {align}

In the case of resonance, according to equation 1, the imaginary part of the impedance or admittance disappears. Consequently, in this arrangement, the susceptances of inductance \(B_L\) and capacitance \(B_C\) cancel each other out. By inserting \(\underline {Y}\) into the resonance condition, the resonance frequency \(\omega _0\) can be determined:

\begin {equation} \label {eq:rlcp:w0} \begin {aligned} \Im \{\underline {Y}\} = \left (\omega C - \frac {1}{\omega L}\right ) &\overset {!}{=} 0 \\ \omega _0 C - \frac {1}{\omega _0 L} &= 0 &&\Rightarrow & \omega _0 &= \frac {1}{\sqrt {LC}} \end {aligned} \end {equation}

Since the susceptance of the capacitance is positive and proportional to the frequency (\(B_C = \omega C \sim \omega \)),

and since the susceptance of the inductance is negative and proportional to the reciprocal of the frequency (\(-B_L \sim \frac {1}{\omega }\)), the inductive component predominates for frequencies below the resonance frequency and the capacitive component predominates for frequencies above the resonance frequency. This results in the following case distinction for the RLC series resonant circuit:

\begin {equation} \Im \{\underline {Y}\} = \begin {cases} \omega C - \frac {1}{\omega L} < 0 & \qquad \omega < \omega _0 \qquad \text {ohmsch-induktiv}\vphantom {\Big |}\\ \omega C - \frac {1}{\omega L} = 0 & \qquad \omega = \omega _0 \qquad \text {rein ohmsch}\vphantom {\Big |}\\ \omega C - \frac {1}{\omega L} > 0 & \qquad \omega > \omega _0 \qquad \text {ohmsch-kapazitiv}\vphantom {\Big |} \end {cases} \label {eq:rlcp:fallunterscheidung} \end {equation}

The case differentiation is illustrated in Figure 13 using a pointer diagram and in Figure 14 using an admittance curve.

5.1 Case differentiation in the pointer diagram

Figure 13 shows an example of a phasor diagram for all three cases.

The diagram shows the currents flowing through \(R\), \(L\) and \(C\), as well as the total current \(\underline {I}\) when excited with a constant voltage \(\underline {U}\). The true-to-scale length ratios \(\frac {I_L}{I_C}\) correspond to: \(\frac {1}{2}\), \(\frac {\sqrt {2}}{\sqrt {2}}\), \(\frac {2}{1}\).

Since this is a passive resonant circuit (consumer counting arrow system), the active power \(P\) of the resonant circuit is always positive with \(P=\Re \{\underline {U}\cdot \underline {I}\}\).

When voltage \(\underline {U}\) is applied (plotted with phase angle \(0\)), the real component of the total current \(\underline {I}\) is always positive. The pointer of the total current \(\underline {I}\) therefore points either to the first quadrant (resistive-inductive), to the real axis (purely resistive) or to the fourth quadrant (resistive-capacitive).

5.2 Case differentiation in admittance curve

For the parallel oscillating circuit shown in Figure 12, the admittance is given by Equation ??: \begin {equation} \begin {aligned}\underline {Y} &= G + \mathrm {j}\left (B_C + B_L\right ) \\ &= G + \mathrm {j}\left (\omega C - \frac {1}{\omega L}\right ) \end {aligned} \end {equation}

The inductive susceptibility (hyperbolic) predominates for frequencies below the resonance frequency, while the capacitive susceptibility (linear) predominates for frequencies above the resonance frequency as shown in equation 20.

Figure 14 shows the admittance curve of an RLC parallel resonant circuit as a function of frequency relative to the resonance frequency \(1/{f}{f_0}\) equivalent to \(1/{\omega }{\omega _0}\). Shown are the magnitudes of the admittance \(|\underline {Y}|\), the susceptance \(|B|\), the susceptance of the capacitance \(|B_C|\), the susceptance of the inductance \(|B_L|\) and the ohmic conductance \(G\).

As can be seen in the admittance curve, the admittance is purely ohmic in the case of resonance and corresponds exactly to the ohmic conductance \(G\): \begin {equation} \underline {Y}_0 = G \end {equation}

The susceptances of the capacitance \(B_C\) (positive) and the inductance \(B_L\) (negative) cancel each other out in the case of resonance (intersection of the curves). The magnitudes of both susceptances are equal in the case of resonance and correspond to the characteristic value \(B_k\):

\begin {equation} \label {eq:rlcp:bk} \begin {aligned} B_k &= |B_{C,0}| = |B_{L,0}| = \sqrt {\frac {C}{L}} \\ &= \omega _0 C \ \ = \frac {1}{\omega _0 L} \end {aligned} \end {equation}

It can also be seen that the admittance approaches the inductive curve for very low frequencies \(\omega << \omega _0\) with \(\lim _{\omega \rightarrow 0}|Y|=|B_L|\) and for very high frequencies \(\omega >> \omega _0\) approaches the capacitive curve with \(\lim _{\omega \rightarrow \infty }|Y|=|B_C|\).

5.3 Resonance curve and quality factor of an RLC parallel resonant circuit

As in equation ??, the admittance of an RLC parallel resonant circuit can be expressed in terms of the component values \(R\), \(L\) and \(C\) as a function of frequency \(\omega \) as follows:

\begin {equation} \underline {Y} = G + \mathrm {j}B = \frac {1}{R} + \mathrm {j}\left (\omega C - \frac {1}{\omega L}\right ) \end {equation}

Using the resonant circuit frequency from equation 19, the circular frequency can be normalised to this. The calculation is performed analogously to the frequency normalisation for the impedance of the RLC series resonant circuit in equation 8. The susceptances \(B_C\) and \(B_L\) are described using the characteristic value \(B_k\) from Eq. 23 and the frequency ratio of \(\omega \) to \(\omega _0\). The following applies to the admittance:

\begin {equation} \label {eq:rlcp:ybk} \begin {aligned} \underline {Y} &= G + \mathrm {j}\left ( \omega C \frac {\omega _0}{\omega _0} - \frac {1}{\omega L}\frac {\omega _0}{\omega _0}\right )\\ &= G + \mathrm {j}\left ( B_{C,0} \frac {\omega }{\omega _0} - B_{L,0} \frac {\omega }{\omega _0}\right )\\ &= G + \mathrm {j}\left ( \frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right ) B_k \end {aligned} \end {equation}

The following applies:

\begin {align} B_C &= +B_k \frac {\omega }{\omega _0} &&= \omega C &&\text {(linear)}\label {eq:rlcp:bc}\\ B_L &= -B_k \frac {\omega _0}{\omega } &&= -\frac {1}{\omega L} &&\text {(hyperbolisch)}\vphantom {\Big |}\label {eq:rlcp:bl}\\ B_k &= \sqrt {\frac {C}{L}}&\text {mit} \omega _0 &= \frac {1}{\sqrt {LC}} &&\text {(konstant)}\label {eq:rlcp:bkw0} \end {align}

As stated in equation ??, the quality \(Q\) of a parallel resonant circuit can be described by the ratio of the characteristic value \(B_k\) to the effective value \(G\). In this respect, the definition differs from that in a series resonant circuit.

The general definition of quality in equation ?? based on the ratio of reactive power to active power can be used to derive the specific definition of quality for the parallel resonant circuit. The same applies to the series resonant circuit, as shown in equation 13:

\begin {equation} \label {eq:rlcp:q} \begin {aligned} Q &= \frac {|Q_{L,0}|}{P_0} = \frac {U \cdot I_{L,0}}{U\cdot I_R} = \frac {I_{L,0}}{I_R} = \frac {|B_{L,0}|}{G}\\ & = \frac {|Q_{C,0}|}{P_0} = \frac {U \cdot I_{C,0}}{U\cdot I_R} = \frac {I_{C,0}}{I_R} = \frac {|B_{C,0}|}{G} = \frac {B_k}{G} \end {aligned} \end {equation}

This allows the impedance of the oscillating circuit to be expressed in terms of its characteristic value \(B_k\) using the quality factor \(Q\):

\begin {equation} \label {eq:rlcp:ynorm} \frac {\underline {Y}}{B_k} = \frac {1}{Q} + \mathrm {j}\left (\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right ) \vphantom {\Big |} \qquad \text {mit} Q = \frac {B_k}{G} \end {equation}

The term \(\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\) is also referred to as relative detuning \(\nu _r\). [?]

Figure 15 shows several resonance curves of an RLC parallel resonant circuit for different quality factors \(Q\) for comparison. The y-axis shows the admittance normalised to the characteristic value \(B_k\) and the x-axis shows the angular frequency normalised to the resonance frequency \(\omega _0\).

The component sizes are the same as those chosen for the impedance curve of the RLC series resonant circuit in Figure 5. Equations 25 and 27 show that the normalised representation of both curves allows a direct comparison of the quality factors.

The most significant difference between the individual admittance curves for different quality factors can be seen in the resonance frequency range. With \(\underline {Y}_0 = G = \frac {B_k}{Q}\), the admittance in the resonance case corresponds directly to the ohmic conductance \(G\) and is inversely proportional to the quality factor.

The quality factor is a measure of the oscillation capacity of the oscillating circuit and a measure of the inverse damping of the oscillating circuit. In contrast to a series resonant circuit, a high quality factor in a parallel resonant circuit means a low effective conductance \(G\) (high effective resistance \(R\)) in relation to a high reactive conductance \(B\) (low reactive resistance \(X\)).

5.4 Current resonance using the example of an RLC parallel resonant circuit

When a parallel resonant circuit is excited by a constant current source \(\underline {I}_q\), a current resonance may occur, characterised by current flow between \(L\) and \(C\).

Consider the RLC parallel resonant circuit shown in Figure 12 with the admittance \(\underline {Y}\) from Equation ??. The current \(\underline {I}\) through the resonant circuit corresponds to the constant source current \(\underline {I}_q\) at its external terminals as shown in Figure 16.

With the knot and mesh equation: \begin {equation*} \begin {aligned} \underline {U} &= \underline {U}_R = \underline {U}_L = \underline {U}_C\\ \underline {I} &= \underline {I}_R + \underline {I}_L + \underline {I}_C = \underline {I}_q \\ \end {aligned} \end {equation*} and the component admittances are followed by the complex current dividers: \begin {align*} \frac {\underline {I}_R}{\underline {I}} &= \frac {\underline {U}}{\underline {U}}\cdot \frac {\underline {Y}_R}{\underline {Y}} =\frac {G}{G+\mathrm {j}\left (\omega C-\frac {1}{\omega L}\right )} \vphantom {\bigg |}\\ \frac {\underline {I}_L}{\underline {I}} &= \frac {\underline {U}}{\underline {U}}\cdot \frac {\underline {Y}_L}{\underline {Y}} =\frac {\frac {1}{\mathrm {j}\omega L}}{G+\mathrm {j}\left (\omega C-\frac {1}{\omega L}\right )} \vphantom {\bigg |}\\ \frac {\underline {I}_C}{\underline {I}} &= \frac {\underline {U}}{\underline {U}}\cdot \frac {\underline {Y}_C}{\underline {Y}} =\frac {\mathrm {j}\omega C}{G+\mathrm {j}\left (\omega C-\frac {1}{\omega L}\right )} \vphantom {\bigg |} \end {align*}

The current ratios of component currents to total current correspond to the frequency-variable ratio of component admittance to total admittance. The calculation of the current divider via the admittances is suitable in parallel circuits due to the simple addition of the individual admittances, similar to the calculation via the impedance in series circuits.

To normalise the frequency-dependent currents, the susceptances can be expressed as in Equation 23 using the characteristic value \(B_k\) and the resonance circuit frequency \(\omega _0\) from Equation 19. With the admittance normalised in this way from equation 25, the following terms result for the voltage dividers:

\begin {equation} \label {eq:rlcp:ibk:iq} \begin {aligned} \frac {\underline {I}_R}{\underline {I}} &= \frac {G}{G+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )B_k}\\ \frac {\underline {I}_L}{\underline {I}} &= \frac {-\mathrm {j}\frac {\omega _0}{\omega }B_k}{G+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )B_k}\\ \frac {\underline {I}_C}{\underline {I}} &= \frac {\mathrm {j}\frac {\omega }{\omega _0}B_k}{G+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )B_k} \end {aligned} \end {equation}

Figure 17 shows the magnitudes of the currents \(I_R\), \(I_L\), \(I_C\) as a function of the angular frequency normalised to the resonance angular frequency.

As can be seen in the figure, in the range of the resonance circuit frequency \(\omega = \omega _0\), there is a current increase at \(L\) and \(C\) compared to the ohmic resistance \(R\). The respective maximum current is not exactly at \(\omega _0\), but slightly below the resonance frequency for \(L\) and slightly above it for \(C\).

As shown in equation 26 and stated in equation ??, the quality corresponds to the ratio of the overcurrent to the total current in the resonance case. In the case of resonance, \(I_0 = I_{R,0} = I_{q,0}\). Therefore, the quality of the current divider rule can be expressed as the ratio of the characteristic conductance \(B_k\) to the effective conductance \(G\):

\begin {equation} Q = \frac {I_{L,0}}{I} = \frac {I_{C,0}}{I} = \frac {B_k}{G} \end {equation}

The frequency-dependent current conditions expressed by the quality factor \(Q\) with \(\underline {Y}\) from equation 27:

\begin {equation} \label {eq:rlcp:ibk:iq:q} \begin {aligned} \frac {\underline {I}_R}{\underline {I}} &= \frac {\frac {1}{Q}}{\frac {1}{Q}+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )}\\ \frac {\underline {I}_L}{\underline {I}} &= \frac {-\mathrm {j}\frac {\omega _0}{\omega }}{\frac {1}{Q}+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )}\\ \frac {\underline {I}_C}{\underline {I}} &= \frac {\mathrm {j}\frac {\omega }{\omega _0}}{\frac {1}{Q}+\mathrm {j}\left (\frac {\omega }{\omega _0}-\frac {\omega _0}{\omega }\right )} \end {aligned} \end {equation}

5.5 Voltage curve of the RLC parallel resonant circuit

When excited with a constant current \(\underline {I}_q\) as shown in Figure 16, the RLC parallel resonant circuit from Figure 16. This is shown in terms of magnitude in Figure 18 for different quality factors \(Q\).

The voltage behaviour of the RLC parallel resonant circuit is analogous to the current behaviour of the RLC series resonant circuit, as described in Chapter 4.5.

With the admittance normalised to the characteristic value from equation 27, Kirchhoff’s rules and Ohm’s law, the following applies to the voltage \(\underline {U}\):

\begin {equation} \label {eq:rlcp:u:iq:bknorm} \begin {aligned} \underline {U} &= \frac {\underline {I}}{\underline {Y}} \\ &= \frac {\underline {I}_q}{G + \mathrm {j}\left (\omega L - \frac {1}{\omega C}\right )} \\ &= \frac {\underline {I}_q}{G + \mathrm {j}\left (\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right )B_k} \end {aligned} \end {equation}

The maximum voltage \(U_0\) occurs in the case of resonance. It corresponds to the quotient of the real source current \(\underline {I}_q = I_q\) by the effective conductance \(G\):

\begin {equation} U_0 = \frac {I_q}{G} \qquad \text {mit} \underline {Y}_0 = G \end {equation}

This limits both the complex total voltage \(\underline {U}\) and its effective component \(\Re \{\underline {U}\}\) by the conductance \(G\) and the source current \(I_q\).

For the voltage \(\underline {U}\) normalised to the maximum voltage \(U_0\), the following results from Eq. 31:

\begin {equation} \label {eq:rlcp:unorm} \begin {aligned} \frac {\underline {U}}{U_0} &= \frac {\frac {I_q}{\underline {Y}}}{\frac {I_q}{G}} = \frac {G}{\underline {Y}}\\ &= \frac {1}{1 + \mathrm {j} \left (\frac {\omega }{\omega _0} - \frac {\omega _0}{\omega }\right ) Q} \end {aligned} \end {equation}

The normalised voltage curve of the RLC parallel resonant circuit therefore corresponds exactly to the normalised current curve of the RLC series resonant circuit with Figure 9 and Equation 17.

5.6 Current curve of the RLC parallel resonant circuit

The current curve of the RLC parallel resonant circuit when excited by a constant voltage source \(\underline {U}_q\) is derived from the admittance \(\underline {Y}\) and the voltage source \(\underline {U}_q\):

\begin {equation*} \underline {I} = \underline {U}_q \cdot \underline {Y} \end {equation*}

This means that the current \(\underline {I}\) is directly proportional to the admittance \(\underline {Y}\) and thus to the angular frequency \(\omega \). The current curve of the RLC parallel resonant circuit therefore corresponds to the admittance curve of the resonant circuit as shown in Figure 15.