LC Resonant Circuits Dr. Roger King June Introduction

LC Resonant Circuits Dr. Roger King June 01 Introduction Second-order systems are important in a wide range of applications including transformerless impedance-matching networks, frequency-selective networks, and control system models. Second-order systems occur whenever two independent energy-storing elements are present. The study of first- and second-order networks is important because of their useful properties, and because they are frequently used as approximations to the main features of even more complex networks. Resonance is usually defined to occur when the imaginary part of the impedance or admittance of a network zeros at a specific resonant frequency, and is usually accompanied by a very large increase in voltage and/or current in some part of the network. Strongly resonant behavior will occur in a network composed of a low-loss inductor and a low-loss capacitor. It can also occur in a two-capacitor or two-inductor system, but only if a controlled source is also present. The purpose of this paper is the review of resonant behavior in a one-capacitor one-inductor system. The primary interest here is the resonance phenomenon, although some note is made of the overdamped condition where the resonance is no longer obvious, but the system behavior remains distinctly second order. Series vs. Parallel LC Resonance Fig. 1 displays two series-resonant LC circuits. Fig. 1(a) shows the classic textbook form of the circuit, including a series-connected inductor, capacitor, and a resistor representing the sum of all internal energy losses in the circuit elements along with any added damping. An independent voltage source is also included. The circuit response may be considered either the inductor current i or the capacitor voltage v. Fig. 1(b) is the same circuit, except that the voltage source and damping resistor are replaced with their Norton equivalent. The key feature of both circuits in Fig. 1 is that with their sources zeroed, the capacitor, inductor and damping resistor are all in series. Fig. displays two parallel-resonant LC circuits. Fig. (a) is the classic form of the circuit. Here the inductor, capacitor and damping resistance are all in parallel, along with an independent current source. Fig. (b) is a Thevenin equivalent to Fig. (a). The key feature of both circuits in Fig. is that with their sources zeroed, the capacitor, inductor and damping resistor are all in parallel. Again, the circuit response may be considered either the inductor current i or the capacitor voltage v. 1 of 14

Fig. 1 (a) Series-resonant LC network with voltage source. (b) Series-resonant LC network with current source. Fig. (a) Parallel-resonant LC network with current source. (b) Parallel-resonant LC network with voltage source. The series-resonant configuration arises naturally in a circuit having a voltage source in which the resistive losses in the inductor windings give the predominant damping effect. It is typically found that in LC resonant circuits the losses in the capacitor are much smaller than those in the inductor. The parallel-resonant configuration arises when the circuit has a current source and the predominant damping effect is the inductor core loss, which is modeled by an equivalent resistance in parallel with the inductor. An LC resonant circuit having some damping resistance in series with the inductor as well as additional damping resistance in parallel with the inductor will also be considered in a later section of this paper. This arrangement is neither series nor parallel resonant, but a reasonable approximate series- or parallel-equivalent may be made. of 14

Table 1 Parameters of an LC Resonant Network Symbol O Q Parameter Name (units) Resonant Frequency (radians/sec) Characteristic Resistance (ohms) Quality Factor (unit less) Damping Ratio (unit less) Damping Factor (radians/sec) o Z o Equation 1 LC L C Q 1 1 Q o Q o Q Zo R s Q Rp Z o Comments series res. parallel res. d Damped Resonant Frequency (radians/sec) d o 1 d o for a lightly damped network Characterization of a Resonant Network Any of the networks in Figs. 1 and can be described by listing the values of L, C and R S (or R P ), although this does not give immediate insight into how the network will behave. A more intuitive description is given by stating three equivalent parameters of the network: the resonant frequency, the characteristic resistance, and the quality factor ( O,, Q). Definitions of these three parameters are given in Table 1. As will be seen, O,, and Q give an immediate sense of what the natural response will be like, and how the network will interact with its independent source. The definitions of O and are the same for both the series- and parallel-resonant networks. The resonant frequency O is the radian frequency at which the circuit will ring if it has no energy dissipation and is excited by some initial stored energy. The characteristic resistance is the expected ratio of voltage amplitude to current amplitude in the ringing response. Resistors represent energy-dissipative elements which tend to damp the ringing of the network. There are three different measures of this same thing: quality factor (Q), damping factor ( ) and damping ratio ( ). As Table 1 indicates, these are mutually interrelated. However, the relationship of damping to the circuit element values is different for series- and parallel-resonant networks. The equations for the Q factor are given in Table 1; the expressions for damping factor and damping ratio are easily derived from these. It will also be seen that a damped network will ring at a frequency lower than O, given by the damped resonant frequency d. 3 of 14

Table. Three Different Damping Conditions Damping Type Undamped (lossless system) Underdamped Critically Damped Overdamped Quality Factor - Q Q Q > 0.5 Q = 0.5 Q < 0.5 Damping Ratio - Damping Factor - O O O The network equations for Figs. 1 and will be solved in the time-domain and the frequency-domain. The form of the solutions will change depending upon the amount of damping. The three cases to be distinguished are under-damping, critical-damping, and over-damping. Applications of the resonant LC network as a narrow-band filter or impedance transformer often prefer light or zero damping, in which case equations are usually written in terms of Q. In control system applications, the expected damping levels often range from just below to just above critical damping, and the expressions are written in terms of. Overdamping will be considered only briefly for the sake of completeness. The damping conditions for each of these three cases are described in Table. Time-Domain Equations for the Resonant Network For the series-resonant network of Fig. 1(a), write the loop equation in inductor current i. The substitution i = C dv/ may be used to get an equivalent equation in v. These equations are: d i Q d v Q di dv o i Z o dv s o v o v s (1) Note that the series-resonant definition for Q has been used. Fig. 1(b) is best analyzed by converting the Norton form of the independent source into its Thevenin equivalent. For the parallel-resonant network of Fig. (a), write the node equation in capacitor voltage v. The substitution v = -L di/ may be used to get an equivalent equation in i. These equations are: d i d v Q Q di dv o i o i s o v o Z o di s () Note the parallel-resonant definition for Q has been used. Fig. (b) is best analyzed by converting the Thevenin form of the independent source into its Norton equivalent. 4 of 14

It may be noted that (1) and () are essentially the same, except for their specific independent source terms. The method for solving (1) or () for any specific driving term and initial conditions is the following. With the driving term (right hand side of equation (1) or ()) set to zero, find the solution of the resulting homogeneous equation. This is the natural response. Then the particular solution to the specific driving term is found. This is known as the forced response. These two solutions are added together to form the general solution. The initial conditions for the network are applied to the general solution to determine the specific values of any unknown constants. The natural response is instructive: It shows what the resonant network will do by itself, given as a starting point the stored energy implied by the initial capacitor voltage and inductor current. These initial conditions are defined as follows: V o I o v(0) i(0) (3) The homogeneous equations in (1) and () are all the same, as shown below using the generic variable x. d x d x Q dx o x 0 or o dx o x 0 (4) The natural solution is given by: x t e ot [A 1 sin d t A cos d t] for 0 <1 x t A 3 e ot A 4 ( o t) e ot for =1 x t A 5 e 1 ot A 6 e 1 ot for >1 where d o 1 (5) The solutions in (5) separate into the three cases of underdamped, critically damped, and overdamped. The underdamped case ( < 1) is the interesting one. 5 of 14

For the series-resonant circuit of Fig. 1(a), and assuming initial conditions V O and I O, (5) is rewritten to give the natural behaviors of the capacitor voltage and inductor current. In each case, constant A is determined by applying the initial condition (I O or V O ). A 1 is determined by applying the initial rate-of-change of current or voltage. This can be calculated by finding the initial inductor voltage and initial capacitor current from the given initial conditions and the circuit in Fig. 1(a). The results are, for the underdamped case ( < 1): i t e ot v t e ot 1 1 V O I O 1 and I O V O 1 sin d t I O cos d t sin d t V O cos d t for 0 1 and damped series resonance (6) These results can be reasonably approximated in the case of light damping ( < 0.1 or Q>5): i t e ot V O I O sin o t I O cos o t and v t e ot [I O V O ] sin o t V O cos o t for 0.1 and damped series resonance (7) For the case of zero damping ( =0 or Q i t V O sin o t I O cos o t and v t I O sin o t V O cos o t for 0, undamped resonance (8) For the parallel-resonant circuit of Fig. (a) the result is similar to (6), but not identical. i t e ot v t e ot 1 1 V O I O 1 and I O V O 1 sin d t I O cos d t sin d t V O cos d t for 0 1 and damped parallel resonance (9) These results can be reasonably approximated in the case of light damping ( < 0.1 or Q>5): 6 of 14

i t e ot V O I O sin o t I O cos o t and v t e ot [I O V O ] sin o t V O cos o t for 0.1 and damped parallel resonance (10) For the case of zero damping ( =0 or Q the behaviors of the parallel-resonant and series-resonant circuits become the same. Both are given by (8). Careful comparison of (6)-(10) shows that both underdamped series- and parallel-resonance produce decaying sinusoidal oscillations in current and voltage if started with some stored energy. However, there are subtle differences in the two ringing behaviors. The less the damping (higher Q), the less these differences. If the two cases are both considered undamped altogether, the difference disappears entirely. This is explored further by PSpice simulation of three resonant circuits, each having an initial stored energy of 1 J, a natural resonant frequency of 1 rad/s, and an impedance of 1. See Fig. 3. One is series resonant with Q=5, another parallel resonant with Q=5, and the third has very little damping (Q=1000). The capacitor voltage and inductor current for each is plotted in Fig. 3, lower waveforms. The voltage/current waveforms for the series- and parallel-resonant circuits with Q=5 are seen to be close, but slightly different. Fig. 3. Three resonant circuits with initial energy of 1 J and resonant frequency of 1 rad/s. One is series resonant with Q=5, one is parallel resonant with Q=5, and the third has Q=1000. 7 of 14

The upper waveforms in Fig. 3 plot the energy stored in each circuit as it rings down. These suggest that the reason for the small differences in the voltage and current waveforms for the parallel- and series-resonant circuits with same Q factor is that the energy losses occur at different times in the ringing cycles, but that as one would expect for two circuits with the same Q, the total energy loss over a complete cycle is the same for both. The high-q circuit is provided as a reference, and has very little energy loss over the time span shown. Some references will note that Q may be defined as times peak energy stored in the circuit divided by energy dissipated over one cycle. This is true if the energy dissipated over one cycle is calculated based on the voltage/current levels remaining the same throughout the cycle. This statement is not true for Fig. 3 with Q = 5 because the voltage/current amplitudes are rapidly decaying. Series-Parallel Equivalents for High-Q Resonant Circuits Sometimes a resonant network includes both series and parallel damping, as illustrated in Fig. 4, left side. In this case, the network is strictly neither series- nor parallel-resonant. However, if the Q factor accounting for all of the damping is higher than about 5, two approximate equivalents are possible. These are shown in Fig. 4 center (series equivalent) and left side (parallel equivalent). The approximate equivalents are given as follows: R P R S R P Q R S Z O P R P (series equivalent) or R P R S Q S R P R S (parallel equivalent) where R S Q P Q S Q eff R P R S 1 1 Q 1 S Q P for Q eff 5 (11) Fig. 4. Resonant network with both series- and parallel-damping, along with an approximate series equivalent, and an approximate parallel equivalent. 8 of 14

Fig. 5. Parallel-resonant network, with two approximately equivalent options for placement of the damping (or load) resistance. Converting all of the damping resistances to either a series or a parallel equivalent is a useful approximation when analyzing a lossy inductor, which may show both ohmic losses in its windings (series damping resistance) and core loss related to its magnetic core material (parallel damping resistance). Fig. 5 shows additional options for placement of the damping resistance in a parallel-resonant network. These approximate equivalents are derived by comparing the impedances (or admittances) between the two nodes indicated. Given a high effective Q in each case, the following approximate equivalents may be found. R C1 1 R R for Q 5 C C 1 C L 1 L 1 L (1) The equivalents in Fig. 5 are particularly useful when the resonant network is used as a signal processing circuit. In this application, an input may be applied as a voltage or current source, and the damping resistance may actually represent a resistive load device. Fig. 5 suggests that splitting the capacitor or inductor into two series-connected parts and extracting the output in parallel with one of those two parts would be useful if the load resistance is too small to get a high Q when connected across the whole network. Other useful tapping schemes may be found for a series-resonant network. In using Fig. 5 and (1), the inductor is assumed to be split into two parts having zero mutual coupling. Resonant Networks as Filters Signal-processing filters may be described by their steady-state sinusoidal responses. For this purpose, refer to Fig. 1(a) and calculate its phasor impedance, or to Fig. (a) for its phasor admittance. These results are the following: 9 of 14

Z j R S j L 1 j C 1 Q j ( O O ) for series resonant network (impedance) Y j 1 R P j C 1 j L Y O 1 Q j ( O O ) for parallel resonant network (admittance) (13) Note that Y O is () -1. The input signal to a series-resonant circuit may be applied as a voltage source, with the output signal extracted as the voltage drop on R S, C, L, or the series combination of C and L. The damping resistance R S is the composite value which includes losses in the inductor and capacitor, as well as any inserted load resistance. The input signal to a parallel-resonant circuit may be applied as a current source, with the outputs extracted as similar choices of element currents. In addition, a load resistance may be inserted into either of these resonant circuits through the technique of tapping-down the capacitor or inductor. This leads to a wide variety of possibilities. A sample set will be illustrated, but many others are used in practice. Fig. 6 shows four filters derived from the series resonant network. In each of these, the single resistor represents the composite of the external load resistance and internal inductor losses, converted into their series equivalent values. Each of the filters in Fig. 6 has the same loop impedance, given by (13) for a series resonant network. Therefore, each of the four transfer functions can be derived by dividing the impedance of the element(s) across which the output voltage is extracted by the loop impedance. A Spice simulation was used to plot the transfer function magnitude for each of the four filters. These are shown in Fig. 7. For these Spice Fig. 6 Four filters derived from the series-resonant network. 10 of 14

Fig. 7 Frequency response for the four filters of Fig. 6, all with f O = 1 Hz, Q = 0.5, 5, and 50. Upper left: bandpass filter. Lower left: bandstop filter. Upper right: lowpass filter. Lower right: highpass filter. simulations, o = (1 Hz), Z o = 1, and Q values of 0.5, 5, and 50 were used. The transfer functions plotted in Fig. 7 are V o /V s in db notation. The frequency response plots for the lowpass and highpass filters suggest that Q values ranging from critical damping to slight underdamping (Q = 0.5 to 1) should be chosen for these types. It can be noted that the peaking in these frequency responses is 0 log(q) above the passband value for values of Q exceeding 5 or so. This is generally undesirable for most applications. There are a number of practical cases where the lowpass filter appears on a dc power bus, either deliberately or created through the interaction of wiring inductance with bus bypass capacitors. In these cases, there is often insufficient damping (excessive Q) of the filter, and it becomes a design problem to add damping by some means to lower the Q to a reasonable value. The bandpass and bandstop filters are intended to act upon a narrow range around the center frequency o. In this type of filter, a high Q is generally desired. It may be seen that the wih of the band of frequencies passed by the bandpass filter, or rejected by the bandstop filter, is inversely proportional to the filter s Q. If this bandwih is defined as the range of frequency for which the bandpass filter transfer function is above -3 db, or the bandstop filter is 11 of 14

below -3 db, it may be calculated as follows. First, 3 db was chosen somewhat arbitrarily because it is the equivalent of, or ( ) 1, a round number, but is an industry standard choice. The lower cutoff frequency 1, and upper cutoff frequency, are then calculated for the loop impedance (13): Z j Z o 1 Q j o 1 Q j ( o ) 1 o 1 1 4Q o 1 1 4Q BW 1 Q Q Q 1 or o Q o Q (14) The approximations given for and are valid for high Q. Bandpass and bandstop filters are used to select, or eliminate, a relatively narrow range of signal components centered on. In these filters, Q is selected in accordance with the desired bandwih. Resonant Networks as Impedance Transformers Resonant networks are often used as impedance transformers. In this application, an external load is attached to the resonant network, which presents a transformed equivalent input resistance to the signal source. This is useful when the signal has a narrow bandwih, and allows the resonant network to combine impedance transformation with filtering. Two possible L-networks are shown in Fig. 8. There are at least six others in which the two arms of the L may be composed of L and C, or two series-connected inductors or capacitors. In Fig. 8, the signal sources are shown as idealized voltage or current sources: If the source s internal impedance is significant, it also becomes part of the network. The two networks in Fig. 8 will contribute both resonant filtering and load resistance transformation. The analysis of Fig. 8, network A, may be done by writing an expression for its input impedance Z eq. See equation (15), first line. Network A is parallel-resonant, with the load resistance providing the damping. The approximation in (15) is introduced by assuming that Fig. 8 Two impedance-transforming L-networks. Network A: Real part of input impedance <. Network B: Real part of input impedance >. 1 of 14

Q is high. The second line of (15) shows that the approximate input impedance of the network is the same as that of a series-resonant network damped by the transformed load resistance. Note that operation in the vicinity of o is assumed. Z eq j L 1 j C j L j j L j 1 ( C) ( C) C Z eq j L 1 j C Z o for Q o C 5 near o the input resistance is R eq Z o Q for Network A C (15) The effective input resistance is seen to be a fraction of the load resistance. Equation (15) also shows that the transformation turns-ratio is equivalently 1:Q. For frequencies off-resonance, the input impedance also includes significant reactive components, indicated by the terms in brackets in (15), second line. A similar analysis of network B in Fig. 8 produces similar results. In this case, the load resistance provides series-resonant damping. The effective input admittance is found to be: Y eq j C 1 j L j C j L R L ( L) Y eq j C 1 j L Z o for Q L 5 near o the input resistance is R eq Z o Q for Network B j C j L ( L) (16) As before, the approximation in the first line of (16) is introduced by assuming that the Q is high. The effective input admittance, second line, is that of a parallel-connected LC network, together with a reflected input conductance. The resistance equivalent of the input conductance is given in line five of (16). Again, the network reflects the load resistance in a manner of a transformer with a turns ratio of Q:1. A straightforward design procedure can be seen from (15) and (16). The resonant frequency of the network is its operating frequency, with a narrow working bandwih. The characteristic resistance ( ) needs to be set equal to the geometric mean of the intended load and reflected equivalent resistances ( Z o R eq ). However, the Q factor is constrained by the ratio of the load and reflected resistances. Q cannot be independently chosen to achieve bandwih objectives, or to ensure a realizable system. Although the L-network is widely used, it is too inflexible to achieve many required results. It should be noticed that the L-network matching criteria can be used to find the series- and parallel-resonant equivalents to networks with several damping resistances, given in (11) and Fig. 4. 13 of 14

Fig. 9 Pi-type resonant matching network, along with two approximately equivalent load resistor placements. Finally, the pi-network will be considered. There are numerous 3-element pi- and tee-networks that can be used to get a more flexible resonant matching network, but one example of the pi-network will be sufficient. Fig. 9 (left side) shows a resonant pi-network for load resistance matching, along with two approximately equivalent load resistor placements. The network shown is assumed to be lightly damped (Q>5), and resonance is expected at the frequency corresponding to L and the series equivalent of C 1 and C. It can be seen that the load is tapped down the series combination of C 1 and C, so the equivalents shown in Fig. 5 may be used to generate the two alternate placements of equivalent damping resistance (middle of Fig. 9) and (right of Fig. 9). Using (1), the equivalent damping resistance values are: C R 1 C L C 1 and C R C L C 1 C RL C 1 (17) The equivalent input admittance of Fig. 9, left side, is then calculated using (17). Y eq 1 C 1 C j C1 j C LC 1 (18) C R L C 1 It can be seen from (18) that the reflected value of the load resistance is. The second term in (18) represents the equivalent susceptance. This term nulls at the resonant frequency determined by L and the series equivalent of C 1 and C. Design of the pi-network is flexible because the resonant frequency is determined by the signal frequency, and the C 1 -C ratio is determined by the desired reflected value of the load resistance, but the characteristic resistance of the resonant network is left undetermined. This may be set to achieve a bandwih objective. When bandwih is not critical, it is common to set Q to the range of 5-10 to keep the losses in the inductor manageable. 14 of 14