LC Resonant Circuits
Dr. Roger King
June 2012
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. 2 displays two parallel-resonant LC circuits. Fig. 2(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. 2(b) is a Thevenin equivalent to Fig. 2(a). The key feature of
both circuits in Fig. 2 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.
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Fig. 1 (a) Series-resonant LC network with voltage source. (b) Series-resonant LC network with current source.
Fig. 2 (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.
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Table 1 Parameters of an LC Resonant Network
Symbol
Q
Parameter Name (units)
Resonant Frequency
(radians/sec)
Characteristic Resistance
(ohms)
Quality Factor (unit less)

Damping Ratio (unit less)


Damping Factor (unit less)

d
Damped Resonant Frequency
(radians/sec)
O
ZO
Equation
o 
1
LC
Zo 
L
C
Q

1
2
1
2Q
o
2Q

o
2

o
Q
  o
d  o 1  
Comments
Q
2
Zo
Rs
Rp
Zo
series res.
parallel res.
d  o
for a lightly damped
network
Characterization of a Resonant Network
Any of the networks in Figs. 1 and 2 can be described by listing the values of L, C and RS
(or RP), 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, ZO, Q). Definitions of these
three parameters are given in Table 1. As will be seen, O, ZO, 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 ZO 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 ZO 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.
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Table 2. Three Different Damping Conditions
Damping Type
Undamped
(lossless system)
Underdamped
Critically Damped
Overdamped
Quality Factor - Q
Q
Damping Ratio - 

Damping Factor - 

Q > 0.5
Q = 0.5
Q < 0.5



O
O
O
The network equations for Figs. 1 and 2 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 2.
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/dt may be used to get an equivalent equation in v. These equations are:
d2i
dt 2

 o di
Q dt

 o dv
Q dt
  2o i 
 o dv s
Z o dt
(1)
d2v
dt 2

 2o
v
 2o
vs
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. 2(a), write the node equation in capacitor
voltage v. The substitution v = -L di/dt may be used to get an equivalent equation in i. These
equations are:
d2i
dt 2

 o di
Q dt
  2o i    2o i s
(2)
d2v
dt 2

 o dv
Q dt
  2o v 
di
 o Z o dts
Note the parallel-resonant definition for Q has been used. Fig. 2(b) is best analyzed by
converting the Thevenin form of the independent source into its Norton equivalent.
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It may be noted that (1) and (2) are essentially the same, except for their specific
independent source terms. The method for solving (1) or (2) for any specific driving term and
initial conditions is the following. With the driving term (right hand side of equation (1) or (2))
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  v( 0 )
(3)
I o  i(0 )
The homogeneous equations in (1) and (2) are all the same, as shown below using the generic
variable “x.”
d2x
dt 2

 o dx
Q dt
  2o x  0
(4)
or
d2x
dt 2

2 o dx
dt

 2o
x0
The natural solution is given by:
xt  e  o t [A 1 sin  d t  A 2 cos  d t ]
xt  A 3 e  o t  A 4 ( o t ) e  o t
xt  A 5 e  
 2 1  o t
 A 6 e   
where
d  o 1  2
2 1
for 0   < 1
for  = 1
(5)
ot
for  > 1
The solutions in (5) separate into the three cases of underdamped, critically damped, and
overdamped. The underdamped case (< 1) is the interesting one.
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For the series-resonant circuit of Fig. 1(a), and assuming initial conditions VO and IO, (5)
is rewritten to give the natural behaviors of the capacitor voltage and inductor current. In each
case, constant A2 is determined by applying the initial condition (IO or VO). A1 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  o t

VO
1
1 2 Z O
 IO

1 2
sin  d t  I O cos  d t
and
vt  e
 o t
IO ZO   VO
sin  d t  V O cos  d t
1 2
(6)
for 0    1 and damped series resonance
These results can be reasonably approximated in the case of light damping (< 0.1 or Q>5):
it  e  o t
VO
ZO

  I O sin  o t  I O cos  o t
and
 o t
[I O Z O   V O ] sin  o t  V O cos  o t
vt  e
for   0.1 and damped series resonance
(7)
For the case of zero damping (=0 or Q
V
it   Z OO sin  o t  I O cos  o t
and
vt  I O Z O sin  o t  V O cos  o t
for   0 , undamped resonance
(8)
For the parallel-resonant circuit of Fig. 2(a) the result is similar to (6), but not identical.
it  e  o t

VO
1
1 2 Z O

 IO
1 2
sin  d t  I O cos  d t
and
vt  e
 o t
IO ZO   VO
1 2
sin  d t  V O cos  d t
for 0    1 and damped parallel resonance
These results can be reasonably approximated in the case of light damping (< 0.1 or Q>5):
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(9)
it  e  o t

VO
ZO
  I O sin  o t  I O cos  o t
and
 o t
[I O Z O   V O ] sin  o t  V O cos  o t
vt  e
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 .
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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 2 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:

RS  RS 
RP
Q 2P
Z 2O
RP
 RS 
(series equivalent)
or

RP  RP 
R S Q 2S
Z2
 R P  ROS (parallel equivalent)
where
R
Q P  Z OP
QS 
Q eff 
(11)
ZO
RS
1
1
1

Q
Q
S
P
for Q eff  5
Fig. 4. Resonant network with both series- and parallel-damping, along with an approximate series equivalent,
and an approximate parallel equivalent.
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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  R
2
C2
C1  C2
2
L1
L1  L2
R L1  R
for Q  5
(12)
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 (12), 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. 2(a) for its phasor
admittance. These results are the following:
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1
Zj  R S  jL  jC
 Z O Q1  j( O 
for series resonant network (impedance)
1
 Y O Q1  j( O 
Yj  R1P  jC  jL
for parallel resonant network (admittance)
O

)
O

)
(13)
Note that YO is (ZO)-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 RS, C, L, or the series combination
of C and L. The damping resistance RS 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.
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Fig. 7 Frequency response for the four filters of Fig. 6, all with fO = 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 = 2(1 Hz), Zo = 1 , and Q values of 0.5, 5, and 50 were used. The transfer
functions plotted in Fig. 7 are Vo/Vs 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 20 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
width 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 “bandwidth” is defined as the range of
frequency for which the bandpass filter transfer function is above -3 dB, or the bandstop filter is
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below -3 dB, it may be calculated as follows. First, “3 dB” was chosen somewhat arbitrarily
1
because it is the equivalent of 2 , or ( 2 ) , a round number, but is an industry standard
choice. The lower cutoff frequency 1, and upper cutoff frequency 2, are then calculated for
the loop impedance (13):
Zj  Z o
2 
1
Q

1
Q
j( o
1  o 1 
1
4Q 2
2  o 1 
1
4Q 2
 j o 

o

)
o


 1 or  2

o
2Q
 o 
o
2Q

o
2Q
 o 
o
2Q
BW   2   1 
(14)
o
Q
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 bandwidth.
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 bandwidth, 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 Zeq. See equation (15), first line. Network A is parallel-resonant, with the load
resistance RL 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 < RL. Network B:
Real part of input impedance > RL.
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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 
RL
1  jR L C
 jL 
Z eq  jL 
R L  jR 2L C
1  (R L C )
Z2
1
 RoL
jC
2
 jL 
R L  jR 2L C
(R L C ) 2
R
for Q  Z OL   o R L C  5
near  o the input resistance is
R eq 
Z 2o
RL

RL
Q2
(15)
for Network A
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
R L  jL
 jC 
Y eq 
R L  jL
 (L ) 2
R
1
jC  jL  Z 2oL
Z
 L
 ROL  RoL  5
R 2L
 jC 
for Q
near  o the input resistance is
R eq 
Z 2o
RL
R L  jL
(L ) 2
(16)
 Q 2 R L for Network B
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 bandwidth. The
characteristic resistance (Z O) needs to be set equal to the geometric mean of the intended load
and reflected equivalent resistances ( Z o  R eq R L ). However, the Q factor is constrained by
the ratio of the load and reflected resistances. Q cannot be independently chosen to achieve
bandwidth 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.
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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 C1 and C2. It can be seen that the load
RL is “tapped down” the series combination of C1 and C2, so the equivalents shown in Fig. 5 may
be used to generate the two alternate placements of equivalent damping resistance RL’ (middle of
Fig. 9) and RL’’ (right of Fig. 9). Using (12), the equivalent damping resistance values are:
C1  C2
C1

RL  RL
and


RL  RL
C2
C1  C2
2
(17)
2
 RL
C2
C1
2
The equivalent input admittance of Fig. 9, left side, is then calculated using (17).
Y eq 
1
RL
C1
C2
2
 jC 1 
jC 2
 2 LC 2  1
(18)
C
2
It can be seen from (18) that the reflected value of the load resistance is R L C21 . The second
term in (18) represents the equivalent susceptance. This term nulls at the resonant frequency
determined by L and the series equivalent of C1 and C2.
Design of the pi-network is flexible because the resonant frequency is determined by the
signal frequency, and the C1-C2 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 bandwidth objective. When bandwidth is not critical, it is common to set
Q to the range of 5-10 to keep the losses in the inductor manageable.
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