Inductor-capacitor "tank" circuit
PARTS AND MATERIALS





Oscilloscope
Assortment of non-polarized capacitors (0.1 µF to 10 µF)
Step-down power transformer (120V / 6 V)
10 kΩ resistors
Six-volt battery
The power transformer is used simply as an inductor, with only one winding
connected. The unused winding should be left open. A simple iron core, singlewinding inductor (sometimes known as a choke) may also be used, but such
inductors are more difficult to obtain than power transformers.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 6: "Resonance"
LEARNING OBJECTIVES



How to build a resonant circuit
Effects of capacitor size on resonant frequency
How to produce antiresonance
SCHEMATIC DIAGRAM
ILLUSTRATION
1
INSTRUCTIONS
If an inductor and a capacitor are connected in parallel with each other, and then
briefly energized by connection to a DC voltage source, oscillations will ensue as
energy is exchanged from the capacitor to inductor and visa-versa. These
oscillations may be viewed with an oscilloscope connected in parallel with the
inductor/capacitor circuit. Parallel inductor/capacitor circuits are commonly known
as tank circuits.
Important note: I recommend against using a PC/sound card as an oscilloscope
for this experiment, because very high voltages can be generated by the inductor
when the battery is disconnected (inductive "kickback"). These high voltages will
surely damage the sound card's input, and perhaps other portions of the
computer as well.
A tank circuit's natural frequency, called the resonant frequency, is determined by
the size of the inductor and the size of the capacitor, according to the following
equation:
2
Many small power transformers have primary (120 volt) winding inductances of
approximately 1 H. Use this figure as a rough estimate of inductance for your
circuit to calculate expected oscillation frequency.
Ideally, the oscillations produced by a tank circuit continue indefinitely.
Realistically, oscillations will decay in amplitude over the course of several cycles
due to the resistive and magnetic losses of the inductor. Inductors with a high "Q"
rating will, of course, produce longer-lasting oscillations than low-Q inductors.
Try changing capacitor values and noting the effect on oscillation frequency. You
might notice changes in the duration of oscillations as well, due to capacitor size.
Since you know how to calculate resonant frequency from inductance and
capacitance, can you figure out a way to calculate inductor inductance from
known values of circuit capacitance (as measured by a capacitance meter) and
resonant frequency (as measured by an oscilloscope)?
Resistance may be intentionally added to the circuit -- either in series or parallel - for the express purpose of dampening oscillations. This effect of resistance
dampening tank circuit oscillation is known as antiresonance. It is analogous to
the action of a shock absorber in dampening the bouncing of a car after striking a
bump in the road.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Rstray is placed in the circuit to dampen oscillations and produce a more realistic
simulation. A lower Rstray value causes longer-lived oscillations because less
energy is dissipated. Eliminating this resistor from the circuit results in endless
oscillation.
3
Netlist (make a text file containing the following text, verbatim):
tank circuit with loss
l1 1 0 1 ic=0
rstray 1 2 1000
c1 2 0 0.1u ic=6
.tran 0.1m 20m uic
.plot tran v(1,0)
.end
RESUME OF THEORY FOR PARTS 1 AND 2
Inductors (L) and Capacitors (C) can actually store electrical energy. This is unlike
Resistors in which energy is only dissipated. Once stored, this energy is available for
release. Ideal Inductors and Capacitors dissipate no energy.
The Inductor and Capacitor, like the Resistor, are Circuit elements with two terminals. As
such, the relationship between voltage across (v) and current (i) through their terminals
can be defined just as Ohms Law defines the relationship between these variables (v and i)
for a resistor.
Circuit
Element
Resistor
Inductor
Capacitor
v =f(i)
i = f(v)
v i R
di
v L
dt
i  v R
t
1
v   idt  vt0
C t0
Power
dissipated
P  i2  R
1
vdt  it0
L t0
i C
dv
dt
Nil
Nil
W
1 2
Li
2
Nil
W
1 2
Cv
2
t
i
Energy stored
4
RESUME OF THEORY FOR PARTS 3 AND 4
Energy delivered by a source can be stored in an inductor or capacitor and this energy can
be returned to the source.
No resistance is inserted into any of the circuits so far, except for the small resistance in the
switches that is made negligible. The question that arises once resistance is inserted is how
would the energy stored in these components be delivered to a resistor.
A resistor dissipates energy (as heat). It stores no energy. It is a fair assumption to make that
the energy stored in an inductor or capacitor would decay with time once a path was
provided for it to flow to a resistor. The object is to specify the mathematical relationship
that this decay with time follows.
Once initial conditions are known, a voltage in an capacitor (V0) and current in an inductor
(I0), the application of Kirchoff’s laws results in the circuit equations that will yield the
voltage and current relationships with time. The definitions of the current/voltage
relationships are applied for the storage elements and for resistors (Ohm’s Law) and first
order ordinary differential equations result.
The solution leads to equations that define an exponential decay with time of any energy
that is stored.
Current
Inductor
i (t )  I 0 e
Capacitor N/A
Voltage
R
 t
L
N/A
v(t )  V0e

t
RC
Time
Power
Constant
2 t
L
t
p  I 02 Re
R
RC
V0 2 t2 t
p
e
R
Energy
2 t
1
LI 02 (1  e t )
2
2 t
1
w  CV02 (1  e t )
2
w
In both instances, the decay in current and voltage is exponential with time.
In both instances, the power delivered to the resistor decays exponentially with time.
In both instances, as time progresses, the energy dissipated in the resistor approaches the
initial energy stored in the inductor or capacitor. If there are no initial conditions the stored
energy in the inductor or capacitor is zero so no energy is dissipated in the resistor.
The time constant t is defined as shown to characterize the exponential decay in energy in
the inductor/capacitor (or growth in the case of energy dissipated in the resistor). It does this
through the factor e-t/t. After a time interval of 5 time constants, the factor is less than 1% of
its initial value.
t
t
t
t
e
3.6788x10-1
t
6t
t
t
e
2.4788x10-3
5
2t
3t
4t
5t
1.3534x10-1
4.9787x10-2
1.8316x10-2
6.7379x10-3
7t
8t
9t
10t
9.1188x10-4
3.3546x10-4
1.2341x10-4
4.5400x10-5
As will be seen, the time constant is usually very much smaller than a second. The
appearance/disappearance of currents and voltages are momentary events and are referred to
as the TRANSIENT RESPONSE of the circuit.
The response that exists a long time after switching has taken place is called the STEADY
STATE RESPONSE
6
RESUME OF THEORY FOR PARTS 5 AND 6
Energy delivered by a source can be stored in an inductor or capacitor.
A resistor resists the flow of current. A fair assumption to make is that there will be a build
up of energy in an inductor or capacitor but the rate will be determined by the size of the
resistor. The object is to specify the mathematical relationship that this build up with time
follows.
Once initial conditions are known, a voltage in an capacitor (V0) and current in an inductor
(I0), the application of Kirchoff’s laws results in the circuit equations that will yield the
voltage and current relationships with time. The definitions of the current/voltage
relationships are applied for the storage elements and for resistors (Ohm’s Law) and again,
first order ordinary differential equations result.
The solution leads to equations that define an exponential build up with time of any energy
that is stored.
Inductor
Current
V
V  ( R )t
i (t )  s  ( I 0  s )e L
R
R
t0
Vs
 ( R )t
(1  e L )
R
t  0 Provided I 0  0
Capacitor N/A
Voltage
N/A
Time Constant
L
R
i (t ) 
v (t )  I s R  (V0  I s R)e
t0
v (t )  I s R(1  e
t0
 ( 1 RC ) t
 ( 1 RC ) t
RC
)
Provided V0  0
In both instances, the growth in current and voltage is exponential with time to a final value
determined by the circuit.
The time constant t is defined as shown to characterize the exponential growth in
current/voltage in the inductor/capacitor. It does this through the factor e-t/t. After a time
interval of 5 time constants, the factor is within 1% of its final value.
7
RESUME OF THEORY FOR PARTS 7
Ideal OPAMP
Input resistance
Input Currents
Rules
Infinite
Zero
Consequence
ip=in
Voltage Gain
Input voltages
Infinite
Equal
vp=vn
Virtual Short Condition
The rules associated with the OPAMP allow the current through Rs and Cf to be summed
to zero using KCL.
is 
i f  is  0
1
vo  
Rs C f
vs
Rs
dv
if  Cf o
dt
dvo
1

vs
dt
RsC f
t
 v dy  v (t )
s
o
o
to
8
Type
Circuit
Section
6.1
Apply Kirchoff
i  0 At a node
I0
2
+
V0
C
_
L
R
Parallel
t
v
dv 1
 C   vdx  I 0  0
R
dt L 0
1
0
Natural
Initial energy is present.
Energy is stored at time t=0.
Section
6.3
I0
2
+
I
V0
_
C
1
0
Initial energy may be
present. DC source I supplies
energy.
Section
6.4
R
1
L
v
dv
C
 I
R
dt
v  0
2
C
di 1
iR  L   idx  V0  0
dt C 0
_
Series
0
Section
6.4
R
1
v  0
2
around a closed loop.
C
i
Divide by LC
d 2i L
1 di L i L
I


2 
dt
RC dt LC LC *
Vo and to make idx into i
0
2
LC
d i
di
i 0
2  RC
dt
dt
Substitute to eliminate I
dvC
di
d 2v
and  C 2C
dt
dt
dt
2
d v
dv
LC 2C  CR C  vC  V
dt
dt
i C
I0
+
V0
L
di L
dv
d 2i
and
 L 2L
dt
dt
dt
2
d i
L di L
LC 2L 
 iL  I
dt
R dt
d 2i R di
i

0
2 
dt
L dt LC
*
Initial energy is present.
Energy is stored at time t=0.
Natural
0
d 2 v L dv
LC 2 
v 0
dt
R dt
Divide by LC
d 2v
1 dv
v

0
2 
dt
RC dt LC
*
Substitute to eliminate v
t
t
i
t
I o and to make vdx into v
around a closed loop. Differentiate to remove
I0
+
V0
i L  iR  iC  I
iL 
Differentiate to remove
v L
L
R
Parallel
Step
i  0 At a node
Mathematics
V
_
Series
0
Initial energy may be
present. DC source V
supplies energy.
iR  L
di
 vC  V
dt
Divide by LC
d 2 vC R dvC vC
V


2 
dt
L dt
LC LC *
Step
* All 4 circuits are described by second order differential equations.
For parallel RLC circuits, the coefficient of the first order differential is
R
L
1
RC .
For Series RLC circuits, the coefficient of the first order differential is .
The challenge now is how to solve a second order differential equation.
9
Resume of Theory for Parallel and Series RLC circuits
All four circuits result in second order differential equations.
The method of solution follows, first for the Natural Response, then for the
Step Response.
Over damped
s1  s2
Under damped
s1  s2
v  A1e  A2 e
s1t
s2 t
Critically Damped
s1  s2
v  A1e  A2 e
s1t
s2 t
s1      2   2 0
s1      2   2 0
s2      2   2 0
1
 
PARALLEL
2 RC
R
 
SERIES
2L
1
0 
LC
2
2
  0 0
s2      2   2 0
1
 
PARALLEL
2 RC
R
 
SERIES
2L
1
0 
LC
2
2
  0 0
v  A1e s1t  A2 e s2t
v  B1e  t cos d t  B2 e  t sin  d t
v  D1te  t  D2 e  t
1
 
PARALLEL
2 RC
R
 
SERIES
2L
 2   02  0
v  D1te  t  D2 e  t
 d   02   2
Constants
v (0 )  V0  A1  A2
Constants
v (0 )  V0  B1
Constants
v (0 )  V0  D2
dv (0 ) iC (0 )

 s1 A1  s2 A2
dt
C
dv (0 ) iC (0 )

   B1   d B2
dt
C
dv (0 ) iC (0 )

 D1   D2
dt
C
Real Roots
Exponential Decay
Imaginary Roots Exponentially
decaying oscillations
Exponential Decay
For the Step Response, simply determine the final value of the current or voltage and add it to
the Natural Response!
10