RLC-circuits
TEP
Keywords
Damped and forced oscillations, Kirchhoff’s laws, series and parallel tuned circuit, resistance,
capacitance, inductance, reactance, impedance, phase displacement, Q-factor, band-width
Application
RLC-circuits are used as frequency filters or resonators in electronic devices; e.g in radio transmitters
and receivers the frequency tuning is accomplished by setting the RLC-circuit to resonate on a special
frequency.
Radio receiver
Experimental set-up
RLC element (series)
Learning objective
The resonance-behaviour of a RLC-circuit is studied and the resonance frequencies f res are
determined and compared with the theoretical values
f res=
1
2 π √L C
.
The resonance curves are measured and the impedance-behaviour of the LC-component is analysed.
Further the bandwidths B and the quality factors Q are determined from the resonance curves and
compared with the theoretical values for a series-tuned circuit, obtained from the parameters of the
electrical components.
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RLC-circuits
Tasks
1. Measure the voltage drop U over the LC-component and the current I through the circuit
and determine the resonance frequency for both combinations of coil and capacitor; compare
with the theoretical values
a)
for the series-tuned circuit with resistors R=47 Ω and
b)
for the parallel-tuned circuit with resistor R=470 Ω .
R=100 Ω .
2. Determine the impedance Z of the LC-component for both circuits with the measurements from
task 1 and compare with the theoretical values.
3. Determine the bandwidth B and Q-factor for the series-tuned circuit from the resonance curve
and compare with the theoretical values (determined by the parameters of the electrical
components).
Equipment
1
1
1
1
1
1
1
1
1
4
2
1
Digital Function Generator, USB
Coil, 900 turns
Capacitor 100 nF/250 V, G1
Capacitor 470 nF/250 V, G1
Resistor 47 Ohm, 1W, G1
Resistor 100 Ohm, 1W, G1
Resistor 470 Ohm, 1W, G1
Multi-range meter/overl.prot.B (Multimeter)
Connection box
Connecting cord, 32 A, 500 mm, black
Connecting cord, 32 A, 250 mm, black
Short-circuit plug,black
13654-99
06512-01
39105-18
39105-20
39104-62
39104-63
39104-15
07026-00
06030-23
07361-05
07360-05
06027-05
Theory
A RLC-circuit (also oscillating, oscillator or resonant circuit) consists of a resistor (R), an inductance (L)
and a capacitor (C) – sometimes it is also refered to as LC-circuit, because the resistor is used to
simulate the loss-resistance of a real circuit. Generally one differs between two kinds of RLC-circuits, the
series- and the parallel-tuned circuit. The circuit diagramms are shown in Fig. 1 and 2, respectively.
Fig. 1:
circuit diagramm for a series-tuned RLC-circuit
2
Fig. 2:
circuit diagramm for a parallel-tuned RLC-circuit
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When a fully charged capacitor is discharged through an inductance coil, the discharge current induces a
magnetic field in the coil, which reaches its maximum, when the capacitor is completely discharged.
Then, due to the decreasing current, the change in the magnetic field induces a voltage which according
to Lenz's law charges the capacitor. Now the current decreases to zero until the capacitor is completely
charged again, but with reversed sign of charges. At this point, the procedure starts again, but with
opposite direction of the current. In absence of any resistance, this charging and discharging would
oscillate forever – but because of ohmic resistances which every real circuit posesses, the oscillation is
damped and so the amplitude of current and voltage decreases by time.
According to Kirchoff's law the total voltage in one loop must add to zero or be equal to an external
potential. Therefore we obtain for the circuit in Fig. 1:
U L +U C +U R=U ext ,
(1)
where
d
I
dt
•
U L=L
•
U C=
•
U R =R I
is the voltage drop across the resistor R,
•
U ext =U FG=U 0⋅exp {i ω t}
is the external voltage, which in our case is the
output of the function generator
is the voltage drop across the inductance L,
Q
C
is the voltage drop across the capacitor C,
Using these identities and differentiating (1) with respect to time t , one obtains with
d2
d
1
L 2 I + R I + I =i ω U 0 exp {i ω t}
dt
C
dt
(2)
d
Q=I :
dt
(3)
This equation can be easily transformed into the inhomogeneous differential equation for the forced
1
R
and the damping coefficient δ=
one obtains
2L
√ LC
Ï +2 δ İ + ω20 I = ω U 0 exp {i(ω t+ π )} .
(4)
L
2
oscillation; by using Euler's formula, ω0=
The real part of the solution for (3) gives the current
I =I 0 ∙ cos(ωt−φ)
(5)
with
I0 =
The phase displacement ϕ is given by
U0
√
(
R2 + ω L−
tan ϕ=−
1
ωC
(
2
.
)
1
1
ω L−
R
ωC
)
(6)
(7)
and the resonance point is found at
ω=ω0=
1
.
√ LC
(8)
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RLC-circuits
The impedance (value) is defined by Z =
U eff
. From (6) one obtains for the LC-component of the
I eff
series-tuned circuit
|
Z s= ω L−
1
ωC
|
(9)
(the absolut value is due to the fact that Z is actually a complex value).
In contrast to the mechanical oscillation, here the resonance frequency is independent of the
dampening. As can be easily shown from relations (6) and (7), at the resonance point the phase
displacement becomes zero in all components of the circuits.
In the case of the parallel-tuned RLC-circuit, we apply Kirchhoff's first law:
I R + I L + I C =0
(10)
Because the function generator represents a constant voltage source (and not constant current), we
differentiate equ. (10) with respect to time, use the identities (2) and we obtain
Ü +
1
1
U̇ +
U =0 .
RC
LC
(11)
With the ansatz U (t)=U 0⋅exp {i ω t} and after discarding the imaginary part one directly obtains the
1
. To determine the impedance for the parallel tuned circuit, one simply
√ LC
U (t )
uses (10) with I R=I and I ( t)=
to obtain
Z
U (t) U (t) U (t )
1
1
=
+i ω C
=
+
.
⇒
(12)
Zp
iω L
Zp
XL
XC
resonance frequency ω0=
|
|
Applying Kirchhoff's first law on the complete circuit and regarding the LC-component as one element
one gets
U ext =U R +U LC
(13)
U 0⋅exp {iω t}=R I +Z LC I
(here Z LC=Z p ). Therefore the solution for the current is, after neglecting the imaginary part,
I ( t)=I 0 cos (ω t + ϕ)
(14)
with
I0 =
The phase displacement ϕ is given by
U0
√
(
ωL
R+
1−ω ω
tan ϕ=
2
2
0
)
1
(
.
1
R
−ωC
ωL
)
(15)
.
(16)
Comparing the calculations from above, the results are the following:
Both circuits (series- and parallel-tuned) have the same resonance frequency
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f res=
TEP
ω0
1
=
.
2 π 2 π √ LC
(17)
In the series-tuned case, the impedance tends to zero when the frequency is approaching the resonance
frequency, which can be seen in the increase of current. In the parallel-tuned case, the impedance of the
LC-component increases while approaching the resonance frequency, which can be seen in the
decrease of current.
Another physical quantity, which describes the behaviour of a resonating system is the bandwith B and
the quality-factor Q . The bandwidth of a resonance curve is simply defined as the distance between
the two points where the maximum amplitude A max = Ares at the resonance drops to a value
Fig. 3), so
B=f 2−f 1 .
A res
(see
√2
(18)
The quality factor Q is given by
Q=
f res
.
B
(19)
In the series-tuned circuit, the quality factor can also be expressed as
Q=
√
1 L
,
R C
(20)
which can be derived from the equations above (but usually one uses
the relation B=2 δ , where δ is the damping, which provides a
much easier and faster way to obtain equ. (20)). One can see, that the
resistor is responsible for the shape of the resonance curve, too.
Fig. 3
In the parallel-tuned circuit, the quality factor, expressed through the parameters of the electrical
components, is given by
Q=R
√
C
.
L
(21)
Set-Up
The experimental set-up for measuring the voltage and current in the series-tuned circuit is shown in Fig.
4a and 4b, respectively. Ri denotes the internal resistance of the digital function generator, which is
given in the technical description as Ri=2 Ω .
The experimental set-up for measuring the voltage and current in the parallel-tuned circuit is shown in
Fig. 5a and 5b, respectively.
For the digital function generator select following settings:
DC-offset:
±0V
Amplitude (USS):
10 V
Frequency:
0 - 10 kHz
Mode:
sinusoidal
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Fig. 4a
Fig. 5a
RLC-circuits
Fig. 4b
Fig. 5b
Select the following measuring ranges on the Multimeter:
series-tuned circuit:
Voltage (~): 3 V
Current (~): 30 mA
parallel-tuned circuit:
Voltage (~): 1 V
Current (~): 10 mA
The settings can be altered according to the experimenter's discretion. But is it important to leave the
settings constant during the experiment. Especially the measuring ranges of the multimeter must remain
the same during the measurement, because different ranges use different internal resistors! It is
recommended to adjust the measuring range of the multimeter at the maximum of the resonance point
(current for series tuned circuit and voltage for parallel tuned circuit) and leave them unchanged during
the measurement.
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Procedure
The voltage U and current I are measured according to the set-up for different frequencies f ,
which can be set and directly read off at the digital function generator. The frequency steps should get
smaller when approaching the resonance frequency.
Nevertheless it is recommended to determine the resonance frequencies for the different values of
electrical components first, in order to have an idea how to choose the steps. For this one should use the
quantity, which reaches its minimum at the resonance frequency.
It is recommended to note all measurements for one quantity (e.g. the voltage) for the different
frequencies first and then measure the other quantity (e.g. current) for the same frequencies.
Results
The resonance frequencies f res are measured as follows:
0.1 µF
0.47 µF
series
3401 Hz
1552 Hz
parallel
3400 Hz
1554 Hz
Now the voltage drops U over the LC-components and the currents I through the circuits are
measured at different frequencies for the different set-ups:
For the series-tuned circuit we obtain the following results:
C = 0.1 µF
f [kHz]
0.5
1.0
1.5
2.0
2.2
2.4
2.6
2.8
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
4.0
4.2
4.4
4.6
5.0
5.5
6.0
6.5
7.0
7.5
8.0
R = 47Ω
I [mA]
0.5
1.0
2.0
3.0
4.0
5.0
6.0
8.5
12.5
15.5
20.0
25.0
27.5
25.0
20.5
16.5
14.0
10.0
8.0
6.5
5.5
4.5
3.5
3.0
2.5
2.0
2.0
2.0
U [V]
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.55
1.45
1.35
1.15
0.75
0.35
0.70
1.10
1.30
1.40
1.50
1.55
1.55
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.60
f [kHz]
0.5
1.0
1.5
2.0
2.2
2.4
2.6
2.8
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
4.0
4.2
4.4
4.6
5.0
5.5
6.0
6.5
7.0
7.5
8.0
C = 0.47 µF
R = 100Ω
I [mA]
0.5
1.0
2.0
3.0
4.0
4.5
6.0
7.5
10.0
11.5
13.0
14.0
14.5
14.0
13.0
12.0
11.0
8.5
7.0
6.0
5.5
4.0
3.5
3.0
2.5
2.0
2.0
2.0
U [V]
1.60
1.60
1.60
1.55
1.55
1.55
1.50
1.40
1.20
1.00
0.75
0.45
0.20
0.40
0.70
0.95
1.10
1.30
1.40
1.45
1.50
1.55
1.55
1.55
1.55
1.55
1.55
1.55
f [kHz]
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
R = 47Ω
I [mA]
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
6.5
8.0
10.0
13.0
17.5
23.0
29.0
30.0
29.0
24.5
20.0
16.0
13.5
12.0
10.5
9.0
8.5
7.5
7.0
6.5
6.0
6.0
5.5
U [V]
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.55
1.45
1.30
1.15
0.50
0.25
0.40
0.95
1.25
1.40
1.45
1.50
1.50
1.55
1.55
1.60
1.60
1.60
1.60
1.60
1.60
f [kHz]
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
R = 100Ω
I [mA]
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
6.0
7.5
9.0
10.5
12.5
14.0
15.0
15.5
15.0
14.5
13.0
12.0
11.0
10.0
9.0
8.0
7.5
7.0
6.5
6.0
5.5
5.5
5.0
U [V]
1.60
1.60
1.60
1.60
1.60
1.60
1.55
1.50
1.50
1.40
1.30
1.20
0.95
0.60
0.25
0.10
0.20
0.55
0.80
1.00
1.15
1.25
1.30
1.35
1.40
1.40
1.45
1.45
1.50
1.50
1.50
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RLC-circuits
For the parallel-tuned circuit we obtain the following results:
f [kHz]
0.50
1.00
1.50
2.00
2.20
2.40
2.60
2.80
3.00
3.10
3.20
3.30
3.35
3.40
3.45
U [V]
0.20
0.40
0.65
0.95
1.05
1.15
1.20
1.30
1.32
1.34
1.35
1.36
1.36
1.36
1.36
C = 0.1 µF
I [mA]
f [kHz]
2.90
3.50
2.80
3.60
2.60
3.70
2.30
3.80
2.10
4.00
1.85
4.20
1.55
4.40
1.20
4.60
0.80
5.00
0.60
5.50
0.40
6.00
0.25
6.50
0.15
7.00
0.10
7.50
0.12
8.00
U [V]
1.36
1.36
1.36
1.35
1.32
1.30
1.25
1.20
1.10
1.00
0.90
0.80
0.75
0.70
0.65
I [mA]
0.20
0.40
0.55
0.75
1.05
1.30
1.50
1.70
2.00
2.25
2.40
2.55
2.60
2.65
2.70
f [kHz]
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.55
U [V]
0.05
0.08
0.11
0.16
0.22
0.28
0.35
0.42
0.51
0.62
0.75
0.90
1.05
1.20
1.30
1.30
C = 0.47 µF
I [mA]
f [kHz]
2.92
1.60
2.92
1.70
2.91
1.80
2.90
1.90
2.90
2.00
2.90
2.10
2.85
2.20
2.80
2.30
2.75
2.40
2.65
2.50
2.50
2.60
2.28
2.70
1.90
2.80
1.30
2.90
0.53
3.00
0.27
U [V]
1.30
1.25
1.15
1.05
0.95
0.85
0.75
0.70
0.65
0.60
0.55
0.53
0.50
0.48
0.45
I [mA]
0.50
1.20
1.73
2.10
2.30
2.45
2.55
2.65
2.70
2.72
2.75
2.80
2.80
2.81
2.82
Evaluation
Task 1: Determine the resonance frequency for both combinations of coil and capacitor and compare
with the theoretical values:
Because the resonance frequency is the same for both series- and parallel-tuned circuits and is
additionally independent of the resistors, one has to consider only two cases for the two capacitors.
From equ. (8) one gets
f res=
1
,
2 π √L C
which in this case leads to the theoretical values ( L=24 mH ):
C
0.1 µF
0.47 µF
fres
3249 Hz
1499 Hz
Comparing the measured and the theoretical values, one finds that the results are within 5% deviation.
The plotted resonance curves are shown in Fig. 6 – Fig. 7:
Fig. 6a:
Resonance curve
of the current in the
series tuned circuit
with C=0.1μ F (t
he values I res / √ 2
for the bandwidth
are
plotted
as
I=const. graphs)
8
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Fig. 6b: Resonance curve of the current in the series tuned circuit with C=0.47μ F (the values
I res / √ 2 for the bandwidth are plotted as I=const. graphs)
Fig. 7a: Resonance curve of the voltage in the Fig. 7b: Resonance curve of the voltage in the
parallel tuned circuit with C=0.1μ F (the values parallel tuned circuit with C=0.47μ F (the values
U res / √ 2 for the bandwidth are plotted as U res / √ 2 for the bandwidth are plotted as
y=const. Graphs)
y=const. Graphs)
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Task 2: Determine the impedance Z of the LC-component for both circuits with the measurements
from task 1 and compare with the theoretical values.
The measured values Z m of the impedance are simply derived by
Zm=
U (f )
,
I (f )
where U (f ) and I (f ) are the voltage and current measured in task 1 at the frequency f . The
theoretical value for the series tuned circuit is given by equ. (9), and one obtains for C=0.1μ F :
C = 0.1 µF
f [kHz]
0.5
1.0
1.5
2.0
2.2
2.4
2.6
2.8
3.0
3.1
3.2
3.3
3.4
3.5
Z_m [Ω]
(R=47Ω)
3200
1600
800
533
400
320
267
182
116
87
58
30
13
28
Z_m [Ω]
(R=100Ω)
3200
1600
800
517
388
344
250
187
120
87
58
32
14
29
Z_th [Ω]
3108
1441
835
494
392
301
220
146
78
46
15
15
45
73
f [kHz]
3.6
3.7
3.8
4.0
4.2
4.4
4.6
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Z_m [Ω] Z_m [Ω]
(R=47Ω) (R=100Ω)
54
54
79
79
100
100
150
153
194
200
238
242
291
273
356
388
457
443
533
517
640
620
800
775
800
775
800
775
Z_th [Ω]
101
128
154
205
254
302
348
436
540
640
735
828
919
1007
Note:
One has to take care of the
correct values of powers of
ten!
The plotted curves for the impendances for the series-tuned circuit with C=0.1μ F are plotted in
Fig. 8:
Fig. 8: theoretical and measured impedances in the series tuned circuits with C=0.1μ F
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For C=0.47μ F
TEP
one gets:
C = 0.47 µF
f [kHz]
Z_m [Ω]
(R=47Ω)
Z_m [Ω]
(R=100Ω)
Z_th [Ω]
f [kHz]
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
3200
1600
1067
800
640
533
400
320
246
200
155
112
74
50
17
8
3200
1600
1067
800
640
533
388
300
250
187
144
114
76
43
17
6
3371
1663
1084
786
602
474
378
303
241
188
142
101
64
31
0
15
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
Z_m [Ω] Z_m [Ω]
(R=47Ω) (R=100Ω)
14
39
63
88
107
125
143
172
182
213
229
246
267
267
291
13
38
62
83
105
125
144
169
187
200
223
242
273
273
300
Z_th [Ω]
30
57
83
108
132
155
178
200
221
242
262
282
301
321
340
The plotted curves for the impendances for the series-tuned circuit with C=0.47μ F are plotted in Fig.
9a and Fig. 9b (different scaling on the y-axis):
Fig. 9a and 9b:
theoretical and measured
impedances in the series
tuned
circuits
with
C=0.47μ F
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In the parallel-tuned circuit we obtain the theoretical values for the impedance of the LC-component by
using equ. (12):
f [kHz]
0.50
1.00
1.50
2.00
2.20
2.40
2.60
2.80
3.00
3.10
3.20
3.30
3.35
3.40
3.45
Z_m [Ω]
69
143
250
413
500
622
774
1083
1650
2233
3375
5440
9067
13600
11333
C = 0.1 µF
Z_th [Ω]
f [kHz]
77
3.50
167
3.60
287
3.70
486
3.80
613
4.00
797
4.20
1091
4.40
1642
4.60
3072
5.00
5225
5.50
16205
6.00
15645
6.50
7979
7.00
5381
7.50
4073
8.00
Z_m [Ω]
6800
3400
2473
1800
1257
1000
833
706
550
444
375
314
288
264
241
Z_th [Ω]
3285
2382
1878
1556
1169
943
795
690
551
444
375
326
290
261
238
f [kHz]
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.55
Z_m [Ω]
17
27
38
55
76
97
123
150
185
234
300
395
553
923
2453
4815
C = 0.47 µF
Z_th [Ω]
f [kHz]
15
1.60
31
1.70
47
1.80
65
1.90
85
2.00
108
2.10
135
2.20
169
2.30
212
2.40
272
2.50
360
2.60
504
2.70
792
2.80
1660
2.90
115274
3.00
3345
Z_m [Ω]
2600
1042
665
500
413
347
294
264
241
221
200
189
179
171
160
Z_th [Ω]
1723
893
613
472
386
329
287
256
231
211
195
181
169
159
150
The plotted curves for the impendances for the parallel-tuned circuit are plotted in Fig. 10a and 10b:
Fig. 10a: theoretical and measured impedances in the parallel-tuned circuits with C=0.1μ F .
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Fig. 10b: theoretical and measured impedances in the parallel-tuned circuits with C=0.47μ F
One can see, that the values of the measured impedances partially differ widely from the theoretical
values, nevertheless the general behaviour is confirmed. The biggest deviations are present near the
resonance frequency, where the ohmic part of the impedance (the ohmic resistance of the coil)
contributes more than at the edges of the plotted curves.
Task 3: Determine the bandwidth B and Q-factor for the series-tuned circuit from the resonance curve
and compare with the theoretical values
The theoretical values for the quality factor Q are given by equ. (20), but before inserting the values,
one must consider the different parts which contribute to the total resistance. These are the ohmic
resistor R itself, the real part of the impedanze at the resonance point, here simply denoted as R LC ,
which is simply given by
U res
,
I res
and the internal resistance of the function generator Ri . Therefore
RLC =
U res
1 L
with Rtot =R+ Ri + R LC=R+ R i+
.
I res
R tot C
The measured value of the quality factor Q m is calculated with equ. (19). The frequencies f 1 and
f 2 for the bandwith are determined from the plot in Fig. 6 & 7.
Qth =
√
For the series-tuned case with C=0.1μ F one gets
Rtot
f1
f2
B
Qm
Qth
R = 47 Ω
61.7 Ω
3.19 kHz
3.63 kHz
0.44 kHz
7.7
7.9
R = 100 Ω
114.4 Ω
3.02 kHz
3.86 kHz
0.84 kHz
4.0
4.3
The theoretical and measured values coincide quite well (deviation within 5% in the circuit with
R=47 Ω and within 7% in the circuit with R=100 Ω ).
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For the series-tuned case with C=0.47μ F one gets
Rtot
f1
f2
B
Qm
Qth
R = 47 Ω
56.7 Ω
1.37 kHz
1.77
0.4 kHz
3.88
3.99
R = 100 Ω
109.1 Ω
1.23 kHz
2.00
0.78 kHz
2.00
2.07
In this case the values coincide even better (within 4%).
For completeness, we would like to compare the values of the quality factor for the parallel-tuned circuit
(but here we consider only the resistor R=470Ω , because the influence of multimeter etc. is much
more complicated than in the series-tuned circuit):
R = 470 Ω
f1
f2
B
Qm
Qth
1.37 kHz
1.77 kHz
0.4 kHz
3.88
3.99
In the parallel-tuned case the theoretical and measured value of the quality factor Q coincide within
3%.
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