RLC Circuit
LEP
4.4.06
-01
Related topics
Kirchhoff’s laws , series and parallel tuned circuit, resistance, capacitance, inductance, phase displacement, 𝑄-factor, band-width, loss resistance, damping
Principle
The current and voltage of parallel and series-tuned circuits are investigated as a function of frequency.
𝑄-factor and band-width of the resonance curves are determined.
Material
1
1
1
1
1
1
2
1
1
1
1
3
3
1
1
Coil, 300 turns
PEK capacitor (case 2) 1.0 µF/250V
PEK capacitor (case 1) 0.1 µF/250V
PEK carbon resistor 1 W 5% 47 Ω
PEK carbon resistor 1 W 5% 100 Ω
PEK carbon resistor 1 W 5% 470 Ω
PEK carbon resistor 1 W 5% 1000 Ω
Connection box
Connection plug
Multimeter
Digital function generator
Connecting cord, l = 250 mm, red
Connecting cord, l = 250 mm, blue
Connecting cord, l = 500 mm, red
Connecting cord, l = 500 mm, blue
Fig. 1:
06513-01
39113-01
39105-18
39104-62
39104-63
39104-15
39104-19
06030-23
39170-00
07128-00
13654-99
07360-01
07360-04
07361-01
07631-04
Experimental set-up with the RLC series-tuned circuit.
www.phywe.com
P2440601
PHYWE Systeme GmbH & Co. KG © All rights reserved
1
TEP
4.4.06
-01
RLC Circuit
Tasks
Analyze the frequency performance of an
1. series-tuned circuit for
a. voltage resonance with 𝑅𝑑 = 0 Ω.
b. current resonance with 𝑅𝑑 = 0 Ω, 47 Ω, 100 Ω.
2. parallel-tuned circuit for
a. current resonance with 𝑅𝑑 = ∞.
b. voltage resonance with 𝑅𝑑 = 470 Ω, 1000 Ω, ∞.
Set-up
The experimental set-up is shown in Fig. 1.
For the digital function generator select following settings:
• DC-offset:
±0V
• Amplitude:
3 to 4 V
• Frequency: 0 - 30 kHz
• Mode:
sinusoidal
None of the settings should be changed during the
experiment.
The circuit diagram for the series-tuned circuit is
shown in Fig. 2, where 𝑅𝑑 denotes the damping resistance. The equivalent diagram for the parallel-tuned
circuit is shown in Fig. 3. A 1 kΩ resistor 𝑅𝑘 is used
for coupling the tuned circuit to the digital function
Fig. 2:
generator.
Equivalent diagram for the series-tuned circuit.
Procedure
The multimeter must be set to AC mode. For measuring the voltage select the range up to 2 V. The
range for the current should be 20 mA. In some
cases it may be necessary to change the range to
200 mA.
In order to evaluate the quality of the resonance
curves, the frequency generator’s internal resistance 𝑅𝑖 and the circuits’ loss resistances 𝑅𝑣 have to
be determined. To calculate the internal resistance
of the function generator measure current and terminal voltage in the series-tuned circuit at the frequency of 0 Hz. For computing the loss resistances
in both circuits measuring current and voltage at the Fig. 3: Equivalent diagram for the parallel-tuned circuit.
resonance point is needed. In the case of the parallel-tuned circuit, the capacitors loss resistance can be ignored, as it amounts to several hundred MΩ .
Task 1:
a) To record the voltage resonance curve for the series-tuned circuit, the voltage across the tuned
circuit is measured consecutively for both capacitances while passing through the frequency
range.
b) In the case of the current resonance the multimeter is connected as ammeter in the circuit and
2
PHYWE Systeme GmbH & Co. KG © All rights reserved
P2440601
LEP
4.4.06
-01
RLC Circuit
the frequency range is again traversed for three damping resistances 𝑅𝑑 = 0 Ω, 47 Ω, 100 Ω and
one capacitance.
Task 2:
a) For recording the current resonance curve in the parallel-tuned circuit, the total current is measured for both capacitances with the multimeter while traversing the frequency range.
b) In the case of the voltage resonance curve, the voltage across the tuned circuit is measured for
one capacitance with the multimeter and the frequency range is again traversed for the parallel
resistances 𝑅𝑑 = 470 Ω, 1000 Ω, ∞.
Note: A change of the measuring range results in an altered internal resistance of the multimeter. This
will lead to different values. Try to prevent such changes during measurements by choosing the measuring range according to the values at the resonance frequency.
Theory
Considering the tuned RLC-circuit one has to differ between series-tuned and parallel-tuned circuits. For
the series-tuned circuit Kirchhoff’s second law states, that all potentials in one loop add to zero, see equation (1).
𝑈𝐿 + 𝑈𝐶 + 𝑈𝑅 − 𝑈𝐺 = 0
(1)
There 𝑈𝐺 is the function generator’s potential, which has the opposite direction in respect to all other potentials. The potentials can be described as follows:
𝑈𝐶 =
𝑄
𝐶
𝑑
𝑈𝐿 = 𝐿 ∙ 𝐼
𝑑𝑡
𝑈𝑅 = 𝑅 ∙ 𝐼
𝑈𝐺 = 𝑈0 ∙ 𝑒 𝑖𝜔𝑡
As the time derivative of the charge 𝑄 gives the current 𝐼, insertion into (1) and differentiation gives relation (2).
𝑑2
𝐿 ∙ 𝑑𝑡 2 𝐼 + 𝑅 ∙
𝑑
𝐼
𝑑𝑡
𝐼
𝐶
+ = 𝑖𝜔𝑈0 ∙ 𝑒 𝑖𝜔𝑡
(2)
This equation can be transformed into the inhomogeneous differential equation for the forced oscillation
1
𝑅
as shown in relation (3). There 𝜔0 =
is the eigenfrequency and 𝛿 = the damping coefficient.
𝐼 ̈ + 2𝛿𝐼 ̇ + 𝜔02 𝐼 =
𝜔
𝐿
∙ 𝑈0 ∙ 𝑒
𝐿𝐶
𝜋
𝑖(𝜔𝑡+ )
2
2𝐿
(3)
The real part of the solution for (3) gives the current
with
𝐼 = 𝐼0 ∙ cos(ωt − φ)
(4)
𝐼0 =
(5)
𝑈0
2
�𝑅2 +�𝜔𝐿− 1 �
.
𝜔𝐶
The phase displacement 𝜑 is given by
1
𝑅
tan 𝜑 = − ∙ �𝜔𝐿 −
1
�
𝜔𝐶
(6)
and the resonance point is found at
𝜔 = 𝜔0 =
1
√𝐿𝐶
.
(7)
www.phywe.com
P2440601
PHYWE Systeme GmbH & Co. KG © All rights reserved
3
TEP
4.4.06
-01
RLC Circuit
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.
For the parallel-tuned circuit Kirchhoff’s first law is applied.
𝐼𝑅 + 𝐼𝐿 + 𝐼𝐶 − 𝐼𝐺 = 0
At the resonance point the current becomes minimal as most of the current circulates as reactive current
in the circuit without flowing through the conductors.
In order to evaluate the quality of the resonance curve the bandwidth 𝐵 of the resonance is calculated
with
𝐵 = 2|𝜔𝑏 − 𝜔0 |
𝐼(𝜔𝑏 )
𝐼(𝜔0 )
=
1
√2
(8)
.
(9)
With 𝜔0 = 2𝜋 ∙ 𝜈0 follows for the quality factor 𝑄 =
𝜐0
.
𝐵
The quality factor can also be calculated directly from resistance, capacitance and inductance of the circuit. We obtain for the series-tuned circuit
1
𝑅
𝑄𝑠 = ∙ �
𝐿
𝐶
(10)
and for the parallel-tuned circuit
𝐶
𝐿
𝑄𝑝 = 𝑅 ∙ � .
(11)
In both cases 𝑅 means the total resistance of the circuit. For the series connection and the parallel con1
𝑝
𝑠
nection we obtain 𝑅𝑡𝑜𝑡
= 𝑅𝑖 + 𝑅𝑣 + 𝑅𝑑 and 𝑅𝑡𝑜𝑡 = 𝑅𝑖 + 𝑅𝑘 + 1 1 respectively. The loss resistance in the
+
𝑅𝑑 𝑅𝑣
parallel circuit lies in series with the coil but is not, however, identical with the coil’s d.c. resistance. In our
case, the coil’s d.c. resistance of 0.8 Ω can be disregarded.
Results and Evaluation
In the following the evaluation of the obtained values is described exemplary. Your results may vary from
those presented here.
Task 1: Analyze the frequency performance of a series-tuned circuit.
For the internal resistance of the digital function generator 𝑅𝑖 = 4.1 Ω was found. The loss resistance
was determined at 𝜐0 = 11 kHz with 𝑅𝑙 = 16.4 Ω for the circuit with 𝐶 = 0.1 µF .
Fig. 4 shows the resonance curves of the current for different resistances. The strong influence of the resistance on the quality of the resonance is obvious. The 𝑄-factors calculated from the bandwidth are given in Tab.1.
Tab. 1:
Current resonance curves: bandwidths and quality factors for the series-tuned circuit.
resistance bandwidth quality factor
2 kHz
5.5
0Ω
8 kHz
1.4
47 Ω
11 kHz
1.0
100 Ω
4
PHYWE Systeme GmbH & Co. KG © All rights reserved
P2440601
RLC Circuit
LEP
4.4.06
-01
As had been shown in the theory-section, the eigenfrequency of a tuned circuit depends on the inductance as well as the capacity in the circuit. Fig. 5 displays that dependence for the voltage resonance in
the series-tuned circuit.
Fig. 5:
Terminal voltage as a function of frequency in the series tuned circuit.
Task 2: Analyze the frequency performance of a parallel-tuned circuit.
A loss resistance of 1460 Ω was found for the parallel-tuned circuit. The 𝑄-factors determined from Fig. 6
are listed in Tab. 2.
Tab.: 2 Voltage resonance curves: bandwidths and quality
factors for the parallel-tuned circuit.
resistance
470 Ω
1000 Ω
∞
bandwidth
8 kHz
6 kHz
4 kHz
quality factor
1.4
1.8
2.8
Fig. 7 shows the current resonance in the parallel-tuned circuit for the second capacitor.
For the resonance frequencies the found values of 11 kHz and 3.25 kHz with 𝐶 = 0.1 µF and 𝐶 = 1.0 µF
respectively agree within 5 %.
www.phywe.com
P2440601
PHYWE Systeme GmbH & Co. KG © All rights reserved
5
TEP
4.4.06
-01
RLC Circuit
Fig. 6:
Fig.: 7
6
Voltage as a function of frequency in the parallel-tuned circuit.
Current as a function of frequency in the parallel-tuned circuit.
PHYWE Systeme GmbH & Co. KG © All rights reserved
P2440601