LEP RLC Circuit

Transcription

1 RLC Circuit LEP 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 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 2 PEK carbon resistor 1 W 5% 1000 Ω Connection box 1 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: Experimental set-up with the RLC series-tuned circuit. P PHYWE Systeme GmbH & Co. KG All rights reserved 1

2 TEP RLC Circuit Tasks Analyze the frequency performance of an 1. series-tuned circuit for a. voltage resonance with R d = 0 Ω. b. current resonance with R d = 0 Ω, 47 Ω, 100 Ω. 2. parallel-tuned circuit for a. current resonance with R d =. b. voltage resonance with R d = 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: ± 0 V 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 R d denotes the damping resistance. The equivalent diagram for the parallel-tuned circuit is shown in Fig. 3. A 1 kω resistor R k is used for coupling the tuned circuit to the digital function generator. Fig. 2: 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 R i and the circuits loss resistances R v 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: 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. Equivalent diagram for the parallel-tuned circuit. 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 P

3 RLC Circuit LEP the frequency range is again traversed for three damping resistances R d = 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 R d = 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). U L + U C + U R U G = 0 (1) There U G is the function generator s potential, which has the opposite direction in respect to all other potentials. The potentials can be described as follows: U C = Q C U L = L d I dt U R = R I U G = U 0 e iωt As the time derivative of the charge Q gives the current I, insertion into (1) and differentiation gives relation (2). L d2 dt 2 I + R d dt I + I C = iωu 0 e iωt (2) This equation can be transformed into the inhomogeneous differential equation for the forced oscillation as shown in relation (3). There ω 0 = 1 is the eigenfrequency and δ = R the damping coefficient. LC 2L I + 2δI + ω 0 2 I = ω L U 0 e i(ωt+π 2 ) (3) The real part of the solution for (3) gives the current I = I 0 cos(ωt φ) (4) with U 0 I 0 =. (5) R 2 + ωl 1 2 ωc The phase displacement φ is given by tan φ = 1 ωl 1 (6) R ωc and the resonance point is found at ω = ω 0 = 1. (7) LC P PHYWE Systeme GmbH & Co. KG All rights reserved 3

4 TEP 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. I R + I L + I C I G = 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 ω b ω 0 (8) I(ω b ) I(ω 0 ) = 1 2. (9) With ω 0 = 2π ν 0 follows for the quality factor Q = υ 0 B. The quality factor can also be calculated directly from resistance, capacitance and inductance of the circuit. We obtain for the series-tuned circuit Q s = 1 R L C (10) and for the parallel-tuned circuit Q p = R C L. (11) In both cases R means the total resistance of the circuit. For the series connection and the parallel connection we obtain R tot = R i + R v + R d and R p tot = R i + R k + 1 respectively. The loss resistance in s the R d 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 R i = 4.1 Ω was found. The loss resistance was determined at υ 0 = 11 khz with R l = 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 0 Ω 2 khz Ω 8 khz Ω 11 khz PHYWE Systeme GmbH & Co. KG All rights reserved P

5 RLC Circuit LEP 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 bandwidth quality factor 470 Ω 8 khz Ω 6 khz khz 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 %. P PHYWE Systeme GmbH & Co. KG All rights reserved 5

6 TEP RLC Circuit Fig. 6: Voltage as a function of frequency in the parallel-tuned circuit. Fig.: 7 Current as a function of frequency in the parallel-tuned circuit. 6 PHYWE Systeme GmbH & Co. KG All rights reserved P