RLC-circuits TEP. f res. = 1 2 π L C.

Transcription

1 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. P PHYWE Systeme GmbH & Co. KG All rights reserved 1

2 TEP 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 R=100 Ω. b) for the parallel-tuned circuit with resistor R=470 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 Digital Function Generator, USB Coil, 900 turns Capacitor 100 nf/250 V, G Capacitor 470 nf/250 V, G Resistor 47 Ohm, 1W, G Resistor 100 Ohm, 1W, G Resistor 470 Ohm, 1W, G 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 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 Fig. 2: circuit diagramm for a parallel-tuned RLC-circuit 2 PHYWE Systeme GmbH & Co. KG All rights reserved P

3 RLC-circuits TEP 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: where U L +U C +U R =U ext, (1) U L =L d I is the voltage drop across the inductance L, dt U C = Q C is the voltage drop across the capacitor C, U R =R I is the voltage drop across the resistor R, U ext =U FG =U 0 exp{i ωt} is the external voltage, which in our case is the output of the function generator (2) Using these identities and differentiating (1) with respect to time t, one obtains with d dt Q=I : L d2 dt 2 I+R d dt I+ 1 C I=iωU 0 exp{i ωt} (3) This equation can be easily transformed into the inhomogeneous differential equation for the forced oscillation; by using Euler's formula, ω 0 = 1 LC The real part of the solution for (3) gives the current with The phase displacement and the damping coefficient δ= R 2 L one obtains Ï+2δ İ+ω 0 2 I= ω L U 0 exp{i(ω t+ π 2 )}. (4) I 0 = ϕ is given by and the resonance point is found at I=I 0 cos(ωt φ) (5) U 0 R2 + ( ω L ωc) 1 2. tan ϕ= 1 R( ω L 1 ω C) (7) ω=ω 0 = 1 LC. (8) (6) P PHYWE Systeme GmbH & Co. KG All rights reserved 3

4 TEP RLC-circuits The impedance (value) is defined by series-tuned circuit Z = U eff I eff. From (6) one obtains for the LC-component of the 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 RC U + 1 U =0. (11) LC With the ansatz U (t)=u 0 exp{i ωt} and after discarding the imaginary part one directly obtains the resonance frequency ω 0 = 1 LC uses (10) with I R =I and I(t)= U (t) Z. To determine the impedance for the parallel tuned circuit, one simply U (t) = U (t) + U (t ) Z p X L X C to obtain 1 = Z p 1 iω L +i ω C. (12) 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{iωt}=r I +Z LC I (here Z LC =Z p ). Therefore the solution for the current is, after neglecting the imaginary part, with The phase displacement I(t)=I 0 cos(ωt+ϕ) (14) U 0 I 0 = R 2 + ( ω L 1 ω ω 0)2 ϕ is given by tan ϕ= 1. (15) R( 1 ω L ωc ). (16) Comparing the calculations from above, the results are the following: Both circuits (series- and parallel-tuned) have the same resonance frequency 4 PHYWE Systeme GmbH & Co. KG All rights reserved P

5 RLC-circuits TEP f res = ω 0 2π = 1 2π 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 = A res at the resonance drops to a value A res 2 Fig. 3), so B=f 2 f 1. (18) The quality factor Q is given by (see Q= f res B. (19) In the series-tuned circuit, the quality factor can also be expressed as Q= 1 R L 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. R i denotes the internal resistance of the digital function generator, which is given in the technical description as R i =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: ± 0 V Amplitude (U SS ): 10 V Frequency: 0-10 khz Mode: sinusoidal P PHYWE Systeme GmbH & Co. KG All rights reserved 5

6 TEP RLC-circuits Fig. 4a Fig. 4b Fig. 5a 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. 6 PHYWE Systeme GmbH & Co. KG All rights reserved P

7 RLC-circuits TEP 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 C = 0.47 µf R = 47Ω R = 100Ω R = 47Ω R = 100Ω f [khz] I [ma] U [V] f [khz] I [ma] U [V] f [khz] I [ma] U [V] f [khz] I [ma] U [V] P PHYWE Systeme GmbH & Co. KG All rights reserved 7

8 TEP RLC-circuits For the parallel-tuned circuit we obtain the following results: C = 0.1 µf C = 0.47 µf f [khz] U [V] I [ma] f [khz] U [V] I [ma] f [khz] U [V] I [ma] f [khz] U [V] I [ma] 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 f res 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 PHYWE Systeme GmbH & Co. KG All rights reserved P

9 RLC-circuits TEP 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 parallel tuned circuit with C=0.1μ F (the values U res / 2 for the bandwidth are plotted as y=const. Graphs) Fig. 7b: Resonance curve of the voltage in the parallel tuned circuit with C=0.47μ F (the values U res / 2 for the bandwidth are plotted as y=const. Graphs) P PHYWE Systeme GmbH & Co. KG All rights reserved 9

10 TEP RLC-circuits 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 Z m = 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 Z_m [Ω] Z_m [Ω] Z_m [Ω] Z_m [Ω] f [khz] (R=47Ω) (R=100Ω) Z_th [Ω] f [khz] (R=47Ω) (R=100Ω) Z_th [Ω] , 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 10 PHYWE Systeme GmbH & Co. KG All rights reserved P

11 RLC-circuits TEP For C=0.47μ F one gets: C = 0.47 µf Z_m [Ω] Z_m [Ω] Z_m [Ω] Z_m [Ω] f [khz] (R=47Ω) (R=100Ω) Z_th [Ω] f [khz] (R=47Ω) (R=100Ω) Z_th [Ω] 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 P PHYWE Systeme GmbH & Co. KG All rights reserved 11

12 TEP RLC-circuits In the parallel-tuned circuit we obtain the theoretical values for the impedance of the LC-component by using equ. (12): C = 0.1 µf C = 0.47 µf f [khz] Z_m [Ω] Z_th [Ω] f [khz] Z_m [Ω] Z_th [Ω] f [khz] Z_m [Ω] Z_th [Ω] f [khz] Z_m [Ω] Z_th [Ω] 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. 12 PHYWE Systeme GmbH & Co. KG All rights reserved P

13 RLC-circuits TEP 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 R LC = U res I res, and the internal resistance of the function generator R i. Therefore Q th = 1 R tot L C with R tot=r+r i +R LC =R+R i + U res I res. 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. For the series-tuned case with C=0.1μ F one gets R tot f 1 f 2 B Q m Q th R = 47 Ω 61.7 Ω 3.19 khz 3.63 khz 0.44 khz R = 100 Ω Ω 3.02 khz 3.86 khz 0.84 khz 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=100Ω ). P PHYWE Systeme GmbH & Co. KG All rights reserved 13

14 TEP RLC-circuits For the series-tuned case with C=0.47μ F one gets R tot f 1 f 2 B Q m Q th R = 47 Ω 56.7 Ω 1.37 khz khz R = 100 Ω Ω 1.23 khz khz 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): f 1 f 2 B Q m Q th R = 470 Ω 1.37 khz 1.77 khz 0.4 khz In the parallel-tuned case the theoretical and measured value of the quality factor 3%. Q coincide within 14 PHYWE Systeme GmbH & Co. KG All rights reserved P