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Chapter 1 RLC Circuit Setup Resonance and Sinusoidal Signal Lab Report

### Question Description

An RLC circuit was assembled according to the circuit diagram shown in Fig. 11.1 of the lab manual. A function generator with the output resistance of about 50 Ohms generating a sinusoidal waive of variable frequency served as an AC source. The resistor box (set for R=950 Ohms) was connected in series with a capacitor (C= 0.522 nF) and inductor (L=22mH). A sinusoidal signal applied to the circuit was registered on Ch1 of the oscilloscope, whereas the signal across the resistor on Ch 2. (Fig.1). A frequency counter was used for the accurate measurements of the frequency output of the function generator.

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RLC circuits: resonance Before attempting the assignment read the description of Experiment11 in the lab manual concentrating on the theory. This assignment is Part II, Steady-state response , only. Experiment. An RLC circuit was assembled according to the circuit diagram shown in Fig. 11.1 of the lab manual. A function generator with the output resistance of about 50 Ohms generating a sinusoidal waive of variable frequency served as an AC source. The resistor box (set for R=950 Ohms) was connected in series with a capacitor (C= 0.522 nF) and inductor (L=22mH). A sinusoidal signal applied to the circuit was registered on Ch1 of the oscilloscope, whereas the signal across the resistor on Ch 2. (Fig.1). A frequency counter was used for the accurate measurements of the frequency output of the function generator. Fig.1. RLC circuit setup. The oscilloscope settings were adjusted to better see several cycles of the sinusoidal wave on both channels (Fig.1). Figure 1 shows the signals from both channels. A phase shift between two sine waves due to the phase shift between the current in the circuit (which is in-phase with the voltage across the resistor registered on Ch 2)) and the driving voltage (Ch 1) is registered by the oscilloscope. Before proceeding with measurements, the range of the frequencies that should be used in experiment was estimated. To do that, a resonance frequency for which there was no phase shift between the signals on the channels was first found, (Fig. 2), then the measurements were performed for the frequencies below and above the resonance. Changing frequency in the vicinity of the resonance show that the amplitude of the current (i.e. the voltage across the resistor VR) is low away from the resonance, and reaches its maximum at the resonance frequency. Fig. 2. Circuit at the resonance frequency. For the very low frequencies (below the resonance frequency), the reactance of the capacitance is very big, dominating the behavior of the circuit, L<<(1/C), tan → - ∞, and = - /2. In this case, current in the circuit leads the voltage on the generator, and the peak of the sinusoid corresponding to the maximum voltage on the resistor (Ch 2) appears before (or to the left of) the peak of the sinusoid corresponding to the voltage on the generator (Ch 1) (Fig.3) Fig. 3. Circuit at the low frequency. For the high frequencies (above the resonance frequency), the reactance of the inductor is very big, dominating the behavior of the circuit, L>>(1/C), tan → ∞, and = + /2. In this case, current in the circuit lags the voltage on the generator, and the peak of the sinusoid corresponding to the maximum voltage on the resistor (Ch 2) appears after (of to the right of) the peak of the sinusoid corresponding to the voltage on the generator (Ch 1) (Fig. 4). Fig. 4. Circuit at the high frequency. Data were taken for the frequency range of 20-85 kHz. The amplitude of the signal on Ch 1 was kept constant throughout the measurements. Figure 5 shows a table with the measured values of the peak-to-peek voltage across the resistor VR (Ch 2) (column 3) and peak-to-peak voltage of the generator V0 (Ch 1) (column 2) at the frequencies listed in column 1. The measured time shift between the signals on Ch 1 and Ch 2, t, is in column 4. Assignment. 1. From the provided data and with the help of the lab manual, calculate T=1/f for each frequency f, gain g=VR/VR max (where VR max is the maximum peak-to-peak voltage across the resistor), and the phase shift =2t/T for each frequency f. Assign correct signs to . Put the measured and calculated values in a table as described in Section 11.3.2 of the lab manual. 2. Plot a graph of g vs. f. Estimate the resonance frequency fo from the graph. Compare it with the theoretical value of f theory (Eq. (11.11). 3. Plot a graph of  vs. f. Estimate the resonance frequency fo from the graph. Compare it with the theoretical value of f theory (Eq. (11.11). Figure 5. Measured peak-to-peak voltage across the resistor VR and time shift t as a function of driving frequency f. ...
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