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Oluwatomiyin Akinbo June 12th, 2023 ECE 301, Lab Section 061 Gennady Friedman TAs: Ahmadzadeh, Mahya Electrical and Computer Engineering Dept. Lab 8 Series RLC Bandpass Filter Introduction Filters are very important in audio and other signal processing applications. Depending on the specifics and complexity of the design, they can be used to remove unwanted high or low frequency noise, select, or reject a specific frequency band or perform other applications.

Fig. 1. Gain vs frequency response for a bandpass filter. The bandpass filter (Fig. 1) is designed to pass a select band of frequencies while blocking all others. It is characterized by its center frequency, ω0, and its bandwidth, B. The bandwidth is the difference between two critical radial frequencies, ωc1 and ωc2, and describes the selectivity of the filter. The critical frequencies are defined to be where the gain MBP drops to a level of 0.707, or 3 dB below the peak value. A filter can be considered a two-port network, such as that in Fig. 2, with the input supplied between nodes a and b, and the output available between nodes c and d. The gain of the system is the ratio of the output voltage to the input. This is also called the transfer function of the network, H(ω). For a passive filter the magnitude of H(ω) is less than or equal to 1. Using an active filter adds the opportunity of having a gain greater than 1.

Fig. 2. Filter as a two-port network [1]. Fig. 3. A linear network performing a bandpass function using a series RLC circuit The techniques used to find the transfer function of the bandpass filter (Fig. 3) are based on the same ac circuit analysis techniques you applied in ECE 201: Transfer sources and components into the phasor domain, analyze the circuit, and determine the transfer function. From the complex transfer function, you can determine the magnitude of the gain (MBP(ω)) and the phase shift introduced by the filter, Φ(ω). These two functions are graphed on a Bode plot. The transfer function for the series RCL circuit, with the output across the resistor, is HBP( 𝜔 )= 𝑉𝑜𝑢𝑡

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𝑉𝑖𝑛 = 𝑗𝑤𝑅𝐶 (1− 𝜔 2)+ 𝑗𝜔𝑅𝐶 The bandwidth and quality factor Q for the filter are given by 2 and 3, respectively. 𝐵 = 𝑅 / 𝐿 𝑄 = 𝜔𝑜𝐿/ R When we take into account “real” components, particularly the inductor and its parasitic resistance, the circuit performance changes (Fig. 4). Fig. 4. RLC bandpass filter with buffered input. Center frequency is 4 kHz, bandwidth is 1 kHz. Component RL represents the parasitic resistance of the inductor L1. The transfer function becomes HBP( 𝜔 )= 𝑉𝑜𝑢𝑡 / 𝑉𝑖𝑛 = 𝑗𝑤𝑅 1 𝐶 / (1− 𝜔 2 𝐿𝐶 )+ 𝑗𝜔𝑅𝑅𝑇𝐶 where RT = R1 + RL. Because of the additional resistance, the magnitude of the gain no longer equals 1 at ω0. MBP( 𝜔 )= 𝑅 1/ 𝑅𝑇 = 𝑅 1/ 𝑅 1+ 𝑅𝐿 The bandwidth and Q also change 𝐵 ′ = 𝑅𝑇 / 𝐿 𝑄 ′ = 𝜔 0/ 𝐵 = 𝜔 0 𝐿 / 𝑅𝑇 The bandwidth is now measured 3 dB down from the reduced peak amplitude

M’BP. Note that equations 4, 5, 6, and 7 revert to the ideal cases if RL = 0. In this lab, you will initially design a series RLC bandpass filter to meet a set of specifications, perform a simulation to verify the design, then match the simulation result to straight line approximations of the Bode plot. Following your analysis, you will modify your design to account for the parasitic resistance of the inductor, then simulate and measure the filter using the ELVISmx Bode Analyzer. This linkage of software and hardware produces overlapping plots of gain and phase, allowing easy comparison. Simulation The circuit of Fig.4 was captured in Multisim (Fig.5). The design produces a bandpass filter with a 10 kHz center frequency and 2 kHz bandwidth considering the real value and parasitic resistance of the inductor. C1 was 3.11nF and R1 was 787 ohm. NI ELVISmx Bode Analyzer block was used to configure the settings to give a suitable sweep range. The Mapping to “logarithmic” to plot temporal frequency f on a log scale. Then the stimulation was run. The gain and phase responses will be plotted (Fig. 6). Figure 5: Multisim Schematic of RLC bandpass filter with buffered input

Fig. 6. Simulated gain and phase response of a series RLC bandpass filter with center frequency of 10 kHz and bandwidth of 2 kHz. Measurement The passive filter with “100mH” inductor and C1 of 3.11nF and R1 of 787 ohms was built (Fig.7). The wiring connections used matches those in “virtual connection” schematic. These are: • AO0(AnalogOutput0) for Stimulus, • AI0(AnalogInput0) to record the Stimulus; connect AI0 + to Stimulus and AI0- to ground, • AI1(AnalogInput1) to record the Response; connect AI1+to the output voltage and AI1- to ground. The frequency sweep of the circuit was run through ELVIS Bode Analyzer. The gain and phase were plotted and was very close to the simulated results (Fig. 8).

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Fig. 8. Real gain and phase response of the 10-kHz center frequency RLC bandpass filter plotted along with the simulated results. Conclusion In the AC Sweep analysis, the center frequency, f0, is 10kHz. For the lab, I performed an AC sweep on the circuit with the parasitic resistance of the inductor included. An AC sweep is required to graph the Bode Plots in Figure 6. The peak bandwidth, B, is 787m which is due to the voltage drop from the parasitic resistance of the resistor