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EXERCISES RESONAT CIRCUITS 5.21 The resonant circuit of the Figure 1 has Q=5 and R=2 . The generator produces resonance in the circuit. Calculate: a) Values of L and C b) Intensity of the current c) Intenisy for =0.001 and 0.05 Results 5.22 Knowing that the quality factor of the coil (L with internal resistance r) in the circuit of the Figure 1 at 0=1Mrad/s is Qb=50, and that the antiresonant circuit receives the maximum power at this frequency, obtain: a) Values of r, L and C. b) Value of the current through the coil if the frequency of the generator is increased in 2% with respect the resonant frequency. c) How should be connected a resistance R into the circuit and what value should it have to increase 3 times the bandwidth. Results 5.23 For the circuit of the Figure 1 the bandwidth at 3dB for th voltage v(t) should be B=200 rad/s. Calculate: a) Value of . b) Temporal expression of v(t) at =10100 rad/s. Results 5.24 In the circuit of the figure 1, the value of the amperimeter is maximum for a specific frequency. If the frequency is separated 5% over this frequency the value of the amperimeter decreases to the value of Aef. Obtain the value of C. Results C= 1 F 5.25 In the circuit of the Figure 1, if the value of the frequency of the generator is changed the coltage v changes respectively as follows: Calculate: a) Values of L and C b) Value of the resistance (RP ) to be connected in parallel to RL to duplicate the bandwidth. Results 5.26 In the circuit of the figure the transformer is perfect. Calculate: a) Value of C2 so that for =10⁷ rad/s the current (IL) through the load resistance RL becomes maximum. b) Value of RL so it receives the maximum power for the frequency =10⁷ rad/s. c) Cut frequencies (2 and 1) for which the attenuation of the current IL is 6 dB. Results a) 5.27 C = 4 nF. In the auto-transformer of the figure 1, the voltage generator is connected to the center of the coil. Knowing that the frequency of the generator eg(t) produces a maximum of the current trough the resistance R. Calculate: a) The current trough the capacitor, iC(t). b) Power dissipated by the resistance R if the frequency of the generator is increased 1%. Results 5.28 For the building the resonant circuit of the figure 1 where Rg=40 and RL =80 , the elements shown in figure 2 are available. Determine: a) Design a resonant circuit RLC serial or RLC parallel to tune a signal comprised between 1=975 rad/s and 2=1025 rad/s. b) How could be duplicated the bandwidth using one of the available elements not used previously and maintaining the same resonant frequency. Results 5.29 It is desired to tune a signal centered at 10 Krad/s and bandwidth at 3 dB of 100 rad/s. For this propose an antenna and a resonant circuit is used as shown in figure 1. The equivalent of the antenna is the circuit left from the terminals A-B, and the resonant circuit is right from these terminals. Calculate: a) Value of the transformation relation (a) and the resistance (R ) to tune the signal. b) Knowing the values of R and C can be changed, justify if R and/or C have to be increased, decreased or remain the same for tune another signal centered at 20 Krad/s, maintaining the bandwidth at 3 dB equal to 100 rad/s. Results 5.30 With the circuit of the figure 1 a signal is to be tuned. The signal is obtained from an antenna whose equivalent circuit is indicated within the dotted rectangle. For the resonant frequency the antenna delivers the maximum power. Calculate: a) The quality factor and bandwidth at 3 dB. b) The voltage at the capacitor at the cut frequencies of the bandwidth. Results 5.31 In the circuit of the figure, the generator (Eg, Rg) delivers the maximum power at the resonance frequency. a) Knowing that v(t)=cos(0t) V, where 0 is the resonance frequency, calculate the value of the amplitude of the generator Eg and the transformation relation (a) of the transformer. b) The resonant frequency, the quality factor and the bandwidth at 3 dB. c) If the resistance RL is substituted by a capacitor, justify is the new resonant frequency increases, decreases or remains the same. Results 5.32 For the circuit shown in figure 1, two resonance curves where obtained one for each of the following two conditions of the circuit: • Condition 1: The load resistance RL is disconnected. • Condition 2: The load resistance RL is connected. The corresponding resonance curves of the amplitude of the normalized voltages are shown in figure 2. From these representations the bandwidth at 3 dB were obtained (Bb = 40 Krad/s Ba=40/3 Krad/s). a) Rezone which resonant curve correspond the each of the conditions. b) Calculate the values of L1 and C. c) Calculate the value of RL. Results 5.33 By varying C in the circuit of the figure 1, it is observed that for the value of 2,5 nF the voltage V1=48V is maximum, being the current I1=3mA. Under these conditions obtain: a) Value of the coefficient L1 and quality factor b) The current trough RL if the frequency is increased 3,125%. c) Value of the resistance to be connected between A and B so that the generator delivers the maximum power at the resonance frequency. d) Relation between the bandwidth of the resonant circuit after and before the resistance between A and B is connected. Results 5.34 By varying the capacitance C in the circuit of the figure 1, it is observed that at C= 5 nF the current through RL becomes maximum and equal to 3 mA. Under these conditions, calculate: a) Value of the transformation relation of the perfect transformer. b) Quality factor of the circuit. c) Voltage at the load resistance vRL(t) when the frequency is increased to 10,5 Mrad/s. Results 5.35 In the circuit of the figure 1 it is observed that the voltage v(t) becomes maximum at 0=500 Krad/s and the bandwidth at 3 dB is B=12,5 Krad/s. Calculate: a) Values of C and RL. b) Temporal expression of v(t) when the frequency is w1=505Krad/s. Results 5.36 In the circuit of the figure 1, the voltage v(t) reaches a maximum peak value of 5 V for a specific frequency. Calculate: a) The value of L2 and the bandwidth at 3 dB. b) The temporal expression of v(t) when the frequency is shifted 1% over the resonance frequency. Results 5.37 In the circuit of the figure 1, a bandwidth at 3 dB of B =100 rad/s is desired. Calculate: a) Value of . b) Voltage v(t) for =0.005. Results 5.38 In the circuit of the figure 1, at the frequency 0= 1000 rad/s the amplitude of the voltage at the capacitor vC(t) is maximum and equal to 7,5 V. By increasing in q0 rad/s the value of the frequency the value of the amplitude of vC(t) becomes 3,75 V. Calculate: a) The quality factor and bandwidth at 3 dB b) The values of R, L and C. Results 5.39 In the circuit of the figure 1 the bandwidth at 3 dB is 1000 rad/s. Calculate: a) Value of RL. b) The current i2(t) if the frequency is decreased 0,5% below the resonance frequency. Results 5.40 In a resonant serial circuit it is known that at 0 = 1000 rad/s the current trough the resistance R is maximum, and by increasing 10 rad/s this frequency an attenuation of 3 dB is observed. Calculate: a) The attenuation in dB for a frequency of 1 = 1020 rad/s. b) The attenuation in dB for a frequency of 1 = 2000 rad/s. c) What will happen with the resonance frequency and with the bandwidth when a resistance RP is connected in parallel to the existing resistance R. d) What will happen with the resonance frequency and with the bandwidth when a resistance RS is connected in serial to the existing resistance R. Results 5.41 In the circuit of the figure 1, at 0 = 10 Krad/s the absolute value of the voltage at R becomes zero. Knowing that at this frequency the value of the absolute value of the voltage at the capacitor is 5 V, calculate: a) Value of L, C and quality factor Q. b) Value of the absolute value of the voltage at the resistance R if the frequency is increased 1%. Results 5.42 With the circuit of the figure 1 we want to tune a signal with frequencies between 1 = 497500 rad/s and 2 = 502500 rad/s. Calculate: a) The values of L1 and L2. b) The instantaneous value of the current, i(t), when the frequency is increased 0,5% with respect the resonant frequency 0. Figure 1. Data: eg(t)=sin(t) V; Rg=100; C=8F. Solutions: a) L1= 2H, L2= 1/2H b) A 5.43 A signal with frequencies between 1 = 192 Krad/s and 2 = 208 Krad/s should be tuned using an ideal capacitor, C, and a real inductor with L =2 mH and r = 8 The signal is provided by an antenna represented by a real voltage generator (Eg, Rg), being Rg = 20 K. For this propose, the two circuits shown in figure 1(a) and (b) are available. a) b) c) With which circuit is it possible to tune the signal: (a), (b) or both ?. For the possible circuit(s) that can be used for tune the signal, calculate the value of the components so that the signal is tuned. For the possible circuit(s), calculate the frequency band that is tuned if the value of Rg where half the value it has now. figure 1. Solutions: a) Circuit b, b) R=10 Kand C= 25/2 nF c) Rg' = Rg/2 = 10 K

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