Parallel Resonant Circuit
Find the Land C values for a parallel resonant circuit that has R = 10 kΩ, f₀ = 1MHz, and B = 100 kHz. If I = 10^-3∠0, draw the phasor diagram showing the currents through each of the elements in the circuit at resonance.

Question

Accepted Answer

First, we compute the quality factor of the circuit. Rearranging Equation 6.46 and substituting values, we have

[latex] B=\frac{f_0}{Q_p} [/latex] (6.46)

[latex]Q_p=\frac{f_0}{B}=\frac{10^6}{10^5}=10[/latex]

Solving Equation 6.41 for the inductance and substituting values, we get

[latex]Q_p=\frac{R}{2\pi f_0 L}[/latex] (6.41)

[latex]L=\frac{R}{2\pi f_0 Q_p}=\frac{10^4}{2\pi imes 10^6 imes 10 }=159.2~\mu \mathrm{H}[/latex]

Similarly, using Equation 6.42, we find that

[latex]Q_p=2\pi f_0 CR[/latex] (6.42)

[latex]C=\frac{Q_p}{2 \pi f_0 R}=\frac{10}{2\pi imes 10^6 imes 10^4}=159.2\mathrm{~pF}[/latex]

At resonance, the voltage is given by

[latex]\mathrm{V_{out}}=\mathrm{I}R=(10^{-3}\angle 0^\circ) imes 10^4 =10\angle 0^\circ[/latex]

and the currents are given by

[latex]\mathrm{I}_R=\frac{\mathrm{V_{out}}}{R}=\frac{10\angle 0^\circ}{10^4}=10^{-3}\angle 0^\circ[/latex]

[latex] \mathrm{I}_L=\frac{\mathrm{V_{out}}}{j2\pi f_0 L}=\frac{10\angle 0^\circ}{j10^3}=10^{-2}\angle -90^\circ [/latex]

[latex] \mathrm{I}_C=\frac{\mathrm{V_{out}}}{-j/2\pi f_0 C}=\frac{10\angle 0^\circ}{-10^3}=10^{-2}\angle 90^\circ [/latex]

The phasor diagram is shown in Figure 6.31. Notice that the currents through the inductance and capacitance are larger in magnitude than the applied source current. However, since I_C and I_L are out of phase, they cancel.

Question 6.6: Parallel Resonant Circuit Find the Land C values for a paral...

Related Answered Questions

Step Response of a First-Order Digital Lowpass Filter Compute and plot the input and output samples for n = 0 to 20, given a = 0.9. The input is a step function defined by x(n) = 0 for n < 0 = 1 for n ≥ 0 Assume that y(n) = 0 for n < 0. ...

Verified Answer:

Bode Plot Using the MATLAB Symbolic Toolbox Obtain a magnitude bode plot of the transfer function H(f ) = Vout/Vin for the circuit of Figure 6.43 with frequency ranging from 100 kHz to 10MHz. Manually check the plotted values for high and low frequencies. ...

Verified Answer:

Using the Transfer Function to Determine the Output The transfer function H(f ) of a filter is shown in Figure 6.4. [Notice that the magnitude |H(f )| and phase ∠H(f ) are shown separately in the figure.] If the input signal is given by vin(t) = 2 cos(2000πt + 40° ) find an expression (as a ...

Verified Answer:

Using the Transfer Function with Several Input Components Suppose that the input signal for the filter of Figure 6.4 is given by vin(t) = 3 + 2 cos(2000πt) + cos(4000πt – 70°) Find an expression for the output signal. ...

Verified Answer:

Calculation of RC Lowpass Output Suppose that an input signal given by vin(t) = 5 cos(20πt) + 5 cos(200πt) + 5 cos(2000πt) is applied to the lowpass RC filter shown in Figure 6.9. Find an expression for the output signal. ...

Verified Answer:

Verified Answer:

Filter Design Suppose that we need a filter that passes components higher in frequency than 1 kHz and rejects components lower than 1 kHz. Select a suitable second-order circuit configuration, choose L = 50 mH, and specify the values required for the other components. ...

Verified Answer:

Verified Answer:

Determination of the Break Frequency for a Highpass Filter Suppose that we want a first-order highpass filter that has a transfer-function magnitude of -30 dB at f = 60 Hz. Find the break frequency for this filter. ...

Verified Answer: