A three-stage staggered tuned amplifier having a Butterworth-type frequency characteristic will be designed. The center frequency of the frequency characteristic is 1 GHz.

The maximum possible effective Q value for the resonance circuits – without any Q -enhancement feature – is given as 20.
(a). What is the realizable bandwidth?
(b). Calculate the tuning frequencies of the stages.
(c). Calculate the appropriate effective Q values of the resonance circuits.

Question

A three-stage staggered tuned amplifier having a Butterworth-type frequency characteristic will be designed. The center frequency of the frequency characteristic is 1 GHz.

The maximum possible effective Q value for the resonance circuits – without any Q -enhancement feature – is given as 20.
(a). What is the realizable bandwidth?
(b). Calculate the tuning frequencies of the stages.
(c). Calculate the appropriate effective Q values of the resonance circuits.

Accepted Answer

The pole–zero diagram of the voltage gain function of the amplifier is shown in Fig. 4.18(b).

From the geometry of the figure it can be easily seen that the magnitude of the real part of the center pole, [latex]p_{12}[/latex], must be equal to the half of the bandwidth:

[latex]\left|\sigma_2 ight|=\left|\frac{\omega_0}{2Q_2} ight|=\Delta \omega \Rightarrow Q_2=\frac{\omega_0}{2\Delta \omega}=\frac{f_0}{2\Delta f} [/latex]

Similarly, the real parts of [latex]p_{11}[/latex] and [latex]p_{13}[/latex] must be equal in magnitude to Δω/2:

[latex]\left|\sigma_1 ight|=\left|\frac{\omega_0}{2Q_1} ight|=\frac{\Delta \omega}{2} \Rightarrow Q_1=\frac{\omega_{01}}{\Delta \omega}=\frac{f_{01}}{\Delta f} [/latex]

[latex]\left|\sigma_3 ight|=\left|\frac{\omega_0}{2Q_3} ight|=\frac{\Delta \omega}{2} \Rightarrow Q_3=\frac{\omega_{03}}{\Delta \omega}=\frac{f_{03}}{\Delta f} [/latex]

[latex]\omega_{01}[/latex] and [latex]\omega_{03}[/latex] can be calculated from the geometry:

[latex]\omega_{01}=\omega_0-\Delta \omega \cos{(\pi/6)}[/latex]

[latex]\omega_{03}=\omega_0+\Delta \omega \cos{(\pi/6)}[/latex]

Since [latex]\omega_{03}[/latex] is the highest among the three tuning frequencies, the quality factor corresponding to this resonance circuit is the highest and must be equal to the possible maximum Q value, which is 20:

[latex]Q_3=\frac{\omega_{03}}{\Delta \omega}=\frac{\omega_0 + \Delta \omega \cos{(\pi/6)}}{\Delta \omega}=\frac{\omega_0}{\Delta \omega}+\cos{(\pi/6)}=\frac{f_0}{\Delta f}+0.866=20[/latex]

which yields Δf = 52.26 MHz (Δω = 328.36 rad/s).

Now the tuning frequencies and the quality factors can be calculated as

[latex]f_{01}=f_0-\Delta f \cos{(\pi/6)}=1000-52.26 imes 0.866=954.74[/latex][MHz]

[latex]Q_1=\frac{f_{01}}{2\Delta f}=\frac{954.74}{52.26}=18.27[/latex]

[latex]f_{02}=f_0=1000\left[MHz ight] Q_2=\frac{f_0}{2\Delta f}=\frac{1000}{2 imes 52.26}=9.57[/latex]

[latex]f_{03}=f_0+\Delta f \cos{(\pi/6)}=1000+52.26 imes 0.866=1045.26[/latex][MHz]

[latex]Q_3=\frac{f_{03}}{\Delta f}=\frac{1045.26}{52.26}=20[/latex]

Note that since the quality factors are not high and the relative bandwidth [latex](2\Delta f/f_0)[/latex] is not small, [latex]Q_1[/latex] and [latex]Q_3[/latex] are not equal.
The normalized frequency characteristic of the circuit calculated and plotted with
MatLab is given in Fig. 4.19.

Question 4.4: A three-stage staggered tuned amplifier having a Butterworth...

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