The low-pass filter in Fig. 12.70 is implemented using a 25-pF capacitor and OTAs with a $g_{m}-I_{A B C}$ sensitivity of 20, a maximum $g_{m}$ of 1 mS, and 3 decades of range. Find the required bias currents for the filter transfer function:

$\frac{\mathbf{V}_{\mathbf{o}}}{\mathbf{V}_{\text {in }}}=\frac{4}{\frac{j \omega}{2 \pi\left(10^{5}\right)}+1}$ 12.87

Question

The low-pass filter in Fig. 12.70 is implemented using a 25-pF capacitor and OTAs with a $g_{m}-I_{A B C}$ sensitivity of 20, a maximum $g_{m}$ of 1 mS, and 3 decades of range. Find the required bias currents for the filter transfer function:

$\frac{\mathbf{V}_{\mathbf{o}}}{\mathbf{V}_{\text {in }}}=\frac{4}{\frac{j \omega}{2 \pi\left(10^{5}\right)}+1}$ 12.87

[latex]\frac{\mathbf{V}_{\mathbf{o}}}{\mathbf{V}_{ ext {in }}}=\frac{4}{\frac{j \omega}{2 \pi\left(10^{5} ight)}+1}[/latex] 12.87

Accepted Answer

Comparing Eq. (12.86) ([latex]\frac{\mathbf{V}_{\mathrm{o}}}{\mathbf{V}_{\mathrm{it}}}=\frac{g_{m 1} / g_{m 2}}{\frac{j \omega C}{g_{m 2}}+1}[/latex]) to the desired function, [latex]g_{m 2} / \mathrm{C}=(2 \pi) 10^{5} .[/latex] For [latex]\mathrm{C}=25 \mathrm{pF}[/latex], [latex]g_{m 2}=15.7 \mu \mathrm{S}[/latex]. Since [latex]g_{m}=20 I_{A B C}[/latex], the bias current for OTA 2 is [latex]I_{A B C 2}=0.785 \mu[/latex] A. Finally, [latex]g_{m 1} / g_{m 2}=4[/latex] yields [latex]I_{A B C 1}=3.14 \mu \mathrm{A}[/latex].

Question 12.33: The low-pass filter in Fig. 12.70 is implemented using a 25-...

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