Active Filters

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Passive RLC filters are simple and easy to design and use. However, can we implement them on-chip? Let us look at a simple example to give us a bit more insight regarding this question.

Example: A passive band-pass filter

Consider the filter shown in Fig. 1.

We can write the transfer function as:

 

 

 

 

(1)

If we let and , then we can rewrite our expression for as:

 

 

 

 

(2)

Notice that the transfer function has two zeros, , and two poles located at:

 

 

 

 

(3)

We get complex conjugate poles if or when , or equivalently, when . If the band-pass filter has , , and :

 

 

 

 

(4)

 

 

 

 

(5)

Let us now consider a lossy inductor with . The loss can then be modeled by the series resistance, , as shown in Fig. 2, with:

 

 

 

 

(6)

We can convert the series RL circuit to its parallel circuit equivalent in Fig. 3 for frequencies around by first writing out the admittance of the series RL circuit as:

 

 

 

 

(7)

Thus, we get:

 

 

 

 

(8)

 

 

 

 

(9)

For , , we can then redraw our band-pass filter with the lossy inductor model, as shown in Fig. 4. Thus, the new transfer function is then:

 

 

 

 

(10)

Note that remains approximately the same, the overall quality factor, , becomes: