Integrator-based Filters

We can convert passive RLC filters into integrator-based active filters in a systematic way using signal flow graphs.

Signal Flow Graphs

Signal flow graphs (SFGs) are topological representations of electrical circuits with two components: (1) nodes, and (2) branches. Any network that can be described by a set of linear differential equations can be represented by a SFG. In a SFG, the nodes represent the variables, e.g. voltages or currents, and the branches represent transfer functions between these node variables, specified by branch multiplication factors (BMFs). These BMFs are obtained from Kirchhoff's equations (KVL and KCL). Note that the branches are directed, as indicated by the arrows. Figs. 1-3 shows a few examples of SFG nodes and branches.

Figure 1: A general impedance where $v_{o}=Z\cdot i_{\text{in}}$.	Figure 2: A resistor where $v_{o}=R\cdot i_{\text{in}}$.
Figure 3: An inductor where $i_{o}={\tfrac {1}{sL}}\cdot v_{\text{in}}$.

Useful SFG Properties

We can use the following properties to transform signal flow graphs into forms that can allow us to convert passive RLC filters into its active integrator-based counterparts.

Figure 4: Parallel branches.

Figure 5: Series branches and node absorption.

Parallel Branches

For two or more parallel branches between two nodes in the same direction, as seen in Fig. 4, the effective branch multiplication factor becomes:

$v_{2}=a\cdot v_{1}=b\cdot v_{1}=\left(a+b\right)\cdot v_{1}=c\cdot v_{1}$

(1)

Node Absorption

We can also remove intermediate nodes as shown in Fig. 5:

${\begin{aligned}v_{3}&=a\cdot v_{1}\\v_{2}&=b\cdot v_{3}=\left(a\cdot b\right)\cdot v_{1}=c\cdot v_{1}\end{aligned}}$

(2)

Figure 6: Node and branch scaling.

Figure 7: Loops in signal flow graphs.

Branch Scaling

Node values in Fig. 6 can be scaled by a factor $k$, but with the corresponding changes on the incoming and outgoing branch:

${\begin{aligned}v_{3}=a\cdot v_{1}&\rightarrow k\cdot v_{3}=\left(k\cdot a\right)\cdot v_{1}\\v_{2}=b\cdot v_{3}&\rightarrow v_{2}={\frac {b}{k}}\cdot \left(k\cdot v_{3}\right)\end{aligned}}$

(3)

Loops

For the loop shown in Fig. 7, we get:

${\begin{aligned}v_{3}&=a\cdot v_{2}-b\cdot v_{3}\\&={\frac {a}{1+b}}\cdot v_{2}\end{aligned}}$

(4)

Examples

• A First-Order Low-Pass Filter
• A Second-Order RLC Filter
• A 5th-Order Butterworth Low-Pass Filter

Differential Integrators

Figure 5: A two-input op-amp integrator.

$v_{o}\left(s\right)=-{\frac {1}{s\cdot R_{1}C}}\cdot v_{1}\left(s\right)-{\frac {1}{s\cdot R_{2}C}}\cdot v_{2}\left(s\right)$

(5)

Figure 6: A two-input fully differential op-amp integrator.

$v_{o}^{+}-v_{o}^{-}=-{\frac {1}{s\cdot R_{1}C}}\cdot \left(v_{1}^{+}-v_{1}^{-}\right)-{\frac {1}{s\cdot R_{2}C}}\cdot \left(v_{2}^{+}-v_{2}^{-}\right)$

(6)