# 5.4: RC Band-Pass Filter

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The circuit shown in Figure \(\PageIndex{1}\) consists of a low-pass filter stage to the left of the voltage follower, and a high-pass filter stage to the right. [The functioning of a high-pass filter is illustrated in homework Problem 5.4.2]. As indicated on Figure \(\PageIndex{1}\), the currents in the two stages are independent of each other, by virtue of the buffering due to the voltage follower. Therefore, the simple ODEs for each type of 1^{st} order filter are still valid for the two-stage circuit of Figure \(\PageIndex{1}\), with mid-circuit voltage \(e_{m}(t)\) being the quantity shared by the two stages, as the output from the low-pass stage and the input to the high-pass stage. These ODEs are Equation 5.2.7 for the low-pass filter,

\[\tau_{L} \dot{e}_{m}+e_{m}=e_{i}, \quad \tau_{L}=R_{L} C_{L}\label{eqn:5.16}\]

and the ODE derived in homework Problem 5.4.1 for the high-pass filter,

\[\tau_{H} \dot{e}_{o}+e_{o}=\tau_{H} \dot{e}_{m}, \quad \tau_{H}=R_{H} C_{H}\label{eqn:5.17}\]

An equation such as Equation \(\ref{eqn:5.17}\) is described as having “right-hand-side (RHS) dynamics” because the right-hand-side includes a derivative of the input (\(\dot{e}_{m}\) in this case), rather than just the undifferentiated input itself.

The combination of these two 1^{st} order circuits turns out to be a 2^{nd} order system, and we shall re-visit this subject in Sections 9.10 and 10.4. We consider the RC band-pass filter circuit now because it illustrates (1) application of a voltage follower, and (2) the important physical characteristic of op-amps that is described next.

Note in Figure \(\PageIndex{1}\) that the feedback wire across the op-amp connects directly to the negative input port; this port has essentially infinite resistance, so there cannot be any current in the feedback wire. But Figure \(\PageIndex{1}\) also shows the non-zero, second-stage current \(i_{H}(t)\) downstream of the op-amp; this appears to contradict the claim of zero feedback current in the op-amp, since the graphical representation of the op-amp suggests that the feedback wire is continuous electrically with the downstream circuit. **In fact, the standard graphical representation of an op-amp with negative feedback, such as that in Figure \(\PageIndex{1}\), is oversimplified to the point of being misleading. For an actual op-amp (as opposed to the standard graphical representation), the downstream current is not continuous with the feedback current, but instead, is completely independent.** In fact, the current downstream of an op-amp is determined only by the output voltage of the op-amp and the downstream electrical components, for example, the second-stage capacitor and resistor in Figure \(\PageIndex{1}\). The technical characteristic of an op-amp that permits this independence of currents is very low

*output impedance*(Horowitz and Hill, 1980, pages 25, 92-95, and 105). For another example of the independence of feedback and downstream currents, see homework Problem 5.10.