## RC Filter Analysis

Referring again to Fig.E.1, let's perform an*impedance analysis*of the simple RC lowpass filter.

### Driving Point Impedance

Taking the Laplace transform of both sides of Eq.(E.1) gives### Transfer Function

Since the input and output signals are defined as and , respectively, the*transfer function*of this analog filter is given by, using

*voltage divider rule*,

*RC time constant*, for reasons we will soon see.

### Impulse Response

In the same way that the impulse response of a digital filter is given by the inverse*z*transform of its transfer function, the impulse response of an

*analog*filter is given by the inverse

*Laplace*transform of its transfer function,

*viz.*,

*Heaviside unit step function*

*partial fraction expansion*(see §6.8) and each term in the expansion inverted by inspection as above.

### The Continuous-Time Impulse

The continuous-time impulse response was derived above as the inverse-Laplace transform of the transfer function. In this section, we look at how the*impulse*itself must be defined in the continuous-time case. An

*impulse*in continuous time may be loosely defined as any ``generalized function'' having

*``zero width''*and

*unit area*under it. A simple valid definition is

More generally, an impulse can be defined as the limit of

*any*pulse shape which maintains unit area and approaches zero width at time 0. As a result, the impulse under every definition has the so-called

*sifting property*under integration,

provided is continuous at . This is often taken as the

*defining property*of an impulse, allowing it to be defined in terms of non-vanishing function limits such as

*distribution*or

*generalized function*[13,44]. (It is still commonly called a ``delta function'', however, despite the misnomer.)

### Poles and Zeros

In the simple RC-filter example of §E.4.3, the transfer function is*pole*at , and we can say there is one

*zero at infinity*as well. Since resistors and capacitors always have positive values, the time constant is always non-negative. This means the impulse response is always an exponential

*decay*--never a growth. Since the pole is at , we find that it is

*always in the left-half plane*. This turns out to be the case also for any

*complex*analog one-pole filter. By consideration of the partial fraction expansion of any , it is clear that, for stability of an analog filter,

*all poles must lie in the left half of the complex plane*. This is the analog counterpart of the requirement for digital filters that all poles lie inside the unit circle.

**Next Section:**

RLC Filter Analysis

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Inductors