### Ideal Spring

Figure 7.4 depicts the ideal spring.

From Hooke's law, we have that the applied force is proportional to the
*displacement* of the spring:

*stiffness*of the spring. Taking the Laplace transform gives

*differentiator*. We can say that the ideal spring differentiates the applied force (divided by ) to produce the output velocity.

The *frequency response* of the ideal spring, given the applied force
as input and resulting velocity as output, is

We call the *compression velocity* of the spring. In more
complicated configurations, the compression velocity is defined as the
difference between the velocity of the two spring endpoints, with positive
velocity corresponding to spring compression.

In circuit theory, the element analogous to the spring is the *capacitor*,
characterized by
, or
.
In an equivalent analog circuit, we can use the value . The
inverse of the spring stiffness is sometimes called the
*compliance*
of the spring.

Don't forget that the definition of impedance requires *zero
initial conditions* for elements with ``memory'' (masses and springs).
This means we can only use impedance descriptions for *steady
state* analysis. For a complete analysis of a particular system,
including the transient response, we must go back to full scale
Laplace transform analysis.

**Next Section:**

Series Combination of One-Ports

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Ideal Mass