Page 52 - Mechanical Engineers' Handbook (Volume 2)

3 Energy, Power, Impedance 41

Clearly, the negative derivative of the voltage (V ) with respect to the current (i )isgiven
o
o
by
R R RR RR
dV o
Z 12 23 13 (16)
di o o R R 3
1
In the second method, the voltage source (V ) can be set to zero, a short circuit, without
s
affecting the impedances, and the current source (I ) can be removed to yield the circuit on
s
the right in Fig. 2. Then the impedances need only be combined as R in series with the
2
parallel pair R and R . Thus,
1
3
RR RR RR RR
Z R 13 12 23 13 as before (17)
o 2
R R 3 R R 3
1
1
3.5 Transforming or Gyrating Impedances
Ideal transformers, transducers, and gyrators play an important part in dynamic systems and
an equally important part in obtaining optimal performance from source–load combinations.
They share several vital features: All are two-port (or four-terminal) devices, all are ener-
getically conservative, and all are considered lumped elements. Each of the many types
requires two equations for its description, always of the same form. Table 2 lists many of
the more common linear two-ports, and Fig. 3 illustrates them.
All of these transducing devices alter the effort–flow relationships of elements connected
at their far end. Consider Fig. 4, which shows a simple model of a front wheel of an
automobile suspension. Because the spring and damper are mounted inboard of the wheel,
their effectiveness is reduced by the mechanical disadvantage of the suspension arm. What
is the impedance of the spring and shock absorber at ( ) as viewed from the wheel at ( ) ?
1 2
Since the spring and shock absorber share a common velocity (both ends share nodes
or points of common velocity), their impedances add to give the impedance of the pair at
their point of attachment, ( ) , to the lever:
1
k bs k
b (18)
(Z ) total Z spring Z damper
1
s s
At the ( ) end of the lever, the force, from Table 2, is F (L /L )F , but from the definition
2
2
1
2
1
of Z for linear elements, F (Z ) v . So we get the following:
1 total 1
1
bs k
L 1
v
F 1 (19)
2
L 2 s
The second equation for the lever is v (L /L )v , and substituting this into Eq. (19) yields
2
2
1
1
L 1 bs k L 1 L 1 2 bs k
v
v
2
2
F 2 (20)
L
s
s
L
L
2
2
2
(Z ) 2
L
1

2 total
1 total
L (Z ) (21)
2
Thus, the impedance of the suspension, observed from the wheel end of the lever arm, is
multiplied by the square of the lever ratio. This general result applies to all transduction
elements.