The transfer function of a WWSSN-LP seismograph

The long-period seismographs of the now obsolete WWSSN (worldwide standardized seismograph network) consisted of a long-period electrodynamic seismometer normally tuned to a free period of 15 sec, and a long-period mirror-galvanometer with a free period around 90 sec. (In order to avoid confusion with the frequency variable $s=j \omega$ of the Laplace transformation, we use the nonstandard abbreviation ``sec'' for seconds in the present section.) The WWSSN seismograms were recorded on photographic paper rotating on a drum. We will now derive three equivalent forms of the transfer function for this system. In our example the damping constants are chosen as 0.6 for the seismometer and 0.9 for the galvanometer. Our treatment is slightly simplified. Actually the free periods and damping constants are modified by coupling the seismometer and the galvanometer together; the above values are understood as being the modified ones.

As will be shown in paragraph 2.9 (Eq. 31), the transfer function of an electromagnetic seismometer (input: displacement, output: voltage) is

$$H_s(s) = E s^3 / (s^2 + 2 s \omega_s h_s + \omega_s^2)$$ (16)

where $\omega_s = 2 \pi/T_s$ is the angular eigenfrequency and h_s the numerical damping. The factor E is the generator constant of the electromagnetic transducer, for which we assume a value of 200 Vsec/m. The galvanometer is a second-order low-pass filter and has the transfer function

$$H_g(s) = \gamma \omega_g^2 / (s^2 + 2 s \omega_g h_g + \omega_g^2)$$ (17)

Here $\gamma$is the responsivity (in meters per volt) of the galvanometer with the given coupling network and optical path. We use a value of 393.5 m/V, which gives the desired overall magnification. The overall transfer function H_d of the seismograph is in our simplified treatment obtained as the product of the factors (16) and (17):

$$H_d(s)=\frac{C s^3}{(s^2 + 2 s \omega_s h_s + \omega_s^2)(s^2 + 2 s \omega_g h_g + \omega_g^2)}$$ (18)

The numerical values of the constants are $C=E \gamma \omega_g^2=383.6\/{\rm /sec}$, $2 \omega_s h_s = 0.5027\/{\rm /sec}$, $\omega_s^2 = 0.1755\/{\rm /sec^2}$, $2 \omega_g h_g = 0.1257\/{\rm /sec}$, and $\omega_g^2=0.00487\/{\rm /sec^2}$.

As the input and output signals are displacements, the absolute value $\vert H_d(s) \vert$ of the transfer function is simply the frequency-dependent magnification of the seismograph. The gain factor C has the physical dimension ${\rm sec}^{-1}$, so H_d(s) is in fact a dimensionless quantity. C itself is however not the magnification of the seismograph! To obtain the magnification at the angular frequency $\omega$, we have to evaluate $M(\omega)=\vert H_d(j\omega) \vert$:

$$M(\omega) = \frac {C \omega^3} {\sqrt{(\omega_s^2 - \omega^2)... ...sqrt{(\omega_g^2 - \omega^2)^2 + 4 \omega^2 \omega_g^2 h_g^2}}$$ (19)

Eq. (18) is a factorized form of the transfer function in which we still recognize the subunits of the system. We may of course insert the numerical constants and expand the denominator into a fourth-order polynomial:

$$H_d(s)= 383.6 s^3 / (s^4 + 0.6283\ s^3 + 0.2435\ s^2 + 0.0245\ s + 0.000855)$$ (20)

but the only advantage of this form would be its shortness.

The poles and zeros of the transfer function are most easily determined from Eq. (18). We read immediately that a triple zero is present at s=0. Each factor $s^2 + 2 s \omega_0 h + \omega_0^2$ in the denominator has the zeros
$$\begin{align*}s_0 &= \omega_0 (-h \pm j \sqrt{1-h^2}) \qquad \textrm{ for } h < ... ... &= \omega_0 (-h \pm \ \sqrt{h^2-1}) \qquad \textrm{ for } h \geq 1 \end{align*}$$
so the poles of H(s) in the complex s plane are (Fig. 2):

$$\begin{align*}s_1 &=\omega_s (-h_s+j \sqrt{1-h_s^2})&=-0.2513+0.3351 j \quad &[{... ...g (-h_g-j \sqrt{1-h_g^2})&=-0.0628-0.0304 j \quad &[{\rm sec}^{-1}] \end{align*}$$

  

Figure 2: Position of the poles of the WWSSN-LP system in the complex s plane.

In order to reconstruct H(s) from its poles and zeros and the gain factor, we write

$$H_d(s)= \frac{C s^3}{(s-s_1)(s-s_2)(s-s_3)(s-s_4)}$$ (21)

It is now convenient to pairwise expand the factors of the denominator into second-order polynomials:

$$H_d(s)=\frac{C s^3}{(s^2-s(s_1+s_2)+s_1 s_2)(s^2-s(s_3+s_4)+s_3 s_4)}$$ (22)

This makes all coefficients real because s₂=s₁^* and s₄=s₃^*. Since $s_1+s_2=2 \omega_s h_s$, $s_1 s_2 = \omega_s^2$, $s_3+s_4=2 \omega_g h_g$, and $s_3 s_4 = \omega_g^2$, Eq. (22) is in fact the same as Eq. (18). We may of course also reconstruct H_d(s) from the numerical values of the poles and zeros. Dropping the physical units, we obtain

$$H_d(s) = \frac{383.6\/ s^3} {(s^2 + 0.5027 s + 0.1755)(s^2 + 0.1257 s + 0.00487)}$$ (23)

as from Eq. (18).

  

Figure 3: Amplitude response of a WWSSN-LP 15-90 electrodynamic seismograph, with asymptotes (Bode plot).

Fig. 3 shows the corresponding amplitude response of the WWSSN seismograph as a function of frequency. The maximum magnification is 750 near a period of 15 sec. The slopes of the asymptotes (thin lines) are at each frequency determined by the dominant powers of s in the numerator and denominator of the transfer function. Generally, the low-frequency asymptote has the slope m (the number of zeros, here =3) and the high-frequency asymptote has the slope m-n (where n is the number of poles, here =4). What happens in between depends on the position of the poles in the complex s plane. Generally, a pair of poles s₁,s₂ corresponds to a second-order corner of the amplitude response with $\omega_0^2=s_1 s_2$ and $2\omega_0 h=-s_1-s_2$. A single pole at s₀ is associated with a first-order corner with $\omega_0=s_0$. The poles and zeros do however not indicate whether the respective subsystem is a low-pass, high-pass, or band-pass filter. In fact this does not matter; the corners bend the amplitude response downward in each case. In the WWSSN-LP system, the low-frequency corner at 90 sec corresponding to the pole pair s₁, s₂ reduces the slope of the amplitude response from 3 to 1, and the corner at 15 sec corresponding to the pole pair s₃, s₄ reduces it further from 1 to -1.

Looking at the transfer function H_s (Eq. 16) of the electromagnetic seismometer alone, we see that the low-frequency asymptote has the slope 3 because of the triple zero in the numerator. The pole pair s₁, s₂ corresponds to a second-order corner in the amplitude response at $\omega_s$ which reduces the slope to 1. The resulting response is shown in a normalized form in Fig. 6, upper right panel. As stated in section 2.6 under point 3, this case of n<m can only be an approximation in a limited bandwidth. In modern seismograph systems, the upper limit of the bandwidth is usually set by an analog or digital cutoff (anti-alias) filter.

As we will see in section 2.9, the classification of a subsystem as a high-pass, band-pass or low-pass filter may be a matter of definition rather than hardware; it depends on the type of ground motion (displacement, velocity, or acceleration) to which it relates. We also notice that interchanging $\omega_s, h_s$ with $\omega_g, h_g$ will change the gain factor C in the numerator of Eq. (19) from $E \gamma \omega_g^2$ to $E \gamma \omega_s^2$ and thus the gain, but will leave the denominator and therefore the shape of the response unchanged. While the transfer function is insensitive to arbitrary factorization, the hardware may be quite sensitive, and certain engineering rules must be observed when a given transfer function is realized in hardware. For example, it would have been difficult to realise a WWSSN seismograph with a 15 sec galvanometer and a 90 sec seismometer; the restoring force of a Lacoste-type suspension cannot be made small enough without becoming unstable.

  

Figure 4:Impulse responses of the seismometer, the galvanometer, and the combination of both. The input is an impulse of acceleration. The length of each trace is 2 minutes.

Fig. 4 illustrates the impulse responses of the seismometer, the galvanometer, and the whole WWSSN-LP system. We have chosen a pulse of acceleration (or of calibration current) as the input, so the figure does not refer to the transfer function H_d of Eq. (18) but to H_a(s)=s^-2H_d(s). H_a has a single zero at s=0 but the same poles as H_d. The pulse was slightly broadened for a better graphical display (the $\delta$ pulse is not plottable). The output signal (d) is the convolution of the input signal to the galvanometer (b) with the impulse response (c) of the galvanometer. (b) itself is the convolution of the broadband impulse (a) with the impulse response of the seismometer. (b) is then nearly the impulse response of the seismometer, and (d) is nearly the impulse response of the seismograph.