Lag angle

The lag angle between stress and strain is defined by the dissipation factor or tan 8 given by... 

The velocities in a real impeller do not follow the ideal Euler impeller pattern, and a degree of slip occurs. The angles of flow and forces deviate from the theoretical values as shown in Figure 8-10 by a lag angle. The slip factor is in fact as the ratio of the measured absolute velocity to the theoretical Euler absolute velocity at the tip diameter of the vanes ... 

This solution holds up to = ui at which point the lag angle 8 reaches ir/4 and the magnetic torque T reaches its first maximum, Eq. (12). The system is then unstable because a small increase in 8 will reduce the torque. It is then favourable to reduce the torque by rapidly in-... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

Fig. 17. Viscoelastic material stress ( and strain (---------) ampHtudes vs time where 5 is the phase angle that defines the lag of the strain behind the...

D = amplitude of the steady-state oseillation 6 = phase angle by whieh the motion lags the impressed foree... 

Inertia foree + Damping foree + Spring foree + Impressed foree = 0 From the previous equation, the displaeement lags the impressed foree by the phase angle 6, and the spring foree aets opposite in direetion to... 

Nacheifer, m. emulation, nacheilen, v.i. lag, retard follow, trail. Nacheilwinkel, m. angle of lag. nacheinander, adv. one after another, successively. 

For the impedance of the resistance and inductance in series, the current will lag the potential phase angle by... 

The rate of heat conduction is further complicated by the effect of sunshine onto the outside. Solar radiation reaches the earth s surface at a maximum intensity of about 0.9 kW/ m. The amount of this absorbed by a plane surface will depend on the absorption coefficient and the angle at which the radiation strikes. The angle of the sun s rays to a surface (see Figure 26.1) is always changing, so this must be estimated on an hour-to-hour basis. Various methods of reaching an estimate of heat flow are used, and the sol-air temperature (see CIBSE Guide, A5) provides a simplification of the factors involved. This, also, is subject to time lag as the heat passes through the surface. 

Solar radiation through windows has no time lag and must be estimated by finite elements (i.e. on an hour-to-hour basis), using calculated or published data for angles of incidence and taking into account the type of window glass (see Table 26.1). 

The ratio is the tangent of a phase angle 8 by which the strain lags behind the stress. 

If the material is anelastic, the stress and the strain will not coincide. The strain will lag behind by an amount which is determined by the phase angle, ( ). Thus ... 

A comparison of Eqs. (8-9) with (8-6) shows that the magnitude and the phase angle of Gp(jco) are exactly the same as the amplitude and phase lag of the normalized "large time" time domain solution. 

With the phase lag, we may see why a first order function is also called a first order lag. On the magnitude log-log plot, the high frequency asymptote has a slope of -1. This asymptote also intersects the horizontal Kp line at co = l/xp. On the phase angle plot, the high frequency asymptote is the -90° line. On the polar plot, the infinity frequency limit is represented by the origin as the Gp(jco) locus approaches it from the -90° angle. 

The magnitude and phase angle plots are sort of "upside down" versions of first order lag, with the phase angle increasing from 0° to 90° in the high frequency asymptote. The polar plot, on the other hand, is entirely different. The real part of G(jco) is always 1 and not dependent on frequency. 

However, the result is immediately obvious if we consider the function as the product of a first order lag and an integrator. Combining the results from Examples 8.2 and 8.7, the magnitude and phase angle are... 

The shape of the magnitude plot resembles that of a PI controller, but with an upper limit on the low frequency asymptote. We can infer that the phase-lag compensator could be more stabilizing than a PI controller with very slow systems.1 The notch-shaped phase angle plot of the phase-lag compensator is quite different from that of a PI controller. The phase lag starts at 0° versus -90°... 

---