10.3 STRAIGHT LINE APPROXIMATIONS
An approximate technique for constructing a gain Bode plot is shown in Figure 10.10. This method involves looking at the transfer function and reducing it to roots in the numerator and denominator. Once in that form, a straight line approximation for each term can be drawn on the graph. An initial gain is also calculated to shift the results up or down. When done, the straight line segments are added to produce a more complex straight line curve. A smooth curve is then drawn over top of this curve.
Figure 10.10 The method for Bode graph straight line gain approximation
Figure 10.11 Why the straight line method works
Figure 10.12 Why the straight line method works (cont'd)
Figure 10.13 Drill problem: Slope of second order transfer functions.
An example of the straight line plotting technique is shown in Figure 10.14. In this example the transfer function is first put into a root form. In total there are three roots, 1, 10 and 100 rad/sec. The single root in the numerator will cause the curve to start upward with a slope of 20dB/dec after 1rad/sec. The two roots will cause two curves downwards at -20dB/dec starting at 10 and 100 rad/sec. The initial gain of the transfer function is also calculated, and converted to decibels. The frequency axis is rad/sec by default, but if Hz are used then it is necessary to convert the values.
Figure 10.14 An approximate gain plot example
The example is continued in Figure 10.15 where the straight line segments are added to produce a combined straight line curve.
Figure 10.15 An approximate gain plot example (continued)
Finally a smooth curve is fitted to the straight line approximation. When drawing the curve imagine that there are rubber bands at the corners that pull slightly and smooth out. For a simple first-order term there is a 3dB gap between the sharp corner and the function. Higher order functions will be discussed later.
Figure 10.16 An approximate gain plot example (continued)
The process for constructing phase plots is similar to that of gain plots, as seen in Figure 10.18. The transfer function is put into root form, and then straight line phase shifts are drawn for each of the terms. Each term in the numerator will cause a positive shift of 90 degrees, while terms in the denominator cause negative shifts of 90 degrees. The phase shift occurs over two decades, meaning that for a center frequency of 100, the shift would start at 10 and end at 1000. If there are any lone 'D' terms on the top or bottom, they will each shift the initial value by 90 degrees, otherwise the phase should start at 0degrees.
Figure 10.17 The method for Bode graph straight line gain approximation
The previous example started in Figure 10.14 is continued in Figure 10.18 to develop a phase plot using the approximate technique. There are three roots for the transfer function. None of these are zero, so the phase plot starts at zero degrees. The root in the numerator causes a shift of positive 90 deg, starting one decade before 1rad/sec and ending one decade later. The two roots in the denominator cause a shift downward.
Figure 10.18 An approximate phase plot example
The straight line segments for the phase plot are added in Figure 10.19 to produce a straight line approximation of the final plot. A smooth line approximation is drawn using the straight line as a guide. Again, the concept of a rubber band will smooth the curve.
Figure 10.19 An approximate phase plot example (continued)
10.3.1 Second Order Underdamped Terms
The previous example used a transfer function with real roots. In a second-order system with double real roots (overdamped) the curve can be drawn with two overlapping straight line approximations. If the roots for the transfer function are complex (underdamped the corner frequencies will become peaked. This can be handled by determining the damping factor and natural frequency as shown in Figure 10.20. The peak will occur at the damped frequency. The peaking effect will become more pronounced as the damping factor goes from 0.707 to 0 where the peak will be infinite.
Figure 10.20 Resonant peaks
The approximate techniques do decrease the accuracy of the final solution, but they can be calculated quickly. In addition these curves provide an understanding of the system that makes design easier. For example, a designer will often describe a system with a Bode plot, and then convert this to a desired transfer function.
Figure 10.21 Drill problem: Draw the straight line approximation
10.3.2 Lone Ds on the Top or Bottom
- cancel the Ds on the top and bottom if possible
- consider the Ds to have corner frequencies at zero. This means that Ds on the top start at infinity and come down. Ds on the bottom start at negative infinity and come up.
- The Ds are on the top, and they are second order so the phase angle starts at 180 degrees, and the gain starts going upwards at 40dB/dec.
- eventually the squared term on the bottom begins to work and the top and bottom cancel out to give a gain of 0dB.
- with these types of problems the top and bottom of the transfer function might go to zero or infinity. Normally we would say this is underdefined. In these cases, L'Hospital's rule may be used.
- in this case the graph is drawn from the right hand side, working towards the left.
- The example below shows
Figure 10.22 A System Gain Starting at Negative Infinity
- The problem solution is similar when the lone Ds are on the bottom.
- All of the examples presented up to now have started or stopped and could be drawn from the left or right. This will not always be the case as shown in the example below.
Figure 10.23 Calculating a point to set a position
