by Les Glatt PhD, ATP/CFI-AI, AGI/IG

As part of the ACS requirements for an Instrument rating, the Pilot must demonstrate an understanding and the required proficiency to fly a holding pattern. There are many training methods to provide the Pilot with the knowledge of how to visualize the holding pattern and enter the holding pattern. One of the required skills, is IR.III.B.S5, which states, “Uses proper wind correction procedures to maintain the desired holding pattern, and to arrive at the holding fix as close as possible to a specified time”. The AIM provides some guidelines for estimating the outbound wind correction angle (OWCA), but there are no guidelines as under what conditions this rule-of-thumb should apply. In addition, there are no guidelines in the AIM for estimating the outbound time other than to fly a one-minute or one-minute and 30 second outbound leg for the initial circuit. The technique utilized to converge to the “Holding Pattern” solution is based on a bracketing technique, which in reality is a “Trial and Error” method. Here we fly a specified outbound OWCA and outbound time and based on the inbound time and whether the aircraft has undershot/overshot the centerline of the inbound course, the Pilot will fly the next circuit with an updated outbound time and OWCA. The process continues until the Pilot converges to the correct “Holding Pattern” solution. Depending on the initial guess for the outbound time and OWCA, the Pilot may require a significant number of circuits before converging to the correct holding pattern. This process of converging to the proper holding pattern can impose a considerable load on the Pilot, especially when attempting to troubleshoot a problem, or while reviewing the approach plate prior to executing the approach.

In order to expel many of the “Myths and Misconceptions” of timing and wind correction in the holding pattern, we derive the exact solution of the “Holding Pattern” problem. This solution is completely analytic and does not use graphical techniques to solve the problem, as utilized in many of the holding pattern calculators previously developed. It provides the following information to the IFR Pilot: (a) Inbound wind correction angle (IWCA), (b) Outbound heading or outbound wind correction angle (OWCA), and (c) outbound time. The solution of the “Holding Pattern” problem is shown to be a function of the following parameters:

(a)
Windspeed ratio, , i.e. the ratio of the
windspeed to the aircraft TAS (V_{TAS})

(b) Wind angle, (degrees) relative to the inbound course to the holding fix

(c) Aircraft rate of turn, (radians/sec)

(d) Required inbound time to the fix (i.e. one-minute or one-minute and 30 seconds)

Note, although the shape of the holding pattern is a function of the above four parameters, the extent of the holding pattern (i.e. what the Radar Controller observes on the radar scope) is also a function of the parameter , since the coordinates of the holding pattern are proportional to this parameter.

The exact solution of the “Holding Pattern” problem allows the Pilot to not only have a better understanding of how to correct both the outbound heading and outbound time, but to be able to converge to the “Holding Pattern” solution in a minimum number of circuits. In addition, the exact solution provides a number of important properties about the holding pattern that have never previously been discussed in the open literature. This information can affect the way we train IFR Pilots in the future. Below are some of the key findings of this Treatise.

(1) There are two important advantages of starting the outbound time when the aircraft has turned to the outbound heading, rather than the abeam point. The first is obvious in that the Pilot does not need to locate the abeam point, and the second is that the outbound time measured from the time the aircraft reaches the outbound heading will be the same, regardless of whether the wind is blowing from either , i.e. from the holding side or the non-holding side. If the Pilot starts the time at the abeam point, the outbound time will be different, depending on whether the wind is coming from the holding or non-holding side.

(2) A completely different type of holding pattern occurs when holding on a strong headwind component. In this type of holding pattern, it is impossible to achieve the one-minute or one-minute and 30 second inbound time unless the aircraft turns less than 90 degrees outbound from the inbound course. We define this holding pattern as a Type-2 holding pattern, as compared to the normal Type-1 holding pattern that we observe in IFR training manuals. We have derived the boundary of this type of holding pattern in windspeed-wind angle space (i.e. space). The boundary line is shown to be a function of the turn rate and the required inbound time to the holding fix. In the case of the one-minute inbound time, the Type-2 holding pattern will occur whenever the windspeed ratio becomes greater than while holding on a direct headwind. The value of increase to 0.38 at , and 0.44 at degrees. The behavior is similar for the one-minute and 30 second inbound time, except at , the value of and increases with in a similar fashion as the one-minute inbound leg case. We show that the Type-2 holding pattern can be extremely difficult to converge to the correct inbound time due to the required outbound turn being less than 90 degrees to the inbound course. In fact, when the outbound turn is between 45 and 90 degrees from the inbound course, the inbound the time is controlled by the outbound heading, whereas, the overshoot/undershoot of the inbound course is controlled by the outbound time. This is exactly opposite to the “Bracketing Method” used for Type-1 holding patterns. Thus, by flying the holding pattern with a windspeed ratio less than , the IFR Pilot can always avoid having to hold with a Type-2 holding pattern. In fact, it is recommended to fly the holding pattern with a value of , in order to have a sufficient amount of outbound time before having to turn to re-intercept the inbound course.

(3) The “Coupling Effect”: The concept of the “Coupling Effect”, which states that “Every Pilot induced change in the outbound time or OWCA causes changes in both the inbound time and the undershooting/overshooting of the inbound course to the fix”. This concept is extremely important in converging to the correct “Holding Pattern” solution using a minimal number of circuits. Using the exact solution of the “Holding Pattern” problem, we developed a “Smart-Convergence” algorithm, to converge to the correct holding pattern in a minimum number of circuits. This algorithm is compared to the current “Bracketing Method” and shows there are significant deficiencies in the “Bracketing Method” that requires additional circuits to converge to the correct holding pattern.

(4) We have developed curves of the exact solution for the standard Type-1 holding patterns for windspeed ratios up to 0.3, which show the outbound time and the ratio of the OWCA to the IWCA (i.e. the M-Factor) as a function of windspeed ratio and relative wind angle. These solutions show that using the AIM recommended M-Factor of 3 for the OWCA holds under a limited set of conditions. These conditions are: (a) For windspeed ratios up to 0.3, the relative wind angle is limited to the range degrees, and (b) For degrees, when the windspeed ratio is less than 0.05. For aircraft holding at a TAS of 100 knots, this would correspond to a wind of less than 5 knots. We identify this as one of the root causes of requiring additional circuits to converge to the correct holding pattern, since the initial circuit can be considerably different than the “Holding Pattern” solution. We have also shown that the bound on the M-Factor is given by

which debunks many of the articles in the open literature which claim that the M-Factor is between 2 and 3.

(5) We have developed some important techniques that can be used when flying Type-1 holding patterns, in order to converge to the “Holding Pattern” solution with a minimal number of circuits. These techniques are shown using actual tracks of the holding pattern while attempting to converge to the “Holding Pattern” solution. These curves are extremely helpful to the CFI-I when using the Simulator to introduce the IFR Student to holding patterns in the presence of a wind. In addition, just eyeballing the outbound time on this chart was shown to reduce the number of required circuits by 40 percent in order to converge to the correct holding pattern.

(6) We also discuss important IFR training methods that will improve the Student’s technique and understanding of wind correction and timing in the holding pattern. These techniques expel many of the “Myths and Misconceptions” of timing and wind correction in the holding pattern.

(7) This simple analysis for determining the outbound time and outbound heading should allow GPS manufacturers to implement a holding pattern page which contains all the information to properly fly the holding pattern. Since the winds can have variability over a period of 5-10 minutes, the GPS will have an update each time the aircraft reaches the holding fix and will provide the IFR Pilot with the outbound time and OWCA for the next circuit. With this GPS capability available, the IFR Pilot load during holding can be considerably reduced.

Finally, important formulas that are useful to both IFR Pilots and CFI-I’s are highlighted in red.

**1.0
****Introduction**

As part of the training requirements for the Airplane Instrument Pilot rating, the Candidate must be proficient in the use of holding procedures. Holding patterns can be necessary for a number of reasons: (a) Delays at the airport of intended landing, (b) Loss of ATC communication, (c) Not prepared to execute the approach due to either equipment malfunction or under Single Pilot Operation, the Pilot may not be ready to execute the approach. However, whatever the need for the hold, the IFR Pilot should use this time in the holding pattern to prepare the aircraft for the approach.

The latest ACS for the Airplane Instrument Pilot rating requires both knowledge and skills in mastering the hold while flying in the presence of a wind. In particular, IR.III.B.S5 states “Uses proper wind correction procedures to maintain the desired pattern and to arrive over the fix as close as possible to a specified time and maintain pattern leg lengths when specified”. The AIM (Par 5-3-8) provides a number of guidelines and rules-of-thumb for flying the hold in the presence of a wind. For example, in terms of the outbound heading, the AIM recommends determining the inbound wind correction angle (IWCA) and multiplying it by 3 (i.e. the M-Factor) to determine the outbound wind correction angle (OWCA). In regard to the outbound time, the AIM recommends on the first circuit, using one minute (or one minute and 30 seconds) for the outbound time measured from the abeam point of the holding fix. If the abeam point cannot be determined, then use the outbound heading as the point to initiate the outbound time. After the first circuit, correct the outbound time to achieve the specified inbound time. Note that this process of converging to the holding pattern is based on a “Bracketing Method” or “Trial and Error Method”. Although the AIM does not recommend any rules-of-thumb for correcting the outbound time for the next circuit, there have been numerous rules-of-thumb proposed in IFR training manuals such as Ref. 1. However, these rules-of-thumb do not come with any specific limitations.

In order to overcome the problem of converging to the correct holding pattern, Holding Pattern Calculators were developed in an attempt to provide the IFR Pilot with both the outbound heading and outbound time, given the windspeed and direction. These calculators were very complex and used graphical methods to generate the outbound time and heading. In addition, as the windspeed increased beyond approximately 0.25, these calculators were found to be inaccurate.

In order to reduce the number of circuits that the IFR Pilot needs to converge to the correct holding pattern, as well as expel many of the “Myths and Misconceptions” of timing and wind correction in the holding pattern, we derive the exact solution of the “Holding Pattern” problem. To the Author’s knowledge, the exact solution of the “Holding Pattern” problem has never been documented in the open literature. This solution is both analytic and exact and thus does not contain any limitations in terms of the windspeed or direction. The work in this Treatise is based on an earlier FAASTeam Seminar on holding patterns, presented in June 2013 in Van Nuys California (Ref. 2).

In Section 2 we derive the exact solution of the ”Holding Pattern” problem. The extent of the Mathematics required involves both elementary Trigonometry and Algebra, plus a small amount of elementary Calculus. Using the exact solution, we discover a number of interesting properties of the holding pattern including a completely different type of holding pattern that arises under a strong wind with a headwind component on the inbound course to the fix. We define this new pattern as a Type-2 holding pattern, compared to the standard Type-1 holding pattern that is documented in many of the IFR training manuals. In addition, in the case of Type-1 holding patterns, we develop simple curves for the M-Factor, OWCA and outbound time as a function of the relative wind angle, for windspeed ratios up to 0.3. We also develop the boundary line between Type-1 and Type-2 holding patterns in windspeed ratio-wind angle space.

In Section 3, we utilize the exact solution to develop a “Smart- Convergence” algorithm that allows the Pilot to converge to the “Holding Pattern” solution in a minimum number of circuits. Here we introduce the concept of the “Coupling Effect” and show that one of the root causes of requiring a large number of circuits to converge to the “Holding Pattern” solution, is due to a lack of understanding of the importance of including the “Coupling Effect” in the convergence process.

In Section 4, we compare the “Smart-Convergence” algorithm with the “Bracketing Method” and show how the “Bracketing Method” is inefficient in converging to the correct holding pattern.

In Section 5 we discuss how to prepare for the hold, and some simple techniques to use which can reduce the number of circuits to converge to the holding pattern while using the “Bracketing Method”.

In Section 6 we develop training techniques that should be included when discussing timing and wind correction in the holding pattern.

In Section 7 we summarize the conclusions drawn from the work in this Treatise, and in Section 8 we list the references.

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