# Calculating the MUFdays

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Let us elaborate the concept of the MUFday by examining how the MUFday factor is being calculated.

## Running HPF-MUF-FOT graph

First we need to run Method 9, HPF-MUF-FOT graph, for our circuit. We are interested in the frequencies at 01 UT (= GMT). Below are the MUF, FOT and HPF values from Method 9:

``` GMT   MUF   FOT   HPF

1.0  13.1  10.5  15.6
2.0  12.2   9.8  14.5
3.0  11.6   9.2  13.8
4.0  12.1   9.6  14.4
5.0  15.1  12.7  17.5
6.0  21.4  18.0  24.8
7.0  29.4  24.7  34.1
8.0  25.2  22.4  28.2
9.0  26.3  23.4  29.4
10.0  36.5  31.7  40.5
11.0  35.0  30.4  38.8
12.0  34.0  29.5  37.7
13.0  33.0  27.8  36.0
14.0  31.4  26.4  34.2
15.0  28.5  23.9  31.1
16.0  24.5  20.6  26.7
17.0  20.4  17.6  24.5
18.0  17.4  14.9  20.8
19.0  15.4  13.3  18.5
20.0  14.1  12.1  16.9
21.0  13.0  10.8  15.5
22.0  12.8  10.6  15.2
23.0  13.1  10.9  15.6
24.0  13.5  11.2  16.0
```

The HPF, FOT and MUF represent the Highest Possible Frequency, Optimum Working Frequency (or Frequency of Optimum Traffic) and Median Maximum Usable Frequency, respectively, for the hourly MOF (median maximum observed frequency) distribution. At the HPF, communication can be supported on 10% of the days (3 days) in the month; at the FOT, 90% of the days (27 days) in the month and, as discussed above, at the MUF, 50% of the days (15 days) in the month.

### A Rule of Thumb

The practical meaning of FOT and HPF [As obtained from Method 9 in VOACAP] is that they determine the frequency range within which we should find 80% of the measured MOFs.

## Two formulas for calculating the Z value

For calculating the fraction (z) of days where we can expect our operating frequency (FREQ) to be below the MUF we have two formulas: one for cases when the FREQ is equal to or below the predicted MUF (1), and the other for cases when the FREQ is above the predicted MUF (2).

```(1) z = (MUF - FREQ) / ((MUF - FOT) / 1.28)

(2) z = ABS(MUF - FREQ) / (ABS(MUF - HPF) / 1.28)
```

Let us choose two frequencies 9.7 and 13.7 MHz for our calculations. 9.7 MHz is below the predicted MUF so we choose the formula 1:

```(1) z = (13.1 - 9.7) / ((13.1 - 10.5) / 1.28)
= 3.4 / 2.03125 = 1.674.
```

## Converting Z to days

To convert the Z to a percentage value and corresponding number of days, let us look at the Z Tables, Table 1: just over 95% or 28 days.

Then for 13.7 MHz, above the predicted MUF, we choose the formula 2:

```(2) z = ABS(13.1 - 13.7) / (ABS(13.1 - 15.6) / 1.28)
= 0.6 / 1.953125 = 0.3072,
```

corresponding to approximately 38% or 11 days (see Table 2).

To confirm these values, below you see that VOACAP has calculated the same values for those two frequencies (0.95 = 95% and 0.38 = 38%):

```   1.0 13.1  6.1  7.2  9.7 11.9 13.7 15.4 17.7 21.6 25.9  0.0  0.0 FREQ
F2F2 F2F2 F2F2 F2F2 F2F2 F2F2 F2F2 F2F2 F2F2 F2F2   -    -  MODE
0.50 1.00 0.99 0.95 0.73 0.38 0.12 0.01 0.00 0.00   -    -  MUFday
```