BURN-THROUGH / CROSSOVER RANGE

The burn-through equations are derived in this section. These equations are most commonly used in jammer type of applications. The following is a summary of the important equations explored in this section:

J/S CROSSOVER RANGE (MONOSTATIC) (J=S) or in dB form: 20log RJ = S = 10log Pt + 10log Gt + 10log - 10log Pj - 10log Gja - 10.99 dB
If simplified radar equations developed in previous sections are used: 20log RJ = S = 10log Pt + 10log Gt + G - 10log Pj - 10log Gja - K1 - 20log f1 (in dB)
BURN-THROUGH RANGE (MONOSTATIC) or in dB form: 20log RBT = 10log Pt + 10log Gt + 10log - 10log Pj - 10log Gja + 10log (Jmin eff/S) - 10.99 dB
If simplified radar equations developed in previous sections are used: 20log R BT = 10log Pt + 10log Gt + G - 10log Pj - 10log Gja - K1 + 10log (Jmin eff/S) - 20log f1 (in dB)
BURN-THROUGH RANGE (BISTATIC) RTx is the range from the radar transmitter to the target and is different from RRx, which is the range from the target to the receiver. Use monostatic equations and substitute RTx for R
Target gain factor; G = 10log + 20log f1 + K2 (in dB)	One-way free space loss; Tx or Rx = 20log (f1 RTx or Rx) + K1 (in dB)
K2 Values (dB) RCS () (units) m 2 ft 2 f1 in MHz K2 = -38.54 -48.86 f1 in GHz K2 = 21.46 11.14	K1 Values (dB) Range (units) NM Km m yd ft f1 in MHz K1 = 37.8 32.45 -27.55 -28.33 -37.87 f1 in GHz K1 = 97.8 92.45 32.45 31.67 22.13

To present the values of J and S, (or J/S) over a minimum to maximum radar to target range of interest, equation [1], in the Constant Power (Saturated) Jamming section which has a slope of 20 log for J vs. range and equation [2], in the same section, which has a slope of 40 log for S vs. range are plotted. When plotted on semi-log graph paper, J and S (or J/S) vs. range are straight lines as illustrated in Figure 1. Figure 1 is a sample graph - it cannot be used for data.

The crossing of the J and S lines (known as crossover) gives the range where J = S (about 1.29 NM), and shows that shorter ranges will produce target signals greater than the jamming signal.

The point where the radar power overcomes the jamming signal is known as burn-through. The crossover point where J = S could be the burn-through range, but it usually isn't because normally J/S > 0 dB to be effective due to the task of differentiating the signal from the jamming noise floor (see receiver sensitivity section). For this example, the J/S required for the ECM to be effective is given as 6 dB, as shown by the dotted line. This required J/S line crosses the jamming line at about 2.8 NM which, in this example, is the burn-through range.

In this particular example, we have:	Pt = 80 dBm	Gt = 42 dB
	Pj = 50 dBm	Gja = 6 dB
	= 18 m2	f = 5.9 GHz (not necessary for all calculations)

A radar can be designed with higher than necessary power for earlier burn-through on jamming targets. Naturally that would also have the added advantage of earlier detection of non-jamming targets as well.

To calculate the crossover range or burn-through range the J/S equation must be solved for range. From equation [3], in the Constant Power (Saturated) Jamming section:

Solving for R:

[1]

BURN-THROUGH RANGE (MONOSTATIC) - Burn-through Range (Monostatic) is the radar to target range where the target return signal (S) can first be detected through the ECM (J). It is usually the range when the J/S just equals the minimum effective J/S.

or in dB form, (using 10log 4 = 10.99 dB):

20log R_BT = 10log P_t + 10log G_t + 10log - 10log P_j - 10log G_ja + 10log (J_{min eff}/S) - 10.99 dB [3]

RANGE WHEN J/S CROSSOVER OCCURS (MONOSTATIC) - The crossover of the jammer's 20 dB/decade power line and the skin return signal's 40 dB/decade power line of Figure 1 occurs for the case where J = S in dB or J/S=1 in ratio. Substituting into equation [1] yields:

or in dB form:

20log R_J = S = 10log P_t + 10log G_t + 10log - 10log P_j - 10log G_ja - 10.99 dB [5]

Note: keep R and in same units in all equations.

CROSSOVER AND BURN-THROUGH EQUATIONS (MONOSTATIC)
USING - ONE WAY FREE SPACE LOSS

The other crossover burn-through range formulas can be confusing because a frequency term is subtracted (equations [6], [7] and [8]), but both ranges are independent of frequency. This subtraction is necessary because when J/S is calculated directly as previously shown, ² or (c/f)² terms canceled, whereas in the simplified radar equations, a frequency term is part of the G term and has to be cancelled if one solves for R. From equation [8], in the Constant Power (Saturated) Jamming section :

10log J/S = 10log P_j + 10log G_ja - 10log P_t - 10log G_t - G + (factors in dB)

or rearranging: = 10log P_t + 10log G_t + G - 10log P_j - 10log G_ja + 10log (J/S)

from the Two-Way Radar Equation (Monostatic) section: = 20log f ₁R₁ + K₁ or 20log R₁ = - K₁ - 20log f₁

then substituting for :

20log R₁ = 10log P_t + 10log G_t + G - 10log P_j - 10log G_ja - K₁ + 10log (J/S) - 20log f₁ (factors in dB) [6]

EQUATION FOR BURN-THROUGH RANGE (MONOSTATIC) - Burn-through occurs at the range when the J/S just equals the minimum effective J/S. G and K ₁ are as defined at the beginning of this section.

20log R _BT = 10log P_t + 10log G_t + G - 10log P_j - 10log G_ja - K₁ + 10log (J_{min eff}/S) - 20log f₁ (factors in dB) [7]

EQUATION FOR THE RANGE WHEN J/S CROSSOVER OCCURS (MONOSTATIC) - The J/S crossover range occurs for the case where J = S, substituting into equation [6] yields:

20log R_J = S = 10log P_t + 10log G_t + G - 10log P_j - 10log G_ja - K₁ - 20log f₁ (factors in dB) [8]

Bistatic J/S crossover range is the radar-to-target range when the power received (S) from the radar skin return from the target equals the power received (J) from the jamming signal transmitted from the target. As shown in Figure 6, of the J/S Ratio - Constant Power (Saturated) Jamming Section, the receive antenna that is receiving the same level of J and S is remotely located from the radar's transmit antenna. Bistatic equations [11], [13], and [14] in that section show that J/S is only a function of radar to target range, therefore J/S is not a function of wherever the missile is in its flight path provided the missile is in the antenna beam of the target's jammer. The missile is closing on the target at a very much higher rate than the target is closing on the radar, so the radar to target range will change less during the missile flight.

It should be noted that for a very long range air-to-air missile shot, the radar to target range could typically decrease to 35% of the initial firing range during the missile time-of-flight, i.e. A missile shot at a target 36 NM away, may be only 12 NM away from the firing aircraft at missile impact.

Figure 2 shows both the jamming radiated from the target and the power reflected from the target as a function of radar-to-target range.

In this particular example, the RCS is assumed to be smaller, 15 m² vice 18m² in the monostatic case, since the missile will be approaching the target from a different angle. This will not, however, always be the case.

In this plot, the power reflected is:

Substituting the values given previously in the example, we find that the crossover point is at 1.18 NM (due to the assumed reduction in RCS).

To calculate the radar transmitter-to-target range where J/S crossover or burn-through occurs, the J/S equation must be solved for range. From equation [11] in J/S Ratio - Constant Power (Saturated) Jamming Section, :

Solving for R_Tx:

Note: Bistatic equation [10] is identical to monostatic equation [1] except R_Tx must be substituted for R and a bistatic RCS () will have to be used since RCS varies with aspect angle. The common explanations will not be repeated in this section.

BURN-THROUGH RANGE (BISTATIC) - Burn-through Range (Bistatic) occurs when J/S just equals the minimum effective J/S. From equation [9]:

or in dB form:

20log R_Tx(BT) = 10log P_t + 10log G_t + 10log - 10log P_j - 10log G_ja + 10log (J_{min eff}/S) - 10.99 dB [11]

If using the simplified radar equations (factors in dB):

20log R_Tx(BT) = 10log P_t + 10log G_t + G - 10log P_j - 10log G_ja - K₁ + 10log (J_{min eff}/S) - 20log f₁ [12]

Where G and K₁ are defined at the beginning of the section.

RANGE WHEN J/S CROSSOVER OCCURS (BISTATIC) - The crossover occurs when J = S in dB or J/S = 1 in ratio.

or in log form:

20log R_Tx(J=S) = 10log P_t + 10log G_t + 10log - 10log P_j - 10log G_ja - 10.99 dB [14]

If simplified equations are used (with G and K₁ as defined at the beginning of this section) we have:

20log R_Tx(J=S) = 10log P_t + 10log G_t + G - 10log P_j - 10log G_ja - K₁ - 20log f₁ (factors in dB) [15]

Note: keep R and in same units in all equations.

Unless you are running a large scale computer simulation that includes maneuvering, antenna patterns, RCS, etc., you will seldom calculate the variation in J/S that occurs during a semi-active missile's flight. Missiles don't fly straight lines at constant velocity. Targets don't either - they maneuver. If the launch platform is an aircraft, it maneuvers too. A missile will accelerate to some maximum velocity above the velocity of the launch platform and then decelerate.

The calculation of the precise variation of J/S during a missile flight for it to be effective requires determination of all the appropriate velocity vectors and ranges at the time of launch, and the accelerations and changes in relative positions during the fly out. In other words, it's too much work for too little return.

The following are simplified examples for four types of intercepts. 

In these examples, all velocities are constant, and are all along the same straight line. The missile velocity is 800 knots greater than the launch platform velocity which is assumed to be 400 kts. The missile launch occurs at 50 NM. For the AAM tail chase, the range from the radar to the target remains constant and so does the J/S. In these examples the maximum variation from launch J/S is +/- 6 dB. That represents the difference in the radar to target range closing at very high speed (AAM head on) and the radar to target range opening at moderate speed (SAM outbound target). The values shown above are examples, not rules of thumb, every intercept will be different.

J/S (dB) J/S (dB) At Launch: 29 n/a Intercept Type At 2 sec. to Intercept: AAM Head-on: 23 -6 SAM Incoming Target: 25 -4 AAM Tail Chase: 29 0 SAM Outbound Target: 35 +6

Even for the simplified linear examples shown, graphs of the J and S will be curves - not straight lines. Graphs could be plotted showing J and S vs. radar to target range, or J and S vs. missile to target range, or even J/S vs. time of flight. If the J/S at launch is just barely the minimum required for effectiveness, and increasing it is difficult, then a detailed graph may be warranted, but in most cases this isn't necessary.