Radar Range Equation Fundamentals

The maximum range (\(R_{max}\)) of a radar system is a critical parameter that determines how far the radar can detect targets. This range is fundamentally governed by the radar range equation. For a monostatic radar (where the same antenna is used for both transmitting and receiving), the maximum range is generally proportional to the fourth root of the peak transmitted power (\(P_t\)) and the square of the antenna gain (\(G\)).

The proportionality can be expressed as:

$R_{max} \propto (P_t G^2)^{1/4}$

Antenna Gain and Diameter Relationship

For many radar antennas, especially parabolic dish antennas, the antenna gain (\(G\)) is directly related to its physical size, specifically its diameter (\(D\)). The gain of such an antenna is proportional to the square of its diameter and inversely proportional to the square of the wavelength (\(\lambda\)).

$G = \eta \left( \frac{\pi D}{\lambda} \right)^2$

Where \(\eta\) represents the antenna's aperture efficiency. From this relationship, it is clear that:

$G \propto D^2$

Range Dependence on Power and Diameter

Now, to determine the overall dependence of the maximum range on both the peak transmitted power and the antenna diameter, we can substitute the relationship of antenna gain with diameter into the radar range proportionality:

• Starting with: $R_{max} \propto (P_t G^2)^{1/4}$
• Substitute $G \propto D^2$ into the expression:

$R_{max} \propto (P_t (D^2)^2)^{1/4}$

$R_{max} \propto (P_t D^4)^{1/4}$

This simplified proportionality shows that the maximum range is proportional to the fourth root of the peak transmitted power and directly proportional to the antenna diameter ($D^4$ raised to $1/4$ simplifies to $D$).

Maximum Range Calculation

Let's consider the initial state of the radar system with peak transmitted power \(P_{t1}\), antenna diameter \(D_1\), and maximum range \(R_{max1}\).

• Initial relation: $R_{max1} \propto (P_{t1} D_1^4)^{1/4}$

The question states that:

• The peak transmitted power is increased by a factor of 16. So, the new power $P_{t2} = 16 P_{t1}$.
• The antenna diameter is increased by a factor of two. So, the new diameter $D_2 = 2 D_1$.

Now, we find the expression for the new maximum range, $R_{max2}$, using these increased values:

$R_{max2} \propto (P_{t2} D_2^4)^{1/4}$

Substitute the new values into the proportionality:

$R_{max2} \propto (16 P_{t1} (2 D_1)^4)^{1/4}$

$R_{max2} \propto (16 P_{t1} (16 D_1^4))^{1/4}$

$R_{max2} \propto (256 P_{t1} D_1^4)^{1/4}$

To determine the factor by which the range increases, we can extract the constant factor:

$R_{max2} \propto (256)^{1/4} (P_{t1} D_1^4)^{1/4}$

Since $256$ is equal to $4 \times 4 \times 4 \times 4$, or $4^4$, its fourth root is 4.

$(256)^{1/4} = 4$

Therefore, the new maximum range can be expressed as:

$R_{max2} \propto 4 \times (P_{t1} D_1^4)^{1/4}$

Comparing this with the initial range $R_{max1} \propto (P_{t1} D_1^4)^{1/4}$, we clearly see that:

$R_{max2} = 4 \times R_{max1}$

Range Increase Conclusion

Based on our calculations, increasing the peak transmitted power by a factor of 16 and the antenna diameter by a factor of two will result in the maximum radar range increasing by a factor of 4.

The final answer is $\boxed{4}$.