Radar Equation Derivation

Radar Equation Derivation:

The radar equation relates the range of the radar to the characteristics of transmitter, receiver, target and environment. It is useful for determining the maximum range at which radar can detect a target. If the transmitting antenna used is isotropic in nature, the power density is given by,

Power density at a range, R from an isotropic antenna is

P_is = P_t/4πR² ----------------- (1)

If a directive antenna of gain, G is used, the power density is given by,

Power density at a range, R from a directive antenna is

P_dic = P_tG/4πR² ----------------- (2)

The radiated back power density is given by,

P_rerad = P_tG/4πR².σ/4πR²

Where σ – radar cross section

The received signal power, Pr = radiated poor density x effected area

ie, Pr = P_tGσA_e/(4π)²R⁴.

The maximum range of radar, R_max is the distance beyond which the target cannot be detected. It occur when the received signal power, Pr = minimum detectable signal, S_min.

Therefore, S_min = P_tGσA_e/(4π)²R_max⁴.

R_max = [P_tGσA_e/(4π)² S_min]^1/4

This is the fundamental form of radar range equation.

If the same antenna is used for transmitting and receiving the relation between gain and effective area is

G = 4πA_e/λ²

A_e=ρ_aA         

Where, ρ_a – aperture

Therefore, R_max = [P_t4πA_eσA_e/ λ²(4π)² S_min]^1/4

Rmax = [PtσAe2/ 4π λ2 Smin]1/4

Pulsed Radar and CW Radar Block Diagram

Pulsed Radar Block Diagram Explanation:

It is a high power, high frequency radar which transmits pulses towards the target object. The range and resolution of radar depends on pulse repetition frequency. The radio frequency transmitted by the pulsed radar consists of series of equally spaced pulses having duration about 1μsec separated by very short intervals.

Pulsed Radar Block Diagram

The transmitter is possibly an oscillator for instance a magnetism that is pulsed by the modulator to generate repetitive train of pulses. The wave form generated by the transmitter travels through the transmission line through the antenna where it is radiated into the space. The trigger source or synchronizer coordinates the timing for range detection. It provides high power to the transmitter to transmit during the transmission period. The output tube may be an oscillator such as a magnetron or amplifier such as klystron amplifier. The pulse modulated wave travels through the duplexer where it is radiated into space. A single parabolic antenna is used for both transmission and reception. The duplexer consist of Transmitter Receiver tube (TR switch) and anti transmitter receiver tube (ATR Switch). The TR switch protects the receiver during the transmission and ATR switch helps in directing the received eco signals to the receiver. The receiver is usually superheterodyne type. The first stage of the receiver is a low noise amplifier. The mixer and LO convert the RF output from the amplifier to comparatively lower frequency level called intermediate frequency state. Thus in a mixer, the carrier frequency is reduced. The detector used is a schottky barrier diode which extracts the modulated pulse waveform from the IF amplifier output. The detector output is then amplified by an amplifier to a stage where it can be correctly displayed on an indicator directly.

Continuous Wave Radar (CW Radar Block Diagram Explanation):

The CW radar transmits high frequency signals continuously. The eco signal is received and processed permanently. It is a form of radar system, where recognized stable frequency continuous wave radio energy is transmitted and received from reflecting objects. These radars determine the target velocity rather than its location and they are simple, compact and less costly. CW radar works on the principle of Doppler effect. The Doppler effect is the change in frequency that occurs when a source and target are in relative motion. The CW radar uses the Doppler effect to determine the velocity with which the target is moving. The transmitter produces a continuous sinusoidal signal at a frequency ‘f_t’ which is radiated by the antenna on signal by a moving object, the transmitted signal is shifted by Doppler effect by an amount ±f_d when the target is moving towards the transmitter or source f_d will be positive. If the target is moving away from the source f_d will be negative. The detector or mixer multiplies the eco system signal at a frequency, f_t ± f_d. The Doppler filter allows the difference frequency component to pass through and rejects the higher frequency. The output of the detector is amplified by a power amplifier.

CW Radar Block Diagram

Comparison between Pulsed Radar and CW Radar

	Pulsed Radar	CW Radar
1	In this system, pulse modulated signal is used for transmission.	This system uses continuous wave modulated signals for transmission.
2	Duplexer is used to use common antenna for transmission and reception.	Circulator is used to separate antenna for transmission and reception.
3	It indicates the range of the radar	It indicates the velocity of the moving target
4	Higher Transmission Power	Lower Transmission Power
5	Complicated and Complex	Simple
6	Does not use Doppler shift	Uses Doppler Shift for working

Radar Block Diagram and Working Principle

Radar Block Diagram and Working Principle

Radar Working Principle:

A transmitter produces an electromagnetic signal that is radiated to the space by an antenna. A part of the transmitted energy is catched by the target and is reradiated in many directions. The radiations directed back towards the radar is collected by the receiving antenna and is delivered to the receiver at the receiver the signal is processed to detect the presence of target and determine its location. A single antenna can act as transmitter and receiver. The range or distance to the target is found out by measuring the time taken by the radar signal to travel to the target and return back to radar.

Block Diagram of Radar:

The transmitter can be a power amplifier such as klystron, travelling wave tube etc. It can also be a power oscillator such as magnetron. The radar signal is produced at low power by a waveform generator which is then amplified by the power amplifier.

The output of the power amplifier is delivered to the antenna by a duplexer which is then radiated into the space. The duplexer allows a single antenna to be used as both transmitter and receiver. Duplexer is a device that produces a short circuit at the input to the receiver when the transmitter is operating so that high power flows to the antenna and not to the receiver. On reception the duplexer directs the echo signal to the receiver and not to the transmitter.

The receiver is always super heterodyne in nature. The input stage is a low noise RF amplifier. The mixer and LO convert the RF Signal to an intermediate frequency which is amplified by the IF amplifier. The IF amplifier is designed as a matched filter, which maximizes the output signal to peak ratio. The matched filter maximizes the detectability of weak echo signals and attenuates unwanted signals. 

The IF amplifier is followed by a detector or demodulator. Its purpose is to extract the modulating signal from the carrier signal. The combination of IF amplifier and detector acts as an envelope detector which transmits the modulating signal and rejects the carrier signal. The signal at the output of detector is amplified by an amplifier to provide sufficient gain to the signal. At the output of the receiver, a decision is made whether or not a target is present if the output is greater than the threshold the decision is that target is present. If the output is less than threshold it is assumed that only noise is present.

Gunn Diode | Advantages and Disadvantages

Microwave devices that operate by transferred electron mechanism are called Gunn Diodes. Some materials like GaAs show the behaviour of negative mobility with increase in electric field. This phenomenon is caused by the transfer of conduction band electrons from lower energy high mobility state to high energy low mobility state. This is known as ‘transferred electron effect’ or Ridley Watkins Hilsum (RWH). The devices based on this effect are called transferred electron devices. Gunn diodes are negative resistance devices which are normally used as low power oscillators at microwave frequencies. The basic structure of Gunn diode consists of n type GaAs semiconductor. Heavily dopped n+ regions are formed over the substrate. If the voltage or electric field is applied to GaAs initially the current will increase with voltage. When the voltage exceeds the threshold voltage a high electric field is produced and the electrons are excited from initial lower state to higher state. If the rate at which the electrons transferred is high the current will decrease with increase in voltage producing negative resistance.

C_j – Diode Capacitance

R_j – Diode Resistance

R_s – Total Resistance

C_p – Package Capacitance

L_p – Package Inductance

The conduction band has two valley’s

1. Central Valley with low energy and high mobility

2. Satellite Valley with high energy and low mobility

Under normal conditions electrons are in the central valley. When the electric field increases, the velocity of electrons increases. This happens only till the energy reaches the threshold value. Above the threshold value the mobility is non linear.

When the electric field increases beyond the threshold value the velocity of electrons in the central valley increases and gain enough energy to transfer to satellite valley. Such a transfer is defined as transferred electron mechanism. The effective mass of electrons in the satellite valley is higher than the effective mass of electron in the central valley. This results in decreased mobility of electrons in the satellite valley. Since the mobility is decreased, velocity decreases.

Current Density, J = qμE = qV.

When velocity decreases, current density decreases. Thus, when the field becomes more than the critical value the reduction in current indicate negative resistance. This is also known as ‘negative differential mobility’ or ‘bulk negative differential conductivity’ or ‘Gunn effect’ or ‘RWH Theory’.

Modes of Operation:

1. Gunn Oscillation Mode:

This mode is defined in the regions where the product of frequency and length is about 10⁷ cm/sec and the product of doping and length is greater than 10¹² cm/sec. In normal conditions, Gunn diode is operated in this mode with E > E_TH.

The three possible modes of Gunn Oscillation are,

i. Transit Time Domain Mode:

Applied Velocity is equal to drift velocity.

Oscillation period is equal to transit time.

ie, τ₀ = τ_t

Efficiency is below 10%.

Length of the domain is 10⁷ cm/sec.

ii. Delayed Domain mode:

In this mode, the length of the domain is between 10⁶ cm/sec and 10⁷ cm/sec.

The domain is created when E < E_TH.

New domain cannot be formed until the field rises above the threshold.

Oscillation period is greater than transit time.

ie, τ₀ > τ_t

It is also known as inhibited mode.

Efficiency is about 20%.

iii. Quenched Domain Mode:

Length of the domain is greater than 2 x 10⁷ cm/sec.

If the field drops below the minimum value, the domain collapses before reaching the anode.

Oscillation period is less than transit time.

ie, τ₀ < τ_t

New domain is created before the field swings back above the minimum value.

Efficiency is about 13%.

2. Limited Space Charge Accumulation Mode:

When the frequency is very high the domains do not have sufficient time to form while the applied electric field is above the threshold value. As a result, the charge carriers accumulate near the cathode and collapse with time.

Efficiency is about 20%

Oscillation time, τ₀ = 3 x τ_t

Gunn Diode Oscillator:

One of the main applications of Gunn diode is Gunn diode Oscillator. It is used to generate and control microwave frequencies. They are mainly applied in relays, radars etc.

When the gunn diode is biased in negative resistance region, it will produce oscillations. These oscillations can be in the range of GHz. The nature of oscillations depends on the diode area.

Advantages of Gunn Diode:

1. Highly Stable

2. Higher band width and reliability

3.  Small Size

4. Low Supply Voltage

5. Low Cost

6. Highly immune to noise.

Disadvantages of Gunn Diode:

1. Efficiency decreases below 10 GHz

2. Poor Temperature Stability

3. Small tuning range

4. More power dissipation

Tunnel Diode Working Principle

The tunnel diode is a PN junction device, that operates in certain regions of V-I characteristics by the tunnelling of electrons across the potential barrier of the junction. This device can be used in high speed switching and logic circuits. Tunnel diodes are useful in many applications such as microwave amplifiers, microwave oscillators etc because of its low cost, light weight, high speed, low power, low noise and high peak current to valley current ratio characteristics.

Principle of Working:

1. Under Equilibrium:

The fermi level is constant throughout the junction. The Fermi level lies below the valence band in the ‘p’ side and above the conduction band in ‘n’ side. Since there are no filled states on one side of the junction which is at the same energy level as the empty state on the other side no flow of charge occurs in either directions and the current is zero. The bands must overlap for the Fermi level to be constant. With a small FB or RB, filled and empty states appear on opposite side separated by width of the depletion region.

2. Under Forward Bias:

When the forward bias is applied across the junction, the potential barrier is decreased by the magnitude of applied voltage. A difference in Fermi levels is created in both sides. While, there are filled states in the conduction band of ‘n’ region at the equal energy level as the authorized empty states in the valance band of p region. Electrons tunnel through the barrier from n region to p region, giving rise to forward tunnelling current from p region to n region. As the forward bias is increased to a max voltage, a maximum number of electrons can tunnel through the barrier producing peak current. If the voltage is increased further, the tunnelling current decreases as there are no empty states available in the p region. When the forward bias voltage is increased further, the current flow increases which is mainly due to minority charge carriers and is known as ‘injection current’.

Tunnel Diode Characteristics:

When the applied forward voltage is between 0 and V_p, the electrons tunnel from n region to p region, thereby increasing the current as the applied forward voltage reaches a value Vp, the current flowing across the junction attains a maximum value called the peak current ‘I_p’. When the applied voltage is increased further a decrease of current occurs this produces a region of negative slope. The maximum value of current achieved in negative resistance region is called Valley Current, I_v at an applied valley voltage V_v. If the voltage is increased beyond the negative resistance region, the current begins to increase due to the flow of minority charge carriers. This characteristic is also known as ‘Voltage controlled negative resistance’ as the current decreases rapidly at a critical voltage.

Tunnel Diode Oscillator:

The value of resistor must be in like a way that it biases the tunnel diode in the middle of negative resistance region. The resistor R₁, is used for setting proper biasing point for the tunnel diode. Resistor R₂, sets proper biasing point for the tank circuit. A parallel combination of resistor Rp, inductor L and capacitor C forms the tank circuit, which resonates at a frequency. Voltage drop in the tunnel diode increases as the applied voltage increases. As a result, the tunnel diode is driven to negative resistance region. In this region, current starts decreasing until the voltage across the diode equals the valley voltage. At this point, further increase of the applied voltage increases the current. This increase in current will raise the voltage drop in the resistor which produces the voltage across the diode.

Microwave Bipolar Transistor

Microwave Bipolar Transistor:

The micro wave bipolar transistor is a non linear device, which is mostly silicon npn type operating up to ‘5 GHz’. The geometry of the transistor can be characterized as interdigitated geometry, over lay geometry and matrix geometry. These geometries have wide emitter area to overcome transit time limitations. The interdigitated geometry is used for small signal, small power circuits. The over lay and matrix types are used for small power only. For high frequency applications, the NPN structure is preferred because the electron mobility is higher than hole mobility. Diffusion and ion implantation are the common methods used for transistor fabrication.

Basic Construction:

An epitaxial n layer is grown over a low resistivity n+ silicon substrate above the epitaxial layer a p region is diffused forming the base and n+ layer is diffused over the p region to form the emitter. The silicon substrate acts as the collector.

The microwave bipolar transistors are active three terminal devices which is commonly used for amplification process and switching phenomena. The three regions of the transistor are emitter, base and collector. The emitter region forms the input of the device and the collector region forms the output of the device. The emitter region of the transistor is heavily doped and has moderate area of cross section. The base of the transistor is thin and lightly doped to reduce the recombination rate. The collector region of the transistor is large and moderately doped. The charge carriers from the emitter are supplied to the collector through the base. When the charge carriers from the emitter reach the base some of them recombine with the charge carriers in the base. The remaining charge carriers are directed towards the collector constituting the collector current or output current.

Modes of Operation:

The microwave transistors have four modes of operation depending on the polarity of the applied voltage.

1. Normal Mode: In this mode, the emitter base junction of the npn transistor is forward biased and collector base junction is reverse biased. Most transistor amplifiers is operated in normal mode.

2. Saturation Mode: When both the emitter base junction and collector base junction are forward biased, the transistor is in saturation mode with low resistance and acts like a short circuit.

3. Cut Off Mode: When both the T^r junctions are reverse biased, the T^r is operated in Cut Off mode. The T^r acts like an open circuit. Both the saturation and cut off modes are used when transistor acts as a switch.

4. Inverted Active Mode: In this mode, the emitter base junction is reverse biased and collector base junction is forward biased.

Power Frequency Limitations:

Microwave transistors have limitations on frequency and power. These limitations can be due to maximum velocity of carriers, maximum electric field and maximum current. The four basic equations for the power frequency limitation are

1. Voltage – Frequency limitation:

V_mf_T = E_mV_s/2 π

Where, f_T - Cut off frequency

f_T = 1/2 πτ

τ = L/V

V_m - maximum voltage

V_s – velocity

E_m – Maximum electric field

When the length decreases, the average time, τ decreases. As a result, the frequency increases. When the frequency increases, the applied max voltage decreases.

2. Current frequency Limitation:

I_m X_cf_T = E_mV_s/2 π

Where, X_c – impedence

I_m – Max Current

If the impedence level is zero, the maximum current is infinite. The impedance value should be maintained in such a way, that maximum current is obtained for producing maximum power.

3. Power - frequency Limitation:

√(P_m X_c)f_T = E_mV_s/2 π

If the value of X_c is zero, the maximum power delivered is infinite.

4. Power Gain frequency Limitation:

√(G_m V_m V_th)f_T = E_mV_s/2 π

G_m – Maximum Gain

V_m –Voltage

V_th – Thermal Voltage

If the frequency increases, the gain of the device decreases.

Equivalent Model of Microwave Bipolar Transistor:

Hybrid Pi equivalent model is commonly used in normal active mode for small signal operations.

A change of emitter voltage V_be at the input terminal will induce a change of collector current at the output terminal. The mutual conductance or transconductance for the small signal model is given by,

g_m = ∂i_c/ ∂V_be ------------- (1)

The density of charge carriers across the junction is given by,

n_p(0) = np_oe^Vbe/VT ----------------- (2)

The collector current is defined based on the charge density as

i_c = qAD_nn_p(o) --------------- (3)

where, q – charge

A – Area of Cross Section

D_n – Diffusion constant

n_p(0) – Charge density

Substitute, eq (2) in eq (3)

i_c = (qAD_n np_oe^Vbe/VT)/L_n ---------------- (4)

Therefore, g_m = ∂i_c/ ∂V_be

g_m = qAD_n np_oe^Vbe/VT)/L_nV_T

= qAD_n np_o/L_nV_T

g_m = i_c/V_T ------------------- (5)

Diffusion Capacitance:

C_be = dQ_b/dV_be

Where, Q_b - charge stored in the base.

Q_b = qn_p(o)ω_bA/2

Sub for np_(o) from (2)

Q_b = qn_poe^Vbe/VTω_bA/2

Therefore,

C_be = qω_bAn_poe^Vbe/VT/2V_T

C_be = qω_bAn_po/2V_T

From equation (3)

C_be = i_cL_nω_b/2V_TD_n

From equation (5)

C_be = g_mL_nω_b/2D_n

If L_n = ω_b

C_be = g_mω_b²/2D_n

The input conductance,

g_b = i_c/h_fe V_T

g_b = g_m/h_fe

where, h_fe – input resistance/Forward emitter resistance