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Monday, 16 September 2019

Tunnel Diode Working Principle

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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 Vp, 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 ‘Ip’. 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, Iv at an applied valley voltage Vv. 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.

Sunday, 15 September 2019

Microwave Bipolar Transistor

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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 Tr junctions are reverse biased, the Tr is operated in Cut Off mode. The Tr 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:

VmfT = EmVs/2 π
Where, fT - Cut off frequency
fT = 1/2 πτ
τ = L/V
Vm - maximum voltage
Vs – velocity
Em – 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:

Im XcfT = EmVs/2 π
Where, Xc – impedence
Im – 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:

(Pm Xc)fT = EmVs/2 π

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

4. Power Gain frequency Limitation:

(Gm Vm Vth)fT = EmVs/2 π
Gm – Maximum Gain
Vm –Voltage
Vth – 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 Vbe 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,

gm = ∂ic/ ∂Vbe ------------- (1)

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

np(0) = npoeVbe/VT ----------------- (2)

The collector current is defined based on the charge density as

ic = qADnnp(o) --------------- (3)

where, q – charge
A – Area of Cross Section
Dn – Diffusion constant
np(0) – Charge density

Substitute, eq (2) in eq (3)

ic = (qADn npoeVbe/VT)/Ln ---------------- (4)

Therefore, gm = ∂ic/ ∂Vbe
gm = qADn npoeVbe/VT)/LnVT
= qADn npo/LnVT
gm = ic/VT ------------------- (5)

Diffusion Capacitance:

Cbe = dQb/dVbe
Where, Qb - charge stored in the base.
Qb = qnp(o)ωbA/2
Sub for np(o) from (2)
Qb = qnpoeVbe/VTωbA/2

Cbe = qωbAnpoeVbe/VT/2VT
Cbe = qωbAnpo/2VT

From equation (3)
Cbe = icLnωb/2VTDn

From equation (5)
Cbe = gmLnωb/2Dn
If Ln = ωb
Cbe = gmωb2/2Dn

The input conductance,

gb = ic/hfe VT

gb = gm/hfe

where, hfe – input resistance/Forward emitter resistance

Directional Coupler S Matrix Derivation

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Directional Coupler is a 4 port device which has primary and secondary waveguides. The primary wave guide is from port 1 to port 2 and secondary waveguide is from port 3 to port 4.

• Directional coupler is used to couple microwave power, which is unidirectional in most of the cases. The properties of a directional coupler are,

1. All the ports are matched.

2. When the power moves from port 1 to port 2, little portion of it would gets coupled to port 4 and not to port 3.

3. When the power moves from port 2 to port 1, little portion of it would gets coupled to port 3 and not to port 4.

4. The coupling factor of a directional coupler is the ratio of incident power to forward power.

Coupling factor = 10 log(Pi/Pf)

• Directivity of the directional coupler is the ratio of forward power to back power.

Directivity = 10 log(Pf/Pb)

• Isolation of a directional coupler is the ratio of incident power to back power.
I = 10 log(Pi/Pb)

• Isolation = Coupling factor + Directivity.

• Two hole directional coupler is same as conventional directional coupler, but with two holes in common between primary and secondary waveguides.

The spacing between these two holes is given by,

L = (2n+1)λg/4

Where, n = an integer
λg = wavelength

• A fraction of energy entering into port 1 passes through holes and is radiated into port 2. The forward waves in port 4 are in the same phase and are added.

• The backward waves in port 3 are out of phase and are cancelled.

The general S matrix of a directional coupler is,

• Since all ports in a directional coupler are matched.

S11 = S22 = S33 = S44 = 0 ----------------- (2)

• Since there is no coupling between ports 1 & 3 and ports 2 & 4

S13 = S31 = S24 = S42 = 0 ------------- (3)

Apply equation (2) & (3) in (1)

By unitary property, [S][S]* = I

R1C1 => |S12|2 + |S14|2 = 1 ------------ (4)
R2C2 => |S12|2 + |S23|2 = 1 ------------ (5)
R3C3 => |S23|2 + |S34|2 = 1 ------------ (6)
R1C3 => S12 S23* + S14 S34* = 0 ----------- (7)

Comparing eq (4) and (5)

|S12|2 + |S14|2 = |S12|2 + |S23|2
S14 = S23 --------------- (8)

Comparing eq (5) and (6)

|S12|2 + |S23|2 = |S34|2 + |S23|2
S12 = S34 --------------- (9)

Let, S12 be real and positive,
ie, S12 = S34 = p --------------- (10)

applying equation (10) in (7)

Therefore, p S23* + S14 p = 0
p [S23* + S14] = 0
p [S23* + S23] = 0
S23* + S23 = 0

To satisfy the above condition, S23 should be a complex value.

Let S23 = jq

Therefore, the S matrix of directional coupler is,

Saturday, 14 September 2019

Magic Tee and Hybrid Ring S Matrix Derivation

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Magic Tee or Hybrid Tee S Matrix Derivation

A combination of E plane Tee and M plane Tee is called Hybrid Tee or Magic Tee. It consists of four ports, if two waves of equal magnitude and same phase are fed into port 1 and port 2, the output will be subtractive and zero at port 3 and will be addictive at Port 4. A wave incident at Port 4 divides equally between port 1 and 2, and will not appear at port 3. A wave incident at Port 3 will produce an output of equal magnitude and opposite phase at ports 1 and 2. The magic tee is matched at ports 3 and 4.

The general matrix of the magic tee is given by,

From the property of Symmetry,
S14 = S41,
S13 = S31,
S23 = S32 ---------------- (2)

Since port 3 acts as the E plane Tee,
S13 = -S23 ------------------- (3)

Since port 4 acts as H plane Tee, S14 = S24 ---------------- (4)

Considering the phase delay in the network,

S34 = -S43 = 0 and
S12 = -S21 = 0 ---------------- (5)

If port 3 and port 4 are matched,
S33 = S44 = 0 ---------------- (6)

Applying equation (2) to (6) in equation (1)

By unitary property, [S][S*] = I

R1C1 => |S11|2+|S13|2+|S14|2 = 1 -------------- (8)
R2C2 => |S22|2+|S13|2+|S14|2 = 1--------------- (9)
R3C3 => 2|S13|2 = 1
S13 = 1/√2 ------------- (10)
R4C4 => 2|S14|2 = 1
S14 = 1/√2 ---------------- (11)

Substitute, Equation (10) and (11) in equation (8)

|S11|2+ (1/√2)2 + (1/√2)2 = 1
|S11|2 = 1 – 1
=> S11 = 0

Equating equation (8) and (9)

We get, S11 = S22

Therefore the s matrix of magic tee is,

Hybrid Ring S Matrix Derivation:

Hybrid ring circuits are also known as ‘Rat Race Coupler’. These junctions overcome the power limitations of magic tee. It is constructed by folding rectangular waveguides into circular waveguides. This junction has 4 ports with upper 3 ports separated by λ/4 and lower two ports separated by 3 λ/4. When a wave is fed into port 1, it will not appear at port 3 due to the phase shifts. Similarly wave fed onto port 2 will not appear at port 4 due to phase difference.

The general matrix of hybrid ring is,

If the ports 1,2,3 and 4 are matched then,
S11 = S22 = S33 = S44 = 0 -------------------- (2)

Considering the input – output conditions,
S13 = S31 = 0
S24 = S42 = 0
S21 = -S41

Therefore, the general matrix can be written as,
[S][S]* = I

R1C1 => 

|S12|2 + |S12|2 = 1
2|S12|2 = 1
S12 = 1/√2

R2C2 => 

|S12|2 + |S23|2 = 1
½ +|S23|2 = 1
S23 = 1/√2

R3C3 => 

|S23|2 + |S34|2 = 1
½ +|S34|2 = 1
S34 = 1/√2

R4C4 => 

|S12|2 + |S34|2 = 1
S12 = 1/√2

Therefore, Matrix of Hybrid Rings is

Waveguide Tees in Microwave

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Wave guide Tees in Microwave:

In microwave circuits, a wave guide with three independent ports is known as a ‘Tee’ junction. The characteristics of Tee junction are,

1. A short circuit can be positioned in one of the arms of 3 port junction in the manner that no power is moved through the other two arms.

2. If the junction is symmetric about one of its arms a short circuit can be placed on that arm. So that, no reflection occurs between other two arms.

3. It is impossible to have matched impedance on all the three arms.

E-Plane Tee in Microwave:

An E plane tee is a waveguide tee junction in which the axis of the side arm is parallel to the electric field. Two arms of the Tee junction are collinear arms. Signal entering at one port is divided among other two ports in such a way that the signals are out of phase with each other. The output of the E-plane will be the difference between input signals. The side arm of E-plane Tee is also known as difference arm.

Since the E-Plane Tee is a 3 port network, the general ‘S’ matrix is represented as

The wave fed into port-3 appears at port 1 and port 2 with equal magnitude and opposite phase.

ie, S13 = -S23 --------------------- (2)

If port 3 is matched, S33 = 0 --------------------- (3)

By the property of symmetry,
S12 = S21
S13 = S31
S23 = S32 ---------------------- (4)

Applying equation (2), (3) & (4) in equation (1)

By unitary property ‘[S][S]* = I’

=> S11S11* + S12S12* + S13S13* = 1
=> |S11|2 + |S12|2 + |S13|2 = 1 ------------ (6)

=> |S12|2 + |S22|2 + |S13|2 = 1 ------------ (7)

=> |S13|2 + |S13|2 + 0 = 1
=> 2|S13|2 = 1
=> S13 = 1/2 ------------ (8)

=> S13 S11* - S13S12*= 0
=> S13 (S11* - S12*) = 0
S11* - S12*= 0
S11* = S12*
ie,  S11 = S12 ------------------ (9)

Equating equations (6) and (7)

We get, |S11|2 = |S22|2

S11 = S22 --------------------- (10)

Substitute equation (8) & (9) in equation (6)

2|S11|2 + ½ = 1

S11 = ½

Therefore, the scattering matrix of E plane Tee is,

H Plane Tee in Microwave:

In H Plane Tee, the side arm or H arm is parallel to the magnetic field. The signal fed to one of the ports will be divided between the other two ports and the signals will be in phase. The output of the H Plane Tee is the sum of input signals.
The general matrix is,

Since, the signals are in phase, S13 = S23 ------------------- (2)

If Port 3 is matched, S33 = 0 -------------- (3)

By Symmetry,
S12 = S21
S13 = S31
S23 = S32 ---------------------- (4)

Applying equation (2), (3) and (4) in (1).

By unitary property, [S][S]* = I

=> |S11|2 + |S12|2 + |S13|2 = 1 ------------ (6)

=> |S12|2 + |S22|2 + |S13|2 = 1 ------------ (7)

=> |S13|2 + |S13|2 + 0 = 1
=> 2|S13|2 = 1
=> S13 = 1/2 ------------ (8)

=> S13 S11* + S13S12*= 0
=> S13 (S11*+ S12*) = 0
S11* + S12*= 0
S11* = -S12*
ie,  S11 = -S12 ------------------ (9)

Equating equation (6) and (7)

We get, |S11|2 =|S22|2

S11 = S22 ------------------ (10)

Substitute eq (8) and (9) in eq (6)

We get,