# Microwave Low Noise Amplifier LNA Computer Science Essay

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In this report the design of Microwave Low Noise Amplifier (LNA) for communication receiver applications are discussed. LNA is used to amplify the small signal received from receiver. LNA has minimum noise figure and high gain. For the design of amplifier GaAs MESFET is used with the help of Microwave office.

## Chapter 2

This chapter describes some basic topics which is necessary to understand for the designing of high frequency circuit. To design microwave low noise amplifier basics of transmission lines theory are important to understand. Some basic of transmission lines are also discussed in this chapter. Two-port networks and impedance matching techniques are also explained in this chapter.

## 2.1 Transmission Lines Theory

Circuit theory and transmission lines theory are mainly differentiated due to electrical size [3]. Transmission lines theory is applied when the physical dimensions of network are greater than electrical wavelength. So transmission lines are distributed parameter network, where voltages and currents can vary in magnitude and phase over its length [3]. Circuit model of transmission lines is shown in figure below

Fig 2.1: Equivalent circuit of Transmission line

In the above figure R represent the series resistance per unit length, L is the series inductance per unit length, G represents shunt conductance per unit length and C represent shunt capacitance per unit length.

## 2.1.1 Characteristics Impedance

The ratio of forward voltage to forward current or reverse voltage to reverse current at any point for a given line is called characteristics impedance.

(2.1)

Where and are forward voltage and forward current. Whileand are reverse voltage and reverse current.

## 2.1.2 Reflection coefficient

When load impedance is different from characteristics impedance than forward and reverse waves exist. The ratio of reverse and forward traveling wave is known as reflection coefficient [4]. The formulae of reflection coefficient is given below

(2.2)

Reflection coefficient at load side

(2.3)

Reflection coefficient at source side

(2.4)

## 2.1.3 Smith Chart

Transmission lines calculations such as reflection coefficient, input impedance and load impedance can be calculated using graphical method known as smith chart. A smith chart is graphical plot of normalized resistance and reactance functions in the reflection coefficient plane [5]. In other word the unit circle || =1 together with its interior is called smith chart [6]. It gives the solution of almost all the problem arising in the design of matching network [4].

## 2.1.4 Physical Transmission Lines

Transmission lines are used to transmit microwave energy from source to load. There are various types of physical transmission lines used for high frequency transmission such as striplines. Examples of striplines are microstrip, slot line, coplanar waveguide [4]. The most commonly used transmission lines are microstrip line for high frequencies from a few megahertz to over 10 GHz [4]. The reason is that geometry of microstrip line is simple and also fabrication is easy.

## 2.1.4.1 Microstrip

A microstrip consists of a metallic strip and a ground metallic sheet separated by a layer of dielectric [4]. Its geometry is shown in figure below [7]

Fig 2.2: Geometry of microstrip circuit

## Characteristics Impedance of microstrip

The characteristics impedance of any TEM-type transmission line is given by [4]

(2.5)

In the above equation L and C are the inductance and capacitance per unit length [4]. If the dielectric permittivity is removed then the microstrip becomes an air-filled line [4]. The characteristics impedance of air-filled line is given by [4]

(2.6)

In the above equation L remains same because inductance does not affect by changing dielectric and C1 is the new inductance per unit length.

The characteristics impedance of microstrip can be calculated using the formula

(2.7)

Equation 2.7 is taken from [4]. In the above equation represents the characteristics of microstrip line as an air filled line. is effective microstrip permittivity and it depends on the width and height of microstrip.

for (2.8)

for (2.9)

Equation 2.8 and 2.9 is taken from [4]. In the above equations represents the permittivity of dielectric.

## 2.2 Two Port Network

The two port network can be shown in figure below

Fig 2.3: Two-port Network

Two port networks can be completely specified by set of four parameters. In the frequency range above 1 GHz it has become costmary to use scattering parameters [8].

## 2.2.1 S-parameter

At higher frequency range it is difficult to measure the voltage and current at the terminal of a device and network [9]. But the power flow can be easily measured. So it is necessary to describe the parameters in terms of power flow. Scattering parameters are defined in terms of square root of power. The S-parameters of two port network are shown in figure below

Fig 2.4: Two-port s-representation

In the above figure,, and are the square root of the incident and reflected powers at port 1 and port 2.

The equations of scattering parameters for two port network are

(2.10)

(2.11)

In the above equations ,,are the scattering parameters. To measure the S-parameters of two port network it is necessary to open or short circuit one port of network. From the above equations it is clear that to measure the only possible if made zero. is made zero by connecting a transmission line of characteristics impedance equal to terminated in a load to port 2-2' of the network [4]. Similarly can be measured when is made zero. For the measurement of, only interchange the positions of port 1-1' and port 2-2' in the measuring system [4].

## 2.3 Impedance Matching

Impedance matching network are inserted between source and load in order to transfer the maximum power to load. There are various techniques used for impedance matching. Here some of techniques are discussed.

Two element L networks

Three- (or more) element networks

Transmission line network

Two or Three element matching networks are mostly designed by using capacitive and inductive element and normally designed for single frequency known as operating frequency. But the network has bandwidth which is determined by quality factor Q. The quality factor of network is approximately equal to reciprocal of the percentage bandwidth [4]. The formula for quality factor is given by

(2.12)

In the above equation is the operating frequency while is the 3-dB bandwidth of the matching network [4].

## Two element L networks

In this method of impedance matching two reactive elements is used in L shape. There are two possibilities to made L shape between source and load. These two configuration are shown below

Fig 2.5-a: L-matching network

Fig 2.5-b: L-matching network

In the above figures R1 and R2 are two resistances to be matched. Element 1 and 2 in the above figures may be inductor and capacitor depends on the load impedance. If the normalized load impedance is inside the circle of 1+jx on the smith chart then circuit (fig 2.5-a) is used. On the other hand when normalized load impedance is outside the circle of 1+jx then circuit (fig 2.5-b) is used [3]. In Two element network we don't have choice of the operating quality factor. The formulae to find the quality factor for two element network is

(2.13)

## Three element networks

In three element network we have choice for operating quality factor. So it is the best technique for high quality factor and narrow band design. In this impedance matching network the element are arranged in T or π shape as shown in figure below

Fig 2.6-a: T-form

Fig 2.6-b: -form

In the above figure two understand the quality factor of -form it is redrawn in the form of two L-network as shown in figure below

Fig 2.7: -network as two back-to-back L-network [4]

In the above figure there is shown virtual resistance say R the task is to match R1 and R2 with this virtual resistance R. Then the quality factor when looking from AA' towards R2

(2.14)

Similarly the quality factor when looking from BB' towards R1 is

(2.15)

The quality factor of overall circuit is the highest Q value. So the quality factor of -form circuit is

(2.16)

In the above equation is the bigger of and .The quality factor of -network can be assigned by fixing the value of R [4]. In the case of -network the R must be smaller than both and. If the value of virtual resistance R is desire to be higher than R1 and R2, -network cannot be used and then T-network is used for matching. The formulae of quality factor Q for T-network is

(2.17)

In the above equation is the lower of and [4].

## Transmission line network

When operating frequency of network increases up to few hundred megahertz then the two or three element network based on lumped element matching network are not designed. The reason is that when the operating frequency is high than the capacitor and inductor values calculated are so small which is impractical. So for higher frequency design matching networks are done by using transmission lines. Here we discuss two methods which are commonly used for amplifier design using transmission lines are

Stub matching

Single stub

Double stub

Quarter-wave transformer

## Single Stub

In this matching technique single open-circuited or short circuited length of transmission line (known as stub) can be connected in series or parallel with the transmission line at a certain distance from a load [3]. Single stub matching network is shown in figure below

Fig 2.8: Single stub matching network (a) Shunt stub (b) Series stub

In the above figure to design matching network using single stub the two parameters are required d (stub distance from the load) and L (length of stub used). In the first case when using shunt stub the distance d is choose such that the value of admittance Y seen looking into line at distance d from load is of the form Yo+jB [3]. Then for match condition choose the susceptance of stub -jB. In the second case when using series stub the distance d is choose such that the value of impedance Z seen looking into the line at distance d from load is of the form Zo+jX[3]. Then to make load matched choose stub reactance equal to -jX.

## Double Stub

In a single stub the stub distance from the load is fixed and it is function of frequency. So if the load is changed then new distance is required to match load. The double stub impedance matching network have two stub separated by fixed distance and located at fixed distance from load can overcome the problem of single stub method. The figure of double stub matching is shown below

Fig 2.9: Double stub matching [4]

In the above figure L1 is the length of stub 1 and L2 is length of stub 2.

## Quarter-wave transformer

The quarter-wave transformer is used to match real load impedance to a transmission line [3]. To design matching network at given frequency the electrical length of matching section is set. Then the formulae of input impedance is

(2.18)

In the above formula is load impedance and is input impedance to be matched with load impedance. The characteristics impedance of quarter wave matching transformer is .

## Microwave Low Noise Amplifier

Low Noise Amplifier (LNA) is used for the amplification of received signal from receiver which is typical very low. These amplifiers are especially important in the telecommunications field [10].The main purpose of LNA in receiver circuit is to amplify the small signal and it has minimum noise figure and high gain.

This chapter describes the terms related to design microwave low noise amplifier using scattering parameters. To design microwave low noise amplifier the knowledge of noise figure is important to understand. So noise figure of cascaded system is also discussed in this chapter. To achieve maximum available gain source and load impedances must be conjugately matched with input and output impedances of transistor. It is also discussed in this chapter.

## 3.1 Gain

The gain of an amplifier is always defined in terms of ratio of two powers [11com]. In two port network design of amplifier various definitions of gain are as follow

## 3.1.1 Power Gain (G)

The ratio of power dissipated in the load to the power delivered to the input of two-port network is known as Power Gain.

## 3.1.2 Operating power Gain (GP)

The ratio of power delivered to the load to the power input to the network is known as operating power gain.

(3.1)

The (3.1) is taken from [4].

## 3.1.3 Transducer Power Gain (GT)

The ratio of power delivered to the load to the power available from the source is known as Transducer Power Gain. The formula for transducer gain is [4]

(3.2)

If input and output impedances are conjugate matched then the transducer gain is equal to operating power gain. The general diagram of single stage amplifier is

Fig 3.1: General Block diagram of single-stage amplifier

## 3.2 Unilateral Amplifier Design

When reverse transmission coefficient (s12) of transistor used is insignificant then unilateral design can be used [4].

## 3.3 Bilateral Amplifier Design

A Bilateral device is one whose reverse transmission coefficient (s12) is not insignificantly small [4].

## 3.4 Stability

In the above figure (3.1) the stability of transistor depends on input and output port impedances.

If the magnitude of input and output impedance have negative values for all passive source and load impedances than network shown above is unconditionally stable.

If the magnitude of input and output port impedance have negative value for only a certain range of passive source and load impedances than network shown above is conditionally stable.

Amplifier design is said to be unconditionally stable if the magnitude of input impedance to be negative. For |Ti| 1 the Rollet coefficient K is greater than 1

(3.3)

with ||=|S11S22 - S12S21| 1 [11com] (3.4)

## 3.5 Noise

Noise can be generally generated internally and externally by the system. Noise generated internally can create a greater problem. On the other hand noise generated externally can be reduced by proper electrical design stage. So our major concern is related to noise generated internally and this cannot be removed unless the system is at absolute zero. The reason for noise generated internally within the system is due to active and passive components present. This has a number of distinct forms such as thermal, shot, and flicker noise [12].

## 3.5.1 Thermal noise

Thermal noise is caused by thermal agitation of the current carriers in the bulk material of the transistor, giving them a random motion [12].

## 3.5.2 Shot noise

Under bias condition, the transistor is no longer in thermal equilibrium and additional noise arises from the flow of electron and hole currents. This constitutes 'shot noise' which has a white spectrum at frequencies which are low in comparison with the reciprocal of the carrier transit time across depletion layer[12]

## 3.5.3 Flicker noise

It is caused by variation of leakage current and surface recombination velocity with surface properties [12].

## 3.6 Noise Figure

It is defined as the ratio of input signal to noise ratio to output signal to noise ratio defined by Friis equation [13].

(3.5)

In the above equation and are input signal and noise powers. While output signal and noise powers are denoted by and.

## 3.6.1 Noise Figure of Cascaded Systems

When two or more devices are connected then the system is called cascaded system. The general block diagram of cascaded system is

Fig 3.2:

The noise figure of cascaded system is given by

(3.6)

F1,2 is overall gain of cascaded system. F1 and F2 is the noise figure of system 1,2 respectively and G is the gain of network 1 [14].

From the above equation it is clear that noise figure of cascaded system is mainly depend on first component of system. So by choosing first component which have low noise figure and maximum gain will reduce the overall noise figure of cascaded system. So LNA is the first component in receiver circuit which has low noise figure and maximum gain to improve the noise figure of the overall system.

## Methodology

In this chapter the design of single stage amplifier is discussed which has been completed in order to develop the understanding how to design microwave low noise amplifier. It operates at 10 GHz, based on the NEC 76038 MESFET. The source and load impedances of 50 Ohms have been used and task is to obtain maximum gain available at 10 GHz. For the design of single stage amplifier Microwave Office (MWO) have been used. The S-parameter has been used to design single stage amplifier.

## 4.1 Single-Stage Amplifier

A model of single-stage amplifier design is show in figure below

In the design of single stage amplifier NEC 76038 MESFET is used as a transistor. S-parameters of MESFET used to design single stage amplifier. Microwave office is used to design single stage amplifier. In order to design single stage amplifier in MWO first import the S-parameter of transistor[]. The figure of S-parameter is shown below

From the above figure the S-parameters of MESFET is

(4.1)

(4.2)

(4.3)

(4.4)

Now add data file to schematic diagram. Add input and output port of 50 Ohms. The figure is shown below

First check the amplifier is stable at the operating frequency. The amplifier is said to be an unconditionally stable if the Rollet coefficient K is greater than 1

(4.5)

with ||=|S11S22 - S12S21| 1 [11com]. (4.6)

The result of MWO for K is shown in the following figure

Fig 4.1: Value of K on rectangular graph

In the above figure it is clear that the value of K is 1.228 at 10GHz. So amplifier is unconditionally stable at 10GHz. The maximum gain of N76038 MESFET at 10 GHz and the gain of schematic 1 shown above at 10 GHz are shown in figure below.

Fig 4.2: Maximum Gain on rectangular graph

In the above figure the maximum gain of N76038 MESFET is 10.15 dB and gain of schematic 1 is 6.888 dB. To make the gain of schematic equal to 10.15 dB source and load impedances must be conjugately matched with input and output impedances of transistor.

(4.7)

(4.8)

If the above condition satisfy then the gain will be maximize. Reverse transmission coefficient (s12) is not insignificantly small so the design implement here discussed is bilateral as discussed in previous chapter. The formulae for input and output reflection coefficient of two port network of transistor is given as [4]

(4.9)

(4.10)

By putting the condition of conjugately matched given above in

(4.11)

(4.12)

By solving two equations simultaneously the optimum source and load termination is given by [4]

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

By solving above equation the optimum source and load reflection coefficient is

(4.20)

(4.21)

Using the smith chart matching network can be determined. Stub matching scheme is used to make the two port network of transistor input and output impedances conjugately matched with the source and load impedances.

First plot on the smith chart for input matching section. The point is shown in figure below and this point represents. means the impedance from the matching network towards the source impedance, . So matching network must transform to the impedance [3]. Open-circuited shunt stub is used for matching network. First mark the point of normalized admittance and the work back towards the load on smith chart to find the line of length that will bring us to 1+jb circle. From the graph the length of line is 0.041. The stub for an open circuit length is 0.186. To match the output side same procedure is repeat and find the line of length that bring us to 1+jb circle. The line of length is 0.15. The length of stub required is 0.156. The smith chart for designing the input and output matching network to find the length of shunt stub is shown in the following figure

To find the electrical length of transmission line using the formula

For input matching circuit

Length of line that bring normalized admittance to 1+jb circle=.041

Length of stub=.186

For output matching circuit

Length of line that bring normalized admittance to 1+jb circle=.15

Length of stub=.156

The schematic diagram of conjugately matched single stage amplifier is shown in figure below