2.3 the smith chart and impedance...

Antennas: from Theory to Practice Antennas: from Theory to Practice 34/87

2.3 The Smith Chart and Impedance Matching


The Smith Chart


The Smith Chart can be utilized to represent many parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and unconditional stability regions.

graphical aid designed for RF engineers to solve transmission line and matching circuit problems

Figure 2.10 The standard Smith Chart

Toward generator

Short

Toward load

Open

Matched


The Smith Chart


the complex reflection coefficient plane in two dimensions

In the standard Smith Chart, only the circle for | Γ | = 1 is shown.

Figure 2.11 The Smith Chart showing the complex reflection coefficient


The Smith Chart


Most information shown on the Smith Chart is actually the normalized complex impedance.

A locus of points on a Smith Chart covering a range of frequencies can be employed to visually represent:

Figure 2.12 The Smith Chart showing the complex impedance

Constant resistance

Constant inductive

Constant capacitive

how capacitive or inductive a load is across the frequency range;

how difficult matching is likely to be at various frequencies;

how well matched a particular component is.


The Smith Chart


Example 2.7: Input impedance and reflection coefficient. Use a Smith Chart to

redo Example 2.1, and also display the reflection coefficient on the chart.

moving along the |Γ|=0.2

circle clockwise


Impedance Matching


impedance matching

to maximize the power transfer and minimize reflections from the load.

the load impedance being the complex conjugate of the source impedance

When the imaginary part is zero,

Normally, we can use either lumped networks or distributed networks to match impedance.


Impedance Matching


Lumped Matching Networks

L network

>1

>1

(b) for Rin GL < 1

(a) for Rin > RL

Figure 2.14 Lumped L networks no degree of freedom to optimize the bandwidth


Impedance Matching


T network

Step 1: according to the load impedance and the desired bandwidth, choose X1 ,

Step 2: since ZL and jX1 are in series,

Step 3: use ZLN and the L network to find B and X2

Figure 2.15 Lumped T network


Impedance Matching


π network

Step 1: according to the load impedance and the desired bandwidth, choose B1,

Step 2: since YL and jB1 are in parallel,

Step 3: use YLN and the L network to find X and B2

Figure 2.16 Lumped π network


Impedance Matching


• Since ZL = RL + j XL = 10 − j100 and n = Rin/RL = 50/10 = 5 > 1, use L network in Figure 2.14(a)

• Because YL=1/(RL+ j XL) ≈ 0.001+ j 0.001 and m =1/(Rin GL )=1/0.05= 20 > 1, use L network in Figure 2.14(b)

Example 2.8: Impedance matching. A load with an impedance of 10 − j100 is to be matched

with a 50 transmission line. Design a matching network and discuss if there are other solutions

available.


Impedance Matching


Distributed Matching Networks

Distributed matching networks can be formed by a λ/4 TL , an open-circuit transmission line, a short-circuit transmission line or their combinations.

The process is best visualized on the Smith Chart.


Impedance Matching


zB1 = 0.0413 − j0.1984 , while yB1 = 1.0 + j4.8318.

Step 1: Move from point A to B1, the rotational angle is about 0.582π.

Step 2: Move from point B1 to the center O.

Example 2.9: Impedance matching and bandwidth. A load with an impedance of 10 − j100 is to be matched with a 50 transmission line. Design two distributed matching networks and

compare them in terms of the bandwidth performance. Assuming the center frequency is 1 GHz.

Figure 2.17 Impedance matching using a Smith Chart

Inductive

Capacitive

This can be achieved easily using a stub

connected in parallel with the line.

l1 = 0.1455λ = 4.365 cm


Impedance Matching


The stub in parallel with the line should produce a susceptance of −4.8318.

Open or short

Open or short Load location

Stub

Ground plane

Shorting pin

Transmission line

(a) parallel stub matching (b) series stub matching

Figure 2.18 Stub-matching networks

A. a short circuit with a stub length l2 = 0.0325λ = 0.975 cm;

B. an open circuit with a stub length l2 = 0.2825λ = 8.475 cm.


Impedance Matching


Impedance Matching


VSWR as a function of the frequency

Both designs have an excellent impedance match at the center frequency 1 GHz.

The stub length of Design A is shorter than that of Design B whilst the bandwidth

of Design A is much wider than that of Design B.


Impedance Matching


Bode-Fano limit

for parallel RC,

Figure 2.20 Four load impedances with LC matching networks

for series RL,

for series RC,

for parallel RL,


The Quality Factor and Bandwidth


quality factor

A low Q is required for wide bandwidths.

Antennas are designed to have a low Q, whereas circuit components are designed for a high Q.

unloaded quality factor at resonance Q0

(average power dissipated)

unloaded quality factor

where BF is the fractional bandwidth.




magnetic and electric energies, and average power

Series resonant circuit

unloaded quality factor of the circuit

Figure 2.21 Series resonant circuit.

Figure 2.22 Relative power dissipated in a series resonant circuit

Frequency (GHz)

ratio of Q at any frequency to that at resonance

,




current in the circuit

ratio of the power dissipated at any frequency to the power dissipated at resonance




fractional bandwidth

This derivation is, therefore, applicable to both high- and low-Q systems.

When p = 0.5,




magnetic and electric energies, and average power

Parallel resonant circuit

unloaded quality factor of the circuit

Figure 2.23 Parallel anti-resonant circuit.


2.4 Various Transmission Lines


2.4 전송선로의 종류

Various transmission lines


Two-wire Transmission Line


per unit length inductance and capacitance

resistance and conductance of a unit length line

Figure 2.25 Two-wire transmission line

where the conductivity of the medium is σ1, and the conductivity of the wire is σ2.


Two-wire Transmission Line


Fundamental Mode

Characteristic Impedance

TEM (transverse electro magnetic) mode – it is nondispersive and the velocity is not changed with frequency.

Loss

The principal loss of the two-wire transmission line is actually due to radiation.

The typical usable frequency is less than 300 MHz.


Coaxial Cable


per unit length parameters of the coaxial line

Figure 2.26 The configuration of a coaxial line

Outer jacket

Copper core

Insulating

material

Braided outer

conductor Protective

plastic

covering


Coaxial Cable


Phase velocity

conductivity of the medium – , conductivity of the wire –


Coaxial Cable


Characteristic Impedance

cut-off frequency and cut-off wavelength

Fundamental Mode

TEM mode

Above the cut-off frequency, some higher modes such as TE11 mode may exist, which is not a desirable situation since the loss could be significantly increased.

Figure 2.27 Field distribution within a coaxial line


Coaxial Cable


Loss

When b/a ≈ 3.592 (which means that the typical characteristic impedance should be around 77 Ωs), the attenuation reaches the minimum.

The breakdown electric field strength in air is about 30 kV/cm (this means the best characteristic impedance should be close to 30 Ωs).

attenuation constant


Coaxial Cable



Microstrip Line


most widely used form of planar transmission line

effective relative permittivity

an empirical expression

Figure 2.28 Microstrip line


Microstrip Line


when W/d < 1

Characteristic impedance

relation between the velocity and per unit length inductance and capacitance

characteristic impedance

when W/d < 1


The first higher mode in a microstrip line is the transverse electric TE10 mode, its cut-off frequency is

The lowest transverse electric mode is TE1 (surface mode) and its cut-off frequency is

The lowest transverse magnetic mode is TM0 and its cut-off frequency is

Figure 2.29 The field distribution of a microstrip

Microstrip Line


Fundamental mode

Quasi-TEM mode – half of the wave is traveling in free space, which is faster than the other half wave traveling in the substrate.


Microstrip Line


surface wave

cut-off frequency of TMn and TEn modes

cut-off frequency of TE1 modes

cut-off frequency of TM0 modes


Microstrip Line


Loss

attenuation constant

( : surface resistivity)


Stripline


better bandwidth, isolation than a microstrip,

It is much harder and more expensive to fabricate than the microstrip.

The strip width is much narrower for given impedance (such as 50 Ωs) and the board is thicker than that for a microstrip.

Figure 2.30 From a coaxial cable to a stripline


Stripline



Fundamental mode

Loss

TEM mode

The smallest wavelength to avoid higher order modes

The loss characteristics of the stripline are similar to the microstrip but have little loss due to radiation.


Coplanar Waveguide (CPW)


It is easy to fabricate and to integrate into circuits.

It can work to extremely high frequencies (100 GHz or more).

Good circuit isolation can be achieved using a CPW.

One disadvantage is potentially lousy heat dissipation.

CPW circuits can be lossier than comparable microstrip circuits if a compact layout is required.

Figure 2.31 Evolution from a coaxial cable to CPW (G: gap: W: width; d: substrate height)





characteristic impedance

effective permittivity

• complete elliptical function of the first kind




Fundamental mode

quasi-TEM mode

Loss

CPW exhibits a higher loss than its microstrip counterpart.

The current on the ground planes is also very focused in a small area, which results in a relatively high conductor loss as well as a heat dissipation problem.

Higher order modes and surface modes may be generated in a CPW just as in a microstrip line.


Waveguide


just one piece of metal, which is tubular, usually with a circular or rectangular cross-section.

Due to the boundary conditions, there are many possible wave patterns, which are called TEmn modes and TMmn modes.

The waveguide can be considered a high-pass filter and is used for microwave and millimeter wave frequency bands.

low loss and high power-handling capacities, which are very important for high-power applications such as radar.

Figure 2.32 Rectangular waveguide

TE10 mode


Waveguide


Fundamental mode

TE10 mode

electric field

magnetic field

TE10 mode


Waveguide


cut-off wavelength and cut-off frequency

standard waveguides


Waveguide


waveguide wavelength

The characteristic impedance is also mode-dependent.

For TE10 mode,


2.5 Connectors


2.5 커넥터

There are many types of industrial standard connectors.

Some adapters have been developed.

SMA, SMB, BNC, Type N, etc.

Since all RF test equipment comes with coaxial connectors (type N and SMA are popular connectors), direct connection with other forms of transmission lines (such as microstrip and CPW) would be tricky.

Connectors are developed as a pair: a male and a female.

The effects of the connector on the system performance and measurements may be quite significant.

Figure 2.33 Male (left) and female (right) N-type connectors Figure 2.34 Wideband antennas fed by CPW and microstrip,

which are directly connected/soldered to SMA connectors


2.5 커넥터

2.3 the smith chart and impedance...

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