RF engineering basic concepts: the Smith chart F.Caspers CERN, Geneva, Switzerland Abstract The Smith chart is a very valuable and important tool that facilitates interpre-tation of S-parameter measurements. This paper will give a brief overview on why and more importantly on how to use the chart. Its definition as well. Smith Charts B/W and colour smith and admittance charts. Smith Chart.pdf: Colour Smith Chart.pdf: References.Phillip Hagar Smith (April 29, 1905–August 29, 1987) Inventor of the 'Smith Chart' - 1939.Phillip H. Smith - 'Electronic Applications of the Smith Chart'. The Smith Chart Utility provides full Smith Chart capabilities, synthesis of matching networks, enabling impedance matching and plotting of constant Gain/Q/VSWR/Noise circles. The Smith Chart Utility is accessed from the Schematic window Tools or DesignGuide menus. The Smith Chart Utility documentation includes these sections. The Smith chart is one of the most important tools in understanding RF impedance and matching networks. This brief tutorial explains what the Smith chart is.
- Smith Chart Basics
- Smith Chart Pdf
- Smith Chart Calculator
- Smith Chart Basics
- Smith Chart Black Magic Design
The Smith Chart is simply a graphical calculator for computing impedance as a function of re ection coe000ecient z = f(ˆ) More importantly, many problems can be easily visualized with the Smith Chart This visualization leads to a insight about the behavior of transmission lines All the knowledge is coherently and compactly represented by the Smith Chart Why else study the Smith Chart?
The Chart known as the Smith Chart was the work of Philip Smith and Mizuhashi Tosaku, who seem to have developed it independently of each other. From the beginning of World War II until the development of digital computers for engineering problems, the Smith Chart was the dominant tool for microwave engineers.
This simplified version of a Smith Chart shows both resistance in ohms (numbers on the horizontal axis that range from 0.2 to 10) and the angle of a quantity called the reflection coefficient, in degrees on the outer edge of the circle.
The microwave engineer’s objective in many instances is to get as much microwave power as possible through electrical obstacles on its way to the goal of being transmitted from an antenna or being amplified in a receiver.
Smith originally designed his chart to solve problems he encountered in transmitting radio waves along a special type of cable called a transmission line that conducts the waves from a radio transmitter to an antenna. Ideally, all the waves would go smoothly along the line in one direction and no reflections would occur. However, with many types of antennas, some of the waves are reflected back into the line. These reflected waves can bounce back and forth along the line and are eventually lost, reducing the amount of power transmitted.
Smith tried to graph the effects of these reflections on an early type of rectangular chart developed by J. A. Fleming for telephone lines, but he kept getting numbers that went off the top and bottom of the chart. In 1936, he had the idea of using a circular chart and transforming the reflections mathematically so that no matter how large they were, the numbers would all fit inside the circular boundary of the chart. For this and other reasons, Smith’s new circular chart made many microwave engineering problems in circuits as well as in transmission lines easy to solve graphically (that is, without doing a lot of math). In the days before electronic digital computers, any technique that allowed an engineer to avoid tedious calculations was very valuable, so Smith’s chart became popular among radio and microwave engineers within a few years after its publication in Electronics magazine in January of 1939.
Mizuhashi Tosaku's article describing his chart appeared in December 1937 in the Journal of Electrical Communications Engineers of Japan
In the 1970s Smith formed a company, Analog Instruments, which merchandised navigational instruments for light aircraft and later supplied Smith Charts and chart-related items. By 1975 he had sold about 9 million copies of his chart to microwave engineers all over the world. Even though computers are now the dominant design tools, the Smith Chart remains vital to the field of microwaves. Its usefulness continues to this day as a method of displaying measured and calculated data produced by computer software and modern measurement instruments.
Examples and Questions:
This article will be dedicated to examples and questions to find out how much you really know about Smith chart and enhance your understanding of this great chart.
But you should learn all Basic Parameters, Equations, and Plots first before continuing reading further here.
We’ll work on examples and then ask questions.
<<<<<<<<<<>>>>>>>>>>
Obtain (Γ_r) and (Γ_i) from given (r) and (x):
(z), normalized load impedance,
$$z={Z_Lover Z_0}={{R_L+jX_L}over {Z_0}}=r+jx….(1)$$
Given (Z_L), and find (Γ), reflection coefficient,
$$Γ=Γ_r+jΓ_i={{Z_L-Z_0}over {Z_L+Z_0}}={{z-1}over {z+1}}….(2)$$
Smith Chart Basics
Or, obtain (Γ_r) and (Γ_i) directly from given (r) and (x):
$${Γ_r}={{r^2-1+x^2}over {{(r+1)^2}+x^2}}….(3)$$
$${Γ_i}={2xover {{(r+1)^2}+x^2}}….(4)$$
Fig. 1 (Γ) and impedance normalization
Fig. 2 (z) circles and (Γ)
Example #1:
Q. Point A in Fig. 2, if the characteristic impedance (Z_0) is 50 ohms, and the load impedance (Z_L={R_L+jX_L}=50+j50),
calculate 1) (z), and 2) (Γ)?
Ans.
1) $$z=r+jx={Z_Lover Z_0}={{50+j50}over 50}=1.0+j1.0$$
So, (r=1.0) and (x=1.0).
2) $$Γ={{Z_L-Z_0}over {Z_L+Z_0}}={{(50+j50)-50}over {(50+j50)+50}}$$
$$={{j50}over {100+j50}}={{(j50)(100-j50)}over {(100+j50)(100-j50)}}$$
$$=0.2+j0.4=Γ_r+jΓ_i$$
Or,
$$Γ={{z-1}over {z+1}}={{1.0+j1.0-1}over {1.0+j1.0+1}}$$
$$={{j1.0}over {2.0+j1.0}}=0.2+j0.4$$
So, ({Γ_r}=0.2), ({Γ_i}=0.4)
Or, applying Equations (3) & (4),
$${Γ_r}={{r^2-1+x^2}over {{(r+1)^2}+x^2}}={{1.0^2-1+1.0^2}over {{(1.0+1)^2}+1.0^2}}=0.2$$
$${Γ_i}={2xover {{(r+1)^2}+x^2}}={{2times 1.0}over {{(1.0+1)^2}+1.0^2}}=0.4$$
The answers match with the plot in Fig. 2.
Question #1:
Q. Point B in Fig. 2, calculate (Γ) if the characteristic impedance (Z_0) is 75 ohms, and the load impedance (Z_L=15+j45).
Ans. (Γ=-0.333+j0.667)
<<<<<<<<<<>>>>>>>>>>
Obtain (r) and (x) from given (Γ_r) and (Γ_i):
For any (Γ), there is a corresponding (z) and we can find (r) and (x) by using these 2 equations:
$$r={{1-{Γ_r}^2-{Γ_i}^2}over {({1-{Γ_r}})^2+{Γ_i}^2}}….(5)$$
$$x={{2{Γ_i}}over {({1-{Γ_r}})^2+{Γ_i}^2}}….(6)$$
Fig. 2 (z) circles and (Γ)
Example #2:
Q. Point C in Fig. 2, (Z_0=50), and (Γ=-0.32-j0.38),
calculate 1) (z), and 2) (Z)?
Ans.
1)
Equation (5),
$$r={{1-{Γ_r}^2-{Γ_i}^2}over {({1-{Γ_r}})^2+{Γ_i}^2}}$$
$$={{1-{(-0.32)}^2-{(-0.38)}^2}over {({1-{(-0.32)}})^2+{(-0.38)}^2}}=0.4$$
Equation (6),
$$x={{2{Γ_i}}over {({1-{Γ_r}})^2+{Γ_i}^2}}$$
$$={{2times (-0.38)}over {({1-{(-0.32)}})^2+{(-0.38)}^2}}=-0.4$$ Bentel security port devices driver.
So,
(z=0.4-j0.4)
As it is showed in Fig. 2.
2)
({R_L}={0.4 times 50}=20 Ω)
({X_L}={-0.4 times 50}=-20 Ω)
So,
(Z_L={20-j20})
Question #3:
Q. Point D in Fig. 2, (Z_0=50), and (Γ=0.25-j0.25),
calculate (z).
<<<<<<<<<<>>>>>>>>>>
Fig. 3 (z) to (y) conversion
Fig. 4 Full Smith chart, (z) & (y) circles
Every impedance (Z) has a corresponding admittance (Y) and,
(Y={1/Z}), also, after normalization,
(y={1/z})
Fig. 5 (z) & (y) conversions
Obtain (g) and (b) from given (r) and (x):
If (z=r+jx) and (y=g+jb), then,
$$y=g+jb={1over z}={1over {r+jx}}$$
And,
$$g={rover {r^2+x^2}}….(7)$$
$$b={-xover {r^2+x^2}}….(8)$$
<<<<<<<<<<>>>>>>>>>>
Obtain (r) and (x) from given (g) and (b):
Conversely,
$$z=r+jx={1over y}={1over {g+jb}}$$
And,
$$r={gover {g^2+b^2}}….(9)$$
$$x={-bover {g^2+b^2}}….(10)$$
Example #4:
Q. Point A in Fig. 5, (z=0.4-j0.3),
calculate 1) (y), and 2) if (Z_0) is 50Ω, what is (Y)?
Ans.
1)
(r=0.4) and (x=-0.3)
Equation (7), $$g={rover {r^2+x^2}}={0.4over {0.4^2+(-0.3)^2}}=1.6$$
Equation (8), $$b={-xover {r^2+x^2}}={0.3over {0.4^2+(-0.3)^2}}=1.2$$
As showed in Fig. 5.
2)
(Y=1/Z), and ({Y_0}={1/{Z_0}}={1/50}=0.02).
Since (y={Y/{Y_0}}), so (Y={Y_0}times y).
$$Y={{Y_0}times (g+jb)}={0.02times (1.6+j1.2)}$$
$$={0.032+j0.024}$$
Example #5:
Q. Point B in Fig. 5, (y=1.0-j1.0), calculate (z)?
Ans.
Equation (9),
$$r={gover {g^2+b^2}}={1.0over {1.0^2+(-1.0)^2}}=0.5$$
Equation (10),
Smith Chart Pdf
$$x={-bover {g^2+b^2}}={1.0over {1.0^2+(-1.0)^2}}=0.5$$
So, (z=0.5+j0.5).
As showed in Fig. 5.
<<<<<<<<<<>>>>>>>>>>
Obtain (Γ_r) and (Γ_i) from given (g) and (b):
$${Γ_r}={{1-g^2-b^2}over {{(g+1)^2}+b^2}}….(11)$$
$${Γ_i}={-2bover {{(g+1)^2}+b^2}}….(12)$$
Example #6:
Q. Point C in Fig. 5, (y=0.6+j0.2), calculate (Γ)?
Ans.
$${Γ_r}={{1-g^2-b^2}over {{(g+1)^2}+b^2}}$$
$$={{1-0.6^2-0.2^2}over {{(0.6+1)^2}+0.2^2}}=0.231$$
$${Γ_i}={-2bover {{(g+1)^2}+b^2}}$$
$$={{-2times 0.2}over {{(0.6+1)^2}+0.2^2}}=-0.154$$
So, (Γ=0.231-j0.154)
As showed in Fig. 5.
<<<<<<<<<<>>>>>>>>>>
Obtain (g) and (b) from given (Γ_r) and (Γ_i):
$$g={{1-{Γ_r}^2-{Γ_i}^2}over {1+{Γ_r}^2+{2Γ_r}+{Γ_i}^2}}….(13)$$
$$b={{-2{Γ_i}}over {1+{Γ_r}^2+{2Γ_r}+{Γ_i}^2}}….(14)$$
Example #7:
Q. Point D in Fig. 5, (Γ=0.4+j0.2), calculate (y)?
Ans.
$$g={{1-{Γ_r}^2-{Γ_i}^2}over {1+{Γ_r}^2+{2Γ_r}+{Γ_i}^2}}$$
$$={{1-0.4^2-0.2^2}over {1+0.4^2+{2times 0.2}+0.2^2}}=0.4$$
$$b={{-2{Γ_i}}over {1+{Γ_r}^2+{2Γ_r}+{Γ_i}^2}}$$
$$={{-2times 0.2}over {1+0.4^2+2times 0.4+0.2^2}}=-0.2$$
So, (y=0.4-j0.2)
As showed in Fig. 5.
Now, you have learned all basics of Smith chart and you know the chart is consisted of 3 very basic parameters, (Γ, z, y), and they can be converted among each other based on a few sophisticated equations.
With any one of these 3 parameters given, you can read the other 2 in the chart simultaneously with a very reasonable accuracy.
Bangho laptops & desktops driver download for windows 10. You don’t need to remember those equations by heart but you should know how to apply them without any difficulty, whenever you need to use it to conveniently solve impedance and reflection issues.
If you can follow those 6 examples easily, then you are good to go to answer the questions below, and, once you get them done correctly, you can continue to learn the most exciting application of Smith chart, impedance matching.
You’ll learn a unique way of impedance matching using Smith chart, and you will also be directed to where you can get a powerful spreadsheet which will help you get the matching job solved within a fraction of a second with very minimal effort.
However, do not force yourself ahead to the next article if you have difficulty to answer the following questions. You should go back to review this article-Basics, Parameters, Equations, and Plots, then come back to practice on this article again.
After that, you will be well prepared for learning impedance matching.
Let’s charge ahead to answer these simple questions below.
<<<<<<<<<<>>>>>>>>>>
Fig. 6 Smith chart Drivers boo-ree port devices.
Unless otherwise stated, (Z_0=50Ω).
Question #3:
If (Γ=-0.4+j0.25), then what are:
- (z)?
- (y)?
- (Z)?
- (Y)?
- Location on the Smith chart?
Ans.
- (z=0.384+j0.247)
- (y=1.842-j1.185)
- (Z=19.22+j12.36)
- (Y=0.037-j0.024)
- Point A in Fig. 7.
Question #4:
If (Z=85-j35), then what are:
Smith Chart Calculator
- (Y)?
- (Γ)?
- Location on the Smith chart?
Ans.
- (Y=0.01+j0.004)
- (Γ=0.306-j0.18)
- Point B in Fig. 7.
Question #5:
If (Y=0.04+j0.028), then what are:
- (Z)?
- (Γ)?
- Location on the Smith chart?
Ans.
- (Z=16.78-j11.74)
- (Γ=-0.453-j0.255)
- Point C in Fig. 7.
Question #6:
Smith Chart Basics
If (z=1.4+j1.2), then, read directly from the Smith chart without using equations, what are the approximate values of:
- (Γ)?
- (y)?
- Location on the Smith chart?
Ans.
- (Γ=0.33+j0.33)
- (y=0.41-j0.35)
- Point D in Fig. 7.
Fig. 7 Q & A Smith chart
Smith Chart Black Magic Design
‘Note: This is an article written by an RF engineer who has worked in this field for over 40 years. Visit ABOUT to see what you can learn from this blog.’