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Impedance Matching and the Smith Chart: The Fundamentals
We’ll discuss the Smith Chart in this sequence and start with the very basic knowledge of this important tool that all RF people should learn and use.
You’ll not learn the mysteries of the Smith Chart, or those sophisticated formula and special usages of this great chart here.
Firstly, you’ll learn these basic parameters such as \(Z\) (impedance), \(z\) (normalized impedance), \(Y\) (admittance), \(y\) (normalized admittance), \(R\) (real part of impedance), \(X\) (imaginary part of impedance), \(r\) (real part of normalized impedance), \(x\) (imaginary part of normalized impedance), \(G\) (real part of admittance), \(B\) (imaginary part of admittance), \(g\) (real part of normalized admittance), \(b\) (imaginary part of normalized admittance), \(Γ\) (reflection coefficient), \(VSWR\) (voltage standing wave reflection), etc.
We’ll briefly mention those basic equations that construct the Smith chart.
Then, we’ll show them out on Smith Chart and learn how to easily use this great chart to help you resolve those difficult RF impedance matching issues.
What is Smith chart and how does it work?
Smith chart was invented by Phillip Smith in 1939 as a graph-based method of simplifying the complex math used to describe the characteristics of RF/microwave components, and solve a variety of RF problems.
Smith chart is really just aplot of complex reflection coefficient overlaid with a normalized characteristic impedance (1 ohm) and/or admittance (1 mho or siemen) grid.
Although calculators and computers can now easily give answers to the problems the Smith chart was designed to solve, this great chart still remains a valuable tool.
It will massively improve your RF skills if you are able to take time to learn how to use this chart .
Understanding those basic parameters
- \(Z\) (impedance, complex number, in ohms), here are 2 examples of Z:
- \(Z_s=R_s+jX_s\), source impedance.
- \(Z_L=R_L+jX_L\), load impedance.
In order to get the best power transfer from a source to a load, the source impedance must equal the complex conjugate of the load impedance:
\(R_s+jX_s=R_L-jX_L\), so \(R_s =R_L\) and \(X_s=-X_L\)
Fig. 1 Best power matching between source and load
- \(Z_0=R_0\), characteristic impedance, is often a real industry normalized value, such as 50Ω (RF/microwave) and 75Ω (cable), etc.
- \(Γ\) (gamma, reflection coefficient):the reflection coefficient is defined as the ratio between the reflected voltage wave and the incident voltage wave:
$$Γ={V_r\over V_i}={{Z_L-Z_0}\over {Z_L+Z_0}}=Γ_r+jΓ_i$$
- \(V_i\) is incident voltage.
- \(V_r\) is reflected voltage.
Fig. 2 Reflection coefficient (Γ)
Fig. 3 4 points on \(Γ\) coordinates
Point A:
\(Γ=0.3 + j 0.5\)
\(|Γ|=\sqrt{(0.3^2+0.5^2)}=0.583\)
\(\angle {Γ}=\arctan (0.5/0.3)=59.0^{\circ}\)
Conversely, if \(|Γ|\) and \(∠Γ\) are provided as most datasheets do, then
\(Γ_r=|Γ|\cos ∠Γ=0.583 \cos 59.0^{\circ}=0.3\)
\(Γ_i=|Γ|\sin ∠Γ=0.583 \sin 59.0^{\circ}=0.5\)
Point B:
\(|Γ|=0.671\)
\(\angle {Γ}=153.4^{\circ}\)
And,
\(Γ_r=|Γ|\cos ∠Γ=0.671 \cos 153.4^{\circ}=-0.6\)
\(Γ_i=|Γ|\sin ∠Γ=0.671 \sin 153.4^{\circ}=0.3\)
So,
\(Γ=-0.6+j0.3\)
Point C:
\(|Γ|=0.721\)
\(\angle {Γ}=-123.7^{\circ}\)
And,
\(Γ_r=|Γ|\cos ∠Γ=0.721 \cos (-123.7^{\circ})=-0.4\)
\(Γ_i=|Γ|\sin ∠Γ=0.721 \sin (-123.7^{\circ})=-0.6\)
So,
\(Γ=-0.4-j0.6\)
Point D:
\(|Γ|=0.86\)
\(\angle {Γ}=-54.5^{\circ}\)
And,
\(Γ_r=|Γ|\cos ∠Γ=0.86 \cos (-54.5^{\circ})=0.5\)
\(Γ_i=|Γ|\sin ∠Γ=0.86 \sin (-54.5^{\circ})=-0.7\)
So,
\(Γ=0.5-j0.7\)
- \(z\), normalized load impedance.$$z={Z_L\over Z_0}={{R_L+jX_L}\over {Z_0}}=r+jx$$
- \(r\) (resistance, real part of \(z\))
- \(x\) (reactance, imaginary part of \(z\))
Fig. 4 \(Γ\) and impedance normalization
So,
$$Γ=Γ_r+jΓ_i={{Z_L-Z_0}\over {Z_L+Z_0}}$$
$$={{z-1}\over {z+1}}={{r+jx-1}\over {r+jx+1}}$$
After a few manipulation steps, we get,
$${Γ_r}={{r^2-1+x^2}\over {{(r+1)^2}+x^2}}…..(1)$$
$${Γ_i}={2x\over {{(r+1)^2}+x^2}}…..(2)$$
$$r={{1-{Γ_r}^2-{Γ_i}^2}\over {({1-{Γ_r}})^2+{Γ_i}^2}}…..(3)$$
$$x={{2{Γ_i}}\over {({1-{Γ_r}})^2+{Γ_i}^2}}…..(4)$$
and,
$${{({Γ_r}-{r\over {r+1}})^2}+{Γ_i}^2}=({1\over {1+r}})^2…..(5)$$
$${({Γ_r}-1)^2}+{({Γ_i}-{1\over x})^2}={1\over {x^2}}………(6)$$
Equations (5) & (6) represent 2 separate sets of impedance circles, \(r\) circles and \(x\) circles.
Fig. 5 r circles
Fig. 6 x circles
For the ease of impedance matching, we need to add on 2 parameter plots on Smith chart, \(g\) & \(b\):
- \(Y\) (admittance, complex number, in siemens).
$$Y={1\over Z}=G+jB$$
- \(G\) (conductance, real part of \(Y\), in siemens).
- \(B\) (susceptance, imaginary part of \(Y\), in siemens).
- \(y\), normalized load admittance.$$y={1\over z}={1\over {r+jx}}$$ $$={r\over {r^2+x^2}}+j{-x\over {r^2+x^2}}$$ $$=g+jb$$
- \(g\) (conductance, real part of \(y\)), $$g={r\over {r^2+x^2}}…..(7)$$
- \(b\) (susceptance, imaginary part of \(y\)), $$b={-x\over {r^2+x^2}}…..(8)$$
Since,
$$Γ={{z-1}\over {z+1}}={{{1/y}-1}\over {{1/y}+1}}$$
$$={{1-y}\over {1+y}}={{1-g-jb}\over {1+g+jb}}$$
After a few manipulation steps, we get,
$${Γ_r}={{1-g^2-b^2}\over {{(g+1)^2}+b^2}}…..(9)$$
$${Γ_i}={-2b\over {{(g+1)^2}+b^2}}…..(10)$$
$$g={{1-{Γ_r}^2-{Γ_i}^2}\over {({1+{Γ_r}})^2+{Γ_i}^2}}…..(11)$$
$$b={{-2{Γ_i}}\over {({1+{Γ_r}})^2+{Γ_i}^2}}…..(12)$$
and,
$${{({Γ_r}+{g\over {g+1}})^2}+{Γ_i}^2}=({1\over {1+g}})^2…..(13)$$
$${({Γ_r}+1)^2}+{({Γ_i}+{1\over b})^2}={1\over {b^2}}………(14)$$
Equations (13) & (14) represent 2 separate sets of admittance circles, \(g\) circles and \(b\) circles.
Fig. 6 g circles
Fig. 7 b circles
If we put together all 4 parameter circle plots (r, x, g, andb), we then have a handy and workable Smith chart.
Fig. 8 The Smith chart
- \(VSWR\) (Voltage Standing Wave Ratio),a function of the reflection coefficient \(Γ\), which describes the power reflected from the antenna.The reflection coefficient is also known as s11 or return loss.
$$VSWR={{1+|Γ|}\over {1-|Γ|}}….(15)$$
And each\(VSWR\) constant represents a circle in the Smith chart.
If a load is purely resistive \({Z_L}={R_L}\), but unequal to the characteristic impedance of the transmission line \(Z_0\), the \(VSWR\) is given simply by their ratio:
$$VSWR={({{R_L}\over {Z_0}})}^{\pm 1}=r…..(16)$$
with the \(±1\) chosen to obtain a value greater than unity.
Combining equations (15) & (16), we are able to easily read a \(VSWR\) circle in the Smith chart.
Let’s take \(VSWR=3.0\) as an example:
This circle passes point (0.5, 0) and touches r=3.0 circle.
\({Γ_r}=0.5\),\({Γ_i}=0\), so
$$|Γ|={\sqrt {0.5^2+0^2}}=0.5$$ and,
$$VSWR={{1+|Γ|}\over {1-|Γ|}}={{1+0.5}\over {1-0.5}}=3.0$$
And at point (0.5, 0),
$$r={{1-{Γ_r}^2-{Γ_i}^2}\over {1+{Γ_r}^2-{2Γ_r}+{Γ_i}^2}}$$
$$={{1-{Γ_r}^2}\over {1+{Γ_r}^2-{2Γ_r}}}={{(1+Γ_r)(1-Γ_r)}\over {{(1-{Γ_r})}^2}}$$
$$={{1+Γ_r}\over {{1-{Γ_r}}}}={{1+0.5}\over {{1-0.5}}}=3.0$$
Therefore, we can determine the value of a \(VSWR\) circle by reading the number of an r circle which it touches at \({Γ_r}\) coordinate.
Fig. 9 \(VSWR\) circles
Or, conversely, we can draw a \(VSWR\) circle by following these steps, here we take \(VSWR=2.0\) as an example:
- Find the r=2.0 circle in the Smith chart.
- Locate the cross point of this r circle with \(Γ_r\) coordinate, and we find it’s 0.33.
- Centered at the origin, draw a circle with radius of 0.33, then it’s the \(VSWR=2.0\) circle.
Let’s verify the validity of this circle:
Since point (0.33,0) is on this circle, \(|Γ|=0.33\).
Apply Equation (5),
$$VSWR={{1+|Γ|}\over {1-|Γ|}}={{1+0.33}\over {1-0.33}}=1.985 \approx 2.0$$
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Now you have learned all Smith chart basics and are excited to find out if you are able to use it in the RF field. Let’s go on to to see a few Examples and Questions to know better about this great chart.
‘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.’
