# Wideband Impedance Matching, Part I-Basics and Misconceptions.

In this article, we’ll discuss wideband impedance matching basics and a few misconceptions.

We have discussed single frequency impedance matching in details in several articles and you should read them if you would like to continue further here.

At the end of this sequence, you will be directed to download a unique proprietary program which you can use to get matching circuits for wideband frequencies and see the matching results both displayed in Smith chart and spreadsheet.

Impedance matching, differences between single frequency and wideband

Single frequency (or narrowband): theoretically you can match any type of impedance into a perfect condition with only two lossless lumped elements, capacitor and inductor, you can then optimize the matching on lab bench using standard parts.

Wideband: “Wideband” doesn’t refer to a specific, discrete frequency range, although it’s common to see it used in descriptions of microwave frequencies above 1 GHz. It’s very difficult, if is not impossible, to match the impedance to perfect condition for all frequencies within a wide frequency range using fairly simple circuits.

Furthermore, depending on how wide the frequency range is and what type of impedance will be matched, you may not be able to get a perfect match no matter how complicate the matching circuit you are willing to implement.

However, you are still able to optimize the output power for the entire frequency band by using the same simple two lossless lumped elements as we did for single frequency impedance matching.

Continue on, you’ll see clearly the results of the impedance matching in Smith chart and spreadsheet.

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Let’s review a few important basic items we mentioned in the single frequency section.

Most RF circuits such as amplifiers, transformers, isolators, couplers, dipexers, duplexers, attenuators, filters, etc. have 2 ports, input port and output port, and each port has its own impedance.

Both input and output ports need to be connected to certain external networks and, therefore, impedance matching to have the best power transfer is indispensable to RF design.

Suppose you find a power amplifier and would like to get the most power output by impedance matching both input port and output port. You most likely obtain input return loss $$S_{11}$$ and output return loss $$S_{22}$$ from the datasheets of this particular component and you wonder what is the next step you should take.

###### Get maximal power output with impedance matching

$$Z$$ (impedance, complex number, in ohms), here are 2 examples of Z:

• $$Z_{in}=R_{in}+jX_{in}$$, input impedance.
• $$Z_{out}=R_{out}+jX_{out}$$, output impedance.
• $$Z_{s}=R_{s}+jX_{s}$$, source impedance.
• $$Z_{L}=R_{L}+jX_{L}$$, load impedance.

In order to get the optimal 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   Optimal power matching between source and load

The impedance of both input and output ports needs to be matched in order to get the maximal power at the output port.

Once the impedance matching is done, both impedance seen from the source and load is 50Ω. Fig. 2   Input and output impedance matching

Common parameters

$$Z_o$$, in ohms, the characteristic impedance of the network, is 50 ohms in most RF applications. Do not leave this space blank.

Operation frequency $$F$$, in MHz, do not leave this space blank.

Return loss amplitude $$|S_{11}| \le 1$$

Return loss angle $$\angle S_{11}$$, it’s between 0° and 360°, or between -180° and 180°.

Reflection coefficient $$Γ=Γ_r+jΓ_i$$, $$Γ_r$$ is equivalent to $$S_{11x}$$ and $$Γ_i$$ is equivalent to $$S_{11y}$$. Both $$Γ_r$$ & $$Γ_i$$ are derived from $$|S_{11}|$$ and $$\angle S_{11}$$.

$$Γ_r=S_{11x}=|S_{11}|\cos \angle S_{11}$$, it is the real part of $$Γ$$ on the Smith chart.

$$Γ_i=S_{11y}=|S_{11}|\sin \angle S_{11}$$, it is the imaginary part of $$Γ$$ on the Smith chart.

Normalized impedance $$z=r+jx$$. $$r$$ and $$x$$ are the only data needed for all other worksheets to get final answers.

Both $$r$$ and $$x$$ are derived from $$Γ$$ by formulas.

Normalized admittance $$y=g+jb$$.

Both $$g$$ and $$b$$ are derived from $$r$$ and $$x$$ by formulas.

###### Impedance matching using lumped elements

For the reason of simplicity and convenience, we normalize $$Z_{in}$$ to $$z_{in}=Z_{in}/50=r+jx$$. Fig. 3   Normalized input impedance

Under a few certain situations, it would be better to transfer the impedance to admittance before applying matching process. Fig. 4   Transfer impedance to admittance

Based on the values of r, g, x, and b, we can roughly categorize the impedance into 4 different types:

• Type #1: r ≥ 1, x any value.
• Type #2: g ≥ 1, b any value.
• Type #3: r < 1, g < 1, x > 0 or b < 0.
• Type #4: r < 1, g < 1, x < 0 or b > 0.

Theoretically all these 4 types of impedance can be perfectly matched to 50Ω by using only 2 lumped elements, inductors and capacitors, if not considering the limited component values we are able to get as well as their tolerances.

Unfortunately, this is only true for single frequency operation.

###### Locate all types of impedance in the Smith chart

Each type of impedance can be conveniently and uniquely located in the Smith chart as showed below.

• Type#1: r ≥ 1, x any value. Fig. 5   Type #1 impedance location in Smith chart

• Type #2: g ≥ 1, b any value. Fig. 6   Type #2 impedance location in Smith chart

• Type #3: r<1, g<1, x>0 or b<0. Fig. 7   Type #3 impedance location in Smith chart

• Type #4: r<1, g<1, x<0 or b>0. Fig. 8   Type #4 impedance location in Smith chart Fig. 9  Four types of impedance in the Smith chart Wideband display on the Smith chart:

Most likely you get the return loss data, $$|S_{11}|$$ and $$\angle S_{11}$$, versus frequencies from the datasheets provided by manufacturer and you are only interested in a small section of frequency range based on your application. Fig. 9  Wideband input data

Figure 9 is an example of $$S_{11}$$ vs. $$F$$ which is a section extracted from a manufacturer provided data. The data cover a frequency range from 2320 MHz to 3070 MHz, a total of 750 MHz bandwidth, evenly spread on 51 points..

You’ll also learn how to provide data via  $$Γ=Γ_r+jΓ_i$$, $$z=r+jx$$, or $$Z=R+jX$$ later.

As you see in the data, $$|S_{11}$$ and $$\angle S_{11}$$ vary with frequency and we plan to match the impedance for the whole range.

For simplicity, we convert $$|S_{11}|$$ and $$\angle S_{11}$$ into $$Γ_r$$ and $$Γ_i$$ before plotting them in the Smith chart.

Refer to “Smith Charts-Basics, Parameters, Equations, and Plots.” to learn all conversions.

Figure 10 shows the data in Smith chart. Fig. 10  Wideband input data Smith chart plot

P_1 is the $$S_{11}$$ of the  lowest frequency 2320 MHz, P_51 is for the highest frequency 3070 MHz, and P_26 is for the middle frequency 2695 MHz, which is the point we will do the impedance matching.

In Part II we’ll discuss how to match wideband impedance by using Smith chart and spreadsheet as we did for single frequency operation. ‘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.’ 