The Volterra series is a mathematical tool to model RF amplifiers with memory and in general, any non linear system with memory. In many ways the Volterra series is analogous to the Taylor series for systems with no memory. The Taylor series has coefficients multiplying each power of the independent variable. The Volterra series has “kernels”. Volterra kernels are difficult to calculate beyond the third order so most analysis only uses that as the maximum limit in analysis. The Volterra series was used to develop the behavioral model of a RF power amplifier. Please visit the SPG website for more articles and items of interest.

# Author Archives: Ain

# Impedance matching using single stubs

A new book was published that describes the process of using a single stub to match a load with a transmission line. The method is simple but requires some understanding. This book describes the methodology, presents an example and presents a completely simulated design using easily available public domain tools. The book can be purchased from Amazon. To browse follow this link:https://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=impedance+matching+using+single+stub

Please also visit the Signal Processing website for more items of interest.

# Cascaded line matching of impedance with ideal transmission lines simulation.

A cascaded transmission line is one of the simplest ways to match impedances. It is simply a quarter wave transmission line with a characteristic impedance given by : Z0 = sqrt(ZS.ZL) where ZS is the source impedance and ZL is the load impedance. A simulation using a public domain simulation CAD tool was performed for s11. The result is shown below:

From the figure the frequency was 5 Ghz. The schematic of this design is shown below:

Note that the impedance of the line is: sqrt(50*100). The length of the line is a quarter wavelength ( 1 wavelength = 60 mm at 5 Ghz).

For more information on impedance matching please refer to the available book : “VSWR and Impedance matching techniques”, available from Amazon,

In addition a forthcoming book that includes many other techniques and scripts is in preparation.

Please visit our website for more information on our offerings and more technical information/articles.

A low noise amplifier module with NF = 1 dB across the 100 Mhz to 3.0 Ghz band with typical 17.7 dB small signal gain over the band.

# Bipolar switching circuit performance parameters

As analog engineers focus on parameters such as the transition frequency of a bipolar device to estimate frequency performance of a transistor ( which is more of a small signal parameter). We sometimes get confused if we have to use a bipolar as a switching device. Especially in a saturated mode of switching. In this case the base of the bipolar is driven hard enough ( lots of base current) to drive the transistor when it is ON to store charge in its base region. The output voltage at the collector drops to its Vsat ( a few hundred millivolts usually) and when the transistor has to leave this state it has to contend with a storage time. So when we want to estimate the switching speed of a bipolar we need to use the datasheet parameters of the device that apply to the switching mode. These parameters are: td = delay time , tr = rise time, tf = fall time and ts = storage time. Then the maximum switching frequency of the bipolar is given by ( estimate), fmax=1/( td+tr+tf+ts). Note that ft, the transition frequency does not enter into this equation at all! For more technical info and services please visit our website.

# RFMW design: Comparing Characteristic impedance results for a microstrip from multiple calculators

We took 4 different available calculators and compared results obtained for Zo on a PCB microstrip. Results were that the SPG package and APPCAD results were very close, TXLINE results were larger than the SPG and APPCAD results, while the wCalc package was lower than the SPG and APPCAD numbers. (wCalc does provide some very valuable parameters as a result of the simulation: R, L, C and G parameters for the microstrip). Here are the comparisons.

Please visit the SPG website for more information and reports.

# Using a directional coupler to measure an unknown impedance

Lets measure RL (return loss) using a bi-directional coupler. In this case we are using the BDCN-20-13+ from Mini-Circuits For this coupler key parameters are:

Coupling port power 20 dB, Directivity 12 dB. Main line loss between 370 Mhz and 470 Mhz is 0.25 dB max. Characteristic impedance is 50 Ohm

Ports are :

1 – the input port

4 – output port

6 – coupled port in the forward direction

3 – coupled port in the reverse direction

Now to further our analysis lets connect a load to port 1 and drive port 4. So this is a reverse direction operation for this coupler.

To use some numbers, lets input 0 dBm at port 4. Then since the loss on the mainline is 0.25 dB the power arriving at port 1 is -0.25 dBm. Also:

lets assume the load is exactly 50 Ohm and we terminate port 3 with also 50 Ohm. Now the power coming out of port 6 is -32 dBm if the coupler is ideal. (directivity + coupling ).

Now lets assume that the load is no longer 50 Ohm but 100 Ohm. Now there will be reflected power also being added to the power out of port 6. The difference between the

– 32 dBm and the total power being measured at port 6 with 100 Ohms is the reflected power PR.

Then the return loss is simply

RL=10log10(PT/PR) dB.

Here PT is 0 dBm. Once RL is known other parameters can be calculated.

# Directivity of Directional Couplers: another look.

The DIRECTIVITY of directional couplers is an important parameter when using a coupler. Simply defined it is the ability of the coupler to distinguish between forward and reverse signals. Referring to the diagram below, we can define the directivity as:

D = s31 + s21 – s32. where s31=coupling ratio, s21=insertion loss and s32 = isolation. All these are defined negatively in dB, but D itself is stated positively. A shorter form is also used for forward measurements as D = s31 – s32. The question is: why is directivity important?

To elaborate a little: power is applied to port 1 and output at port 2. Some of the applied power is coupled to port 3 with a coupling factor of K^2. At port 3 part of the input power is sampled. Port 3 is the coupling port. Port 4 is the isolation port where ideally, no input power should appear. Any of the ports can be defined as the input port provided the other ports are appropriately labeled.

Mathematically speaking we can define: Coupling = 10Log(P1/P3) = -20LogK ( dB). Also in a similar fashion the other parameters can be defined: Directivity = 10Log(P3/P4) = + 20Log[K/ABS(s14)]. Isolation = I = 10Log(P1/P4) = -20Log[ABS(s14)] dB. Insertion loss = L = 10Log(P1/P2) = -20Log[ABS(s12)] dB.

The coupling shows how much of the input power is coupled to port 3. The directivity shows how well the coupler isolates forward and reverse signals. Isolation shows how much of power is delivered to the isolated port. The insertion loss provides the calculation for the amount of power delivered to the output port, less the power delivered to the coupled and isolated ports. In an ideal coupler the directivity and isolation are infinitely high.

Using the example of measuring an unknown impedance with a signal source, a detector and a coupler, we see that if the unknown impedance is perfectly matched to the source impedance then there is no reflection of power at all, and there is no reverse wave. So in this case we will measure the impedance perfectly. Its simple enough ! However, if the unknown impedance is not matched to the source impedance then there will be a reflected wave. We sample this using port 4. In this case however, the coupler not being ideal will transmit part of the reverse signal to port 3, the coupling port that we also use to sample the forward signal. Now there will be an error in the measurement. We need as little of the reverse signal in the coupling port as possible. A characteristic of the coupler. This is why directivity is important. The higher the directivity the less error we will get. A very good article on this topic is provided by REF1: ” Directivity and VSWR measurements”, by Doug Jorgesen and Christopher Marki of Marki Microwave Inc. They also provide curves of error versus directivity etc. A very readable article. Please visit our website for more articles of interest, calculators and a description of what we do and some of our products.

# RF Transformer: Relationship between the primary and secondary inductances with the turns ratio

The transformer primary and secondary inductance ratio is equal to the square of the turns ratio of the RF transformer. Simple, but useful identity for multiple applications and specifications. So Ls/Lp= N*N where N is the turns ratio of the transformer. Please visit our website for more information about us and other items of interest.

# Class F RFPA design: Waveforms of the amplifier

A class F amplifier is highly efficient and uses waveform shaping at the drain of a power device. This shaping takes the form of using ( usually) the second and third harmonic of the waveform to generate a drain to source ( collector to emitter) voltage that is roughly a square wave ( ideally a square wave, but we all know that ideal is a not a word we can use loosely). The drain current is then shaped to be very small when the D/S voltage is high and high when the D/S voltage is low. This causes a non-overlapping condition in operation and ensures a high efficiency. A pre-distortion or pre-conditioning circuit is also required to get the desired linearity. ( This is the subject of a later post). The waveforms below are what we got using Microwave Office as the simulator. The active device was a Cree 40010. It is simply instructive to examine the waveforms in case the reader is starting to design a Class F. The figure below represents the drain source waveform.

The figure above is the D.S current in mA.

The figure below is the simulated power gain and PAE.

We hope these will be of help to aspiring designers of Class F amplifiers. Please visit the Signal Processing Group Inc., website for more information and articles of interest.

# RF/Microwave design: The spacing circle in double stub matching

The Spacing Circle in double stub matching

In double stub matching the two stubs are spaced a predetermined distance away from each other. These distances are typically λ/8, λ/4, 3λ/8, 5λ/8 etc. Knowing what we know about a length of line acting as a transformer, we know that the length of line between the two stubs acts as a transformer. The action of this transformer is to convert the admittance at the position of stub 2 to a different admittance at the position of stub 1. So we start from the position of stub 2.

In order that stub 2 can be finally used to match the line admittance, the real part of the admittance at the position of stub 2, on the line, has to be 1.0 (normalized value). Its susceptance is then jB. jB is the susceptance that is cancelled using stub 2 to ultimately get the matching to the line admittance. The admittance at the position of stub 2, (without the stub) lies on the constant conductance, g = 1 circle. The admittances on the g = 1 circle are all the possible admittances at the stub 2 position for a match to take place.

To reiterate, as a result of these deliberations, that some point of the VSWR circle formed by the position of stub 2, must intersect the g=1 or the unity conductance circle on the Smith Chart.

We also conclude, that the admittance at the position of the first stub, must lie on

a circle of equal radius but having its center rotated (moved to or displaced) by the spacing between the stubs towards the load. Lets call this spacing ‘d’.

This circle is called the spacing circle.

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