Smith Chart

Rectilinear impedance plane Smith Chart Review +R +jX -jX ¥ . 90 o Polar plane 1.0 .8 .6 ¥ .4 180 o + - .2 o Rectilinear impedance plane -90 o Constant X Z = Zo L Constant R G = Smith Chart maps rectilinear impedance plane onto polar plane The amount of reflection that occurs when characterizing a device depends on the impedance the incident signal sees. Let's review how complex reflection and impedance values are displayed. Since any impedance can be represented as a real and imaginary part (R+jX or G+jB), we can easily see how these quantities can be plotted on a rectilinear grid known as the complex impedance plane. Unfortunately, the open circuit (quite a common impedance value) appears at infinity on the x-axis. The polar plot is very useful since the entire impedance plane is covered. But instead of actually plotting impedance, we display the reflection coefficient in vector form. The magnitude of the vector is the distance from the center of the display, and phase is displayed as the angle of vector referenced to a flat line from the center to the rightmost edge. The drawback of polar plots is that impedance values cannot be read directly from the display. Since there is a one-to-one correspondence between complex impedance and reflection coefficient, we can map the positive real half of the complex impedance plane onto the polar display. The result is the Smith chart. All values of reactance and all positive values of resistance from 0 to ¥ fall within the outer circle of the Smith chart. Loci of constant resistance now appear as circles, and loci of constant reactance appear as arcs. Impedances on the Smith chart are always normalized to the characteristic impedance of the test system (Zo, which is usually 50 or 75 ohms). A perfect termination (Zo) appears in the center of the chart. Z = 0 (short) Z = (open) L L G G = 1 ±180 O = 1 O Smith Chart

Circuits components

Smith Chart main points

From S parameters to impedance

IF BW and averaging Heterodyne detection scheme IF BW reduction Dynamic Range (definition)

Smoothing trace Smoothing (similar to video filtering) averages the formatted active channel data over a portion of the displayed trace. Smoothing computes each displayed data point based on one sweep only, using a moving average of several adjacent data points for the current sweep. The smoothing aperture is a percent of the swept stimulus span, up to a maximum of 20%. Rather than lowering the noise floor, smoothing finds the mid-value of the data. Use it to reduce relatively small peak-to-peak noise values on broadband measured data. Use a sufficiently high number of display points to avoid misleading results. Do not use smoothing for measurements of high resonance devices or other devices with wide trace variations, as it will introduce errors into the measurement.

Averaging trace Averaging computes each data point based on an exponential average of consecutive sweeps weighted by a user-specified averaging factor. Each new sweep is averaged into the trace until the total number of sweeps is equal to the averaging factor, for a fully averaged trace. Each point on the trace is the vector sum of the current trace data and the data from the previous sweep. A high averaging factor gives the best signal-to-noise ratio, but slows the trace update time. Doubling the averaging factor reduces the noise by 3 dB.

IF BW reduction IF bandwidth reduction lowers the noise floor by digitally reducing the receiver input bandwidth. It works in all ratio and non-ratio modes. It has an advantage over averaging as it reliably filters out unwanted responses such as spurs, odd harmonics, higher frequency spectral noise, and line-related noise. Sweep-to-sweep averaging, however, is better at filtering out very low frequency noise. A tenfold reduction in IF bandwidth lowers the measurement noise floor by about 10 dB. Bandwidths less than 300 Hz provide better harmonic rejection than higher bandwidths.

Impedance Measurements

Which Value Do We Measure? TRUE EFFECTIVE Before proceeding to practical measurements, we need to understand the concept of True, Effective and Indicated values. This is essential since we all tend to forget that the instrument does NOT necessarily measure what we want to measure. By the way, which value do instruments measure? The TRUE value excludes all parasitics and is given by a math relationship involving the component's physical composition. If you think of a 50 Ohm PC board stripline, it is built up assuming that the dielectric constant K is constant. But in the real world this is not true. The TRUE value has only academic interest. The EFFECTIVE value is what we generally want to measure because it takes into consideration the parasitics and dependency factors, as this figure shows. When designing and simulating circuits, only EFFECTIVE values should be used to reflect the actual circuit behavior. But the INDICATED value given by the instrument takes into account not only the real world device, but also the test fixture and accessories as well as the instrument inaccuracies and losses. What is the difference between TRUE and EFFECTIVE values? The quality of the component. And what is the difference between EFFECTIVE and INDICATED values? The quality of the instrument and above all the quality of the MEASUREMENT. Our goal is to make the INDICATED value as close as possible to the EFFECTIVE value INDICATED % +/- Instrument Test fixture Real world device Kobe Instrument Division Back to Basics - LCRZ Module

Frequency vs. Measurement Techniques Network Analysis 100KHz RF I-V 1 MHz 1.8 GHz I-V 10KHz 110MHz Resonant 22KHz 30MHz 70MHz Auto Balancing Bridge This chart will help you visualize the frequency range for 5 measurement techniques. The frequency range numbers are a mix of practical and theoretical limits and should be used as a reference only. The autobalancing bridge basic accuracy is 0.05% while the network analysis one is 1.5%. This already uncovers possible trade-offs. 5HZ 40MHz 1 10 100 1K 10K 100K 1M 10M 100M 1G 10G Frequency (Hz) Kobe Instrument Division Back to Basics - LCRZ Module

Auto Balancing Bridge Theory of Operation Virtual ground H R L 2 DUT I I = I 1 2 - + V = I R 2 2 V 2 2 V V R 1 1 2 Z = = I 2 V 2 Kobe Instrument Division Back to Basics - LCRZ Module

Advantages and Disadvantages Auto Balancing Bridge Advantages and Disadvantages Most accurate, basic accuracy 0.05% Widest measurement range C,L,D,Q,R,X,G,B,Z,Y,O,... Widest range of electrical test conditions Simple-to-use Let us summarize the advantages and disadvantages of each of the measurement techniques. The Autobalancing bridge technique is by far the best technique for measurements below 40 MHz. It provides the most accurate measurements possible and has the widest impedance measurement range. Both of these are critical for accurate component analysis. A wide range of AC and DC stimulus can be applied to the component. In addition, because this is a low frequency technique, it is the simplest measurement technique to use. Low frequency, f < 40MHz

Resonance (Q - Meter) Technique Theory of Operation Tune C so the circuit resonates At resonance X = -X , only R remains D C D DUT L (X ), R D D e Z Tuning C (X c) ~ e I= V V OSC X = = (at resonance) C V I R V e D |X | R D |X | |V| e C Q = = = R D Kobe Instrument Division Back to Basics - LCRZ Module

Advantages and Disadvantages Resonant Method Advantages and Disadvantages Very good for high Q - low D measurements Requires reference coil for capacitors Limited L,C values accuracy Vector Scalar 75kHz - 30MHz 22kHz - 70MHz automatic and fast manual and slow easy to use requires experienced user The Resonant technique, or Q-meter, used to be a very manual measurement technique. However, the design of automatically tunable air capacitor standards allows today fast and error free measurement of high Q or low D components. In low D capacitor test, it is still difficult to achieve high accuracy measurements due to the need for very stable reference inductors, which are difficult to design. Testing chip or SMD capacitors requires specific test fixtures which have strays, essentially stray capacitance, that influence the value of the tuning capacitance. With the new automatic technique, test fixture parasitics can be compensated for by offset compensation. This requires accurate design and evaluation of the stray capacitance of the test fixture. No compensation limited compensation

I - V Probe Technique Theory of Operation R V 2 V = I R V V V R I Z = 1 V V R I 1 1 2 2 Z = = I V DUT 2 2 Kobe Instrument Division Back to Basics - LCRZ Module

Advantages and Disadvantages I-V (Probe) Advantages and Disadvantages Medium frequency, 10kHz < f < 110MHz Moderate accuracy and measurement range Grounded and in-circuit measurements Simple-to-use The I-V, or "probe technique", provides very good mid-frequency range performance, extending up to 100 MHz. Another key feature of this technique is that it is a floating measurement technique, thus grounded and in-circuit measurements are very easy.

RF I-V Theory of Operation Low Impedance Test Head High Impedance Test Head Low Impedance Test Head Current Voltage Current Detection Detection Vi Detection Voltage Vi Detection Ro Ro Vv Ro DUT Vv Ro DUT Kobe Instrument Division Back to Basics - LCRZ Module

Advantages and Disadvantages RF I-V Advantages and Disadvantages High frequency, 1MHz < f < 1.8GHz Most accurate method at > 100 MHz Grounded device measurement The RF I-V technique provides very good high-frequency range performance, extending up to 1.8 GHz. This is the most accurate technique at frequencies higher than 100 MHz. Although this is a 50 Ohm system, the technique has a very good impedance measurement range with quite good accuracy

Network Analysis (Reflection) Technique Theory of Operation V INC DUT V R V Z - Z R L O = = V Z + Z INC L O Kobe Instrument Division Back to Basics - LCRZ Module

Advantages and Disadvantages Network Analysis Advantages and Disadvantages High frequency - Suitable, f > 100 kHz - Best, f > 1.8 GHz Moderate accuracy Limited impedance measurement range (DUT should be around 50 ohms) Network analysis is the best solution for very high frequency measurements, extending up to tens of GHz. Measurements as low as 100 KHz are possible with this technique (directional bridge low-end limit). Given the existence of the autobalancing bridge, I-V probe, and RF I-V techniques, it is advised that the network analysis technique be used for measurements above 1.8 GHz. Above 1.8 GHz, the reflection technique is the only measurement technique currently available.

TDR Theory of Operation Oscilloscope V V DUT Z Step Generator H V INC R Z L Series R & L Parallel R & C t Step Generator V INC R Z - Z L O Z + Z = Kobe Instrument Division Back to Basics - LCRZ Module

Advantages and Disadvantages TDNA (TDR) Advantages and Disadvantages Reflection and transmission measurements Single and multiple discontinuities or impedance mismatches ("Inside" look at devices) DUT impedance should be around 50 ohms Not accurate for m or M DUTs or with multiple reflections Although this is a 50 Ohm system, the technique has a very good impedance measurement range with quite good accuracy. Good for test fixture design, transmission lines, high frequency evaluations

Measurement examples

1 nF 1 mH 10 nF 10 mH 1 kOhm Corto circuito

1 nF 1 mH 10 nF 10 mH 1 kOhm Corto circuito

RF Device Characterization

Group delay

Use electrical delay to remove linear portion of phase response Deviation from Linear Phase Use electrical delay to remove linear portion of phase response Linear electrical length added Deviation from linear phase RF filter response Phase 45 /Div o (Electrical delay function) Phase 1 /Div o + yields Frequency Frequency Frequency Looking at insertion phase directly is usually not very useful. This is because the phase has a negative slope with respect to frequency due to the electrical length of the device (the longer the device, the greater the slope). Since it is only the deviation from linear phase which causes distortion, it is desirable to remove the linear portion of the phase response. This can be accomplished by using the electrical delay feature of the network analyzer to cancel the electrical length of the DUT. This results in a high-resolution display of phase distortion (deviation from linear phase). Low resolution High resolution

What is group delay? f w f w p w Dw f Df = = * f -d f Group Delay (t ) Frequency g Group Dw Delay t f o Phase Average Delay Df Group Delay (t ) -d f d w = g Frequency d f d f -1 Deviation from constant group delay indicates distortion = * 360 o Another useful measure of phase distortion is group delay. Group delay is a measure of the transit time of a signal through the device under test, versus frequency. Group delay is calculated by differentiating the insertion-phase response of the DUT versus frequency. Another way to say this is that group delay is a measure of the slope of the transmission phase response. The linear portion of the phase response is converted to a constant value (representing the average signal-transit time) and deviations from linear phase are transformed into deviations from constant group delay. The variations in group delay cause signal distortion, just as deviations from linear phase cause distortion. Group delay is just another way to look at linear phase distortion. f in radians Average delay indicates transit time w in radians/sec f in degrees in Hz 2 = ( ) f w p f

Why measure group delay? Phase Phase f f -d f d w -d f d w Group Delay Group Delay Why are both deviation from linear phase and group delay commonly measured? Depending on the device, both may be important. Specifying a maximum peak- to-peak value of phase ripple is not sufficient to completely characterize a device since the slope of the phase ripple is dependent on the number of ripples which occur per unit of frequency. Group delay takes this into account since it is the differentiated phase response. Group delay is often a more accurate indication of phase distortion. The plot above shows that the same value of peak-to-peak phase ripple can result in substantially different group delay responses. The response on the right with the larger group-delay variation would cause more signal distortion. f f Same p-p phase ripple can result in different group delay