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♦ PDF Cover Page ♦ ♦ Verso Filler Page ♦ To receive a hard copy of this handy guide, just Call or write QuadTech at 1-800-253-1230 100 Nickerson Road Marlboro, MA 01752 Contents
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Impedance is the basic electrical parameter electronic components are not pure resistors, used to characterize electronic circuits, inductors or capacitors, but a combination of components, and materials. It is defined as all three. Today’s generation of LCR meters the ratio of the voltage applied to the device are capable of displaying these parameters and the resulting current through it. To put and can easily calculate and display many this another way, impedance is the total other parameters such as Z, Y, X, G, B, D, opposition a circuit offers to the flow of an etc. This primer is intended as an aid in alternating current (ac) at a given frequency, understanding which ac impedance and is generally represented as a complex measurements are typically used and other quantity, which can be shown graphically. factors that need to be considered to obtain The basic elements that make up electrical accurate and meaningful impedance impedances are resistance, inductance and measurements. capacitance, R, L, and C. In the real world
Definitions The mathematical definition of resistance for However, if capacitance or inductance are dc (constant voltage) is the ratio of applied present, they also affect the flow of current. voltage E to resulting current I. This is Ohms The capacitance or inductance cause the Law: R = E/I. An alternating or ac voltage is voltage and current to be out of phase. one that regularly reverses its direction or Therefore, Ohms law must be modified by polarity. If an ac voltage is applied to a circuit substituting impedance (Z) for resistance. containing only resistance, the circuit Thus for ac, Ohm’s Law becomes : Z = E/I. resistance is determined from Ohms Law. Z is a complex number: Z = R + jX .
E E For dc, Resistance, R = For ac, Impedance, Z = = R +jX I I
The phase shift can be drawn in a vector number having a real part Gp (conductance) diagram which shows the impedance Z, its and an imaginary part jBp (susceptance) real part Rs, its imaginary part jXs with a phase angle Φ. Note Φ = - θ. (reactance), and the phase angle θ. Because Because admittances in parallel add, an series impedances add, an equivalent circuit equivalent circuit for an admittance would for an impedance would put Rs and Xs is put Gp and Bp in parallel. Note from the series hence subscript s . The reciprocal of formulas below that, in general, Gp ≠ 1/Rs Z is Admittance, Y which is also a and Bp ≠ -1/Xs . complex
3 Impedance Terms
Table 1
Parameter Quantity Unit Symbol Formula Z Impedance ohm, Ω 1 θ ZR=+ jX ==||Z ε j SSY |Z| Magnitude of Z ohm, Ω 1 ||ZR=+=22 X SS||Y R or ESR Resistance, ohm, Ω G s R = P real part of Z S 22+ GBPP X Reactance, ohm, Ω B s X =− P imaginary part of Z S 22+ GBPP Y Admittance siemen, S 1 φ YG=+ jB ==||Y ε j PPZ |Y| Magnitude of Y siemen, S 1 ||YG=+=22 B (was mho) PP||Z G Real part of Y siemen, S R p G = S P 22+ RXSS B Susceptance siemen, S X p B =− S P 22+ RXSS C Series capacitance farad, F 1 s C =− =CD()1 + 2 S ω P X S C Parallel capacitance farad, F p ==B C S C P ω 1+ D 2 L Series inductance henry, H 2 s ==X Q LSpL ω 1+ Q 2 L Parallel inductance henry, H 1 1 p L =− =L ()1 + P ω S 2 BP Q Ω Rp Parallel resistance ohm, ==1 +2 RP RQS ()1 GP Q Quality factor none 1 X G Q ==S =P =tanθ D RS BP D, DF or Dissipation factor none ==1 RS =GP =0 −=θ δ tan δ D tan(90 ) tan Q X S BP θ phase angle of Z degree or θ =−φ radian φ phase angle of Y degree or φ =−θ radian Notes: f = frequency in Hertz, j = −1 , ω ω = 2π f R and X are equivalent series quantities unless otherwise defined, G and B are equivalent parallel quantities unless otherwise defined. We sometimes use parallel R (Rp) but rarely use parallel X, and very rarely series G or series B. C and L each have two values, series and parallel. If not defined usually we mean the series values, but not necessarily, especially for C (Cp is common, Lp is less used). We define Q as being positive if it is inductive, negative if it is capacitive, We define D as positive if capacitive. Thus D = -1/Q. Some people (particularly in Europe) use tan δ instead of D, tan δ = D.
4 Phase Diagrams
+jX +jX +jBX +jBX Rs Gp +R Z Y +G θ θ δ jωCp δ jωLs 1 1 -j δ δ -j ω ωCs θ θ Ls +R +G Y Z Rs Gp -jX -jX -jB -jB
Rs Rs Rp or Cp Rp or Lp Gp Gp Cs Ls
Impedance Admittance Capacitive Inductive Capacitive Inductive
Figure 1
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5 1 #! Y = Z #2$ ) 3 $ 3 !! :; 1 < < # $% & : 1(3*+!(3*+ω0 ; 3.+/ω- #4$
5 Rs Gp 1 D === Xs Bp Q 6 #=$
7 : ; 3 /)! /,# . /,$ ; : # 6 : δ 1 1 Rs $ Y == = − Z Rs+ jXs Rs22+ Xs Xs j =+Gp jBp Rs22+ Xs : ; #8$ :; ' 1 3( /) ,(9 #5 !(. /,# " $ )(9 $ ; 3 0- # $ ! # $ ) 3 :; )( /3)
6 5 : ; 1kohm
7 .05uF 2kohm .1uF # 2$
Connection Methods between the two leads, both of which affect Two-Terminal Measurements the measurement results. Because of these The impedance of a device is defined by Ohm’s Law as the ratio of the voltage across it to the current through it. This requires at error sources, the typical impedance least two connections and therefore the measurement range for a two–terminal arithmetic of terminals starts with two. With connection is limited to 100Ω to 10kΩ. only two terminals, the same terminals must be used for both applying a current and Four-Terminal Measurements measuring a voltage as shown in Figure 3. First let’s jump into four-terminal measurements, which are simpler to explain and more commonly used than a three- terminal measurement. With a second pair of terminals available, one can measure IL PL PH IH voltage across the device with one pair and apply current to the device with the other pair. This simple improvement of independent leads for voltage and current Z effectively removes the series inductance and resistance error factor (including contact Figure 3 resistance) and the stray capacitance factor Two-Terminal discussed with two-terminal measurements. Accuracy for the lower impedance When a device is measured in this way it measurement range is now substantially might not be an accurate measurement. improved down to 1Ω and below. There There are two types of errors and these are will be some mutual inductance between the the errors that measurements with more current leads and voltmeter leads which will connections will avoid, one is the lead introduce some error, but much of this is inductance and lead resistance in series with eliminated by using shielded coaxial cabling. the device and the other is stray capacitance The most famous use of the four-terminal
7 connection is the Kelvin Bridge which has been widely used for precision DC resistance measurements. This circuitry A three-terminal connection usually associated Lord Kelvin’s name so closely employs two coaxial cables, where the outer with the four-terminal connection technique shields are connected to the guard terminal that “Kelvin” is commonly used to describe of the LCR meter. The guard terminal is this connection. electrically different from the instrument ground terminal which is connected to chassis ground. Measurement accuracy is usually improved for higher impedances, but not lower because lead inductance and IL PL PH IH resistance are still present.
Ix Oscillator Z Zx Figure 4 I1 I2 Four-Terminal Z1 Z2
Three-Terminal (or Guarded) Measurements Voltmete Ammeter While the four terminal-terminal measurement applies a current and measures the resulting open-circuit voltage, the three – Figure 5 terminal measurement does the opposite, it Guarded Configuration applies a voltage and measures the short circuit current. The extra terminal, or third terminal, is called the guard. Any components shunting the unknown can effectively be removed by connecting some point along the shunt to this guard terminal. IL PL PH IH When the guard is properly connected as shown in Figure 5, it reduces the input signal current but does not affect the measurement of the DUT, (Zx). Zx is calculated using the Z current Ix because I2 is negligible compare to Ix since the internal impedance of the Figure 6 ammeter is very low compared to Z2. Three-Terminal (Guarded)
8 Impedance Measuring Instruments
Methods measurement which differs significantly There are many different methods and from that employed by the traditional techniques for measuring impedance. The most familiar is the nulling type bridge method shown in Figure 7. When no current measuring instruments. In particular the flows through the detector (D), the value of 7000 uses digital techniques for signal the unknown impedance Zx can be obtained generation and detection. In the elementary by the relationship of the other bridge measurement circuit as shown in Figure 8, Z1 both the voltage across the device under test elements, where Zx = Z3 Z2 (Zx) and the voltage across a reference Various types of bridge circuits, employing resistor (Rs) are measured, which essentially combinations of L, C, and R as bridge carry the same current. elements, are used for different instruments with varying applications. Ix K Ex
Z1 Zx Zx
Detector D Rs K Es
Z2 Z3 Figure 8 7000 Measurement Circuit
Oscillator The voltage across Zx is Ex and the voltage across Rs is Es. Both voltages are simultaneously sampled many times per Figure 7 cycle of the applied sine wave excitation. In Bridge Method the case of the 7000, there are four reference resistors. The one used for a particular Most recently instruments have been measurement is the optimal resistor for the developed which employ elaborate software- device under test, frequency, and amplitude driven control and signal processing of the applied ac signal. For both Ex and Es techniques. For example, the QuadTech a real and imaginary (in phase and 7000 LCR Meter uses a principle of quadrature) component are computed mathematically from the individual sample measurements. The real and imaginary
9 components of Ex and Es are by themselves When a test device is connected, the voltage meaningless. Differences in the voltage and applied to the device depends on the value current detection and measurement process of the source resistor (Rs) and the are corrected via software using calibration impedance value of the device. This is data. The real and imaginary components of shown in Figure 10, where the programmed Ex (Exr and Exi) are combined with the real voltage is 1V but the voltage to the test and imaginary components of Es (Esr and device is 0.5V. Some LCR meters, such as Esi) and the known characteristics of the the QuadTech 7000 have a constant voltage reference resistor to determine the apparent mode, where the voltage to the device is impedance of the complex impedance of Zx monitored and maintained at the using complex arithmetic. programmed level.
Rs = 25Ω
Vs = 1V Vx = 0.5V Zx = 25Ω
Figure 9 QuadTech 7600 LCR Meter Figure 10 Source Impedance Factors Functions The demand on component testing is much Ranging more than a resistance, capacitance or In order to measure both low and high inductance value at a given test frequency impedance values measuring instrument and stimulus voltage. Impedance meters must have several measurement ranges. must go beyond this with the flexibility to Ranging is usually done automatically and provide multi-parameters over wide selected depending on the impedance of the frequency and voltage ranges. Additionally, test device. Range changes are an easily understood display of test results accomplished by switching range resistors and the ability to access and use these results and the gain of detector circuits. This helps has become increasingly important. maintain the maximum signal level and highest signal-to-noise ratio for best Test Voltages measurement accuracy. The idea is to keep the measured impedance close to full scale The ac output of most LCR meters can be for any given range, again, for best accuracy. programmed to select the signal level Range holding, rather than autoranging, is a applied to the DUT. Generally, the feature sometimes used in specific programmed level is obtained under an open applications. For example, when repetitive circuit condition. A source resistance (Rs, testing of similar value components, range internal to the meter) is effectively holding can speed up the test time. Another connected in series with the ac output and use of range hold occurs when measuring there is a voltage drop across this resistor.
10 components whose value falls within the Many testers today must be equipped with overlap area of two adjacent ranges, where if some type of standard data communication allowed to autorange an instrument display can sometimes result in operator confusion. interface for connection to remote data Averaging processing, computer or remote control. For The length of time that an LCR meter an operation retrieving only pass/fail results spends integrating analog voltages during the Programmable Logic Control (PLC) is the process of data acquisition can have an often adequate, but for data logging it’s a important effect on the measurement results. different story. The typical interface for this If integration occurs over more cycles of the is the IEEE-488 general purpose interface test signal the measurement time will be bus or the RS-232 serial communication longer, but the accuracy will be enhanced. line. These interfaces are commonly used This measurement time is usually operator for monitoring trends and process control in controlled by selecting a FAST or SLOW a component manufacturing area or in an mode, SLOW resulting in improved environment where archiving data for future accuracy. A further enhancement to reference is required. For example when accuracy can be obtained by selection of testing 10% components, the yield is fine median value or averaging. In a median when components test at 8% or 9%, but it mode 3 measurements might be made and does not take much of a shift for the yield to two thrown away. In an averaging mode plummet. The whole idea of production many measurements are made and the monitoring is to reduce yield risks and be average of these calculated for the end able to correct the process quickly if needed. result. All of this is a way of reducing An LCR Meter with remote interface unwanted signals and effects of unwanted capability has become standard in many test noise, but does require a sacrifice of time. applications where data logging or remote control have become commonplace. Computer Interface
Fixturing and Cables by a single residual component the Compensation compensation is simple. Compensation reduces the effects from error sources existing between the device under test and the calibrated connection to the Take the case of stray lead capacitance (C measuring instrument. The calibrated stray) in parallel with the DUT capacitance connection is determined by the instrument (C x), shown in Figure 11. The value of the manufacturer, which can be front or rear stray capacitance can be measured directly panel connections, or at the end of a with no device connected. When the device predefined length of cable. Compensation is connected the actual DUT value can be will ensure the best measurement accuracy determined by subtracting the stray possible on a device at the selected test capacitance (C stray) from the measured conditions. When a measurement is affected value (C measure). The only problem is, its
11 not always this simple when stray residuals Load Correction is a compensation are more than a single component. technique which uses a load whose impedance is accurately known and applies a correction to measurements of similar components to substantially improve C measure C measure measurement accuracy. The purpose being to correct for non-linearity of the measuring instrument and for test fixture or lead effects which may be dependent on the test C stray C stray frequency , test voltage, impedance range, or other factors. Criteria for selecting the appropriate load include: C x a. Load whose impedance value is C x = C measure – C stray accurately known (NIST traceable if possible) Figure 11 b. Load whose impedance value is very Compensation close to the DUT (this ensures that the measuring instrument selects the same Open/Short measurement range for both devices). Open/Short correction is the most popular c. Load whose impedance value is stable compensation technique used in most LCR under the measurement conditions. instruments today. When the unknown d. Load whose physical properties allow it terminals are open the stray admittance to be connected using the same leads or (Yopen) is measured. When the unknown fixture as the DUT. terminals are shorted the residual impedance (Zshort) is measured. When the device is A prerequisite for load correction is to measured, these two residuals are used to perform a careful open/short compensation calculate the actual impedance of the device as previously discussed. A unique feature under test. When performing an OPEN found on the QuadTech 7000 LCR Meter measurement it is important to keep the provides for an automatic load correction, distance between the unknown terminal the where the load’s known value is entered into same as they are when attached to the memory, the load then measured, and this device. It’s equally important to make sure difference then applied to ongoing that one doesn’t touch or move their hands measurements. near the terminals. When performing a SHORT measurement a shorting device Z actual = Z measure +/- delta Z (shorting bar or highly conductive wire) is connected between the terminals. For very where delta Z = the difference low impedance measurements it is best to between known and measured value connect the unknown terminals directly of the load together. Through the use of load correction it is Load Correction possible to effectively increase the accuracy of the measuring instrument substantially,
12 but this is only as good as the know accuracy and traceability of the load used in determining the correction.
13 Capacitance Measurements 0 # $ : " 6) 6) 2 0 0 > 6) : : 6)(:/ω0 ω(2 6) π 0
ALUMINUM ELECTROLYTIC
TANTALUM ELECTROLYTIC
METALIZED PLASTIC
CERAMIC
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14 C "A7 &% 0 L Rs 99 µ0 9µB Figure 13 Rp ? @ Small C Rp Large C Rp C C Large Xc More Small Xc Less Significant Significant " Rs Rs > Less More Significant Significant Figure 14 Figure 15 ; C999-0)@ " )) "# 8 ) -0: 0 #)$ $ % ) " ! " ! A ! = 0 7 " ! !! & " @ - 0 .. ! ' " # ! "! & • 9"Ω. • ! 9Ω. " " # • ! . D (" ) *
15 / 99 :% 99 :% E IH / 7 PH 0 ! 0 ! 1
V DUT , (, * ; PL 6)#6 IL A ) $ Figure 16 6) ! . # > )$ % 6) + ,-. /+ / F 6) ) # $ E / &'() *+, " ) ,# C$ , E −1 &Cs = " ωXs ! ! # ? # " 0 1 . # $
16 6) 6)()() ' A # G$ 6)
DUT Rs ) '()
1 = Rp Xs 1 + ω222 Cs + jCpω 1CpRp Rp
Figure 17 G G : )(9 energy lost Real part of Z D == ! 6)H9 − energy stored ( Imaginary part of Z) 6)H)
Rs D = ==Rsωω C (ESR) C ()Xs− "
Rs = Rp 1Cp+ ω 222Rp
Cp Rp = + 1 Cs = 1 Cp ( ω 2 Cp 2 Rp 2 )
ab
2
17 Inductance Measurements
An inductor consists of wire wound around a accuracy. For mid range values of core material. Air is the simplest core inductance a more detail comparison of material for inductors, but for physical reactance to resistance should be used to efficiency, magnetic materials such as iron help determine the mode. and ferrites are commonly used. The most important thing to remember Series or Parallel whenever a measurement correlation As with capacitor measurements, inductor problem occurs, is to use the test conditions measurements can be made in either a series specified by the component manufacturer. or parallel mode, use of the more suitable Independent of any series/parallel decision, mode results in a value that equals the actual it is not uncommon for different LCR meters inductance. In a typical equivalent circuit to give different measured results. One good for an inductor, the series resistance (Rs), reason for this is that inductor cores can be represents loss of the copper wire and test signal dependent. If the programmed parallel resistance (Rp) represents core output voltages are different the measured losses as shown in Figure 19. inductance will likely be different. Even if the programmed output voltage is the same, two meters can still have a different source Lx Rs impedance. A difference in source impedance can result in a difference in current to the device, and again, a different measured value. Rp
Figure 19
In the case where the inductance is large, the reactance at a given frequency is relatively large so the parallel resistance becomes more significant than any series resistance, hence the parallel mode should be used. For very large inductance a lower measurement frequency will yield better accuracy.
For low value inductors, the reactance also becomes relatively low, so the series resistance is more significant, thus a series measurement mode is the appropriate choice. For very small inductance a higher measurement frequency will yield better
18 $
Series or Parallel Of the three basic circuit components, So how does one choose the series or resistors, capacitors and inductors, resistors parallel measurement mode? For low values cause the least measurement problems. This of resistors (below 1kΩ) the choice usually is true because it is practical to measure becomes a low frequency measurement in a resistors with a dc signal applied or at series equivalent mode. Series because the relatively low ac frequencies. In contrast to reactive component most likely to be present this, capacitors and inductors always in a low value resistor is series inductance, experience ac signals and thus generally which has no effect on the measurement of measured under these conditions. Resistors series R. To achieve some degree of are usually measured at dc or low frequency precision with low resistance measurements ac where Ohm’s Law gives the true value it is essential to use a four-terminal under the assumption that loss factors are connection as discussed earlier. This accounted for. The thing to keep in mind is technique actually eliminates lead or contact that if resistors are used in high frequency resistance which otherwise could elevate the circuits they will have both real and reactive measured value. Also, any factor that affects components. This can be modeled as shown the voltage drop sensed across a low in Figure 20, with a series inductance (Ls) resistance device will influence the and parallel capacitance (Cp). measurement. Typical factors include contact resistance and thermal voltages Ls Rx (those generated by dissimilar metals). Contact resistance can be reduced by contact cleanliness and contact pressure.
Cp For high values of resistors (greater than several MΩ) the choice usually becomes a Figure 20 low frequency measurement in a parallel equivalent mode. Parallel because the For example, in the case of wire-wound reactive component most likely to be present resistors (which sounds like an inductor) its in a high value resistor is shunt capacitance, easy to understand how windings result in which has no effect on the measurement of this L term. Even though windings can be parallel R. alternately reversed to minimize the inductance, the inductance usually increases with resistance value (because of more turns). In the case of carbon and film resistors conducting particles can result in a distributed shunt capacitance, thus the C term.
19 Precision Impedance Meters in a Cal Lab 9 $,,, 0-( *B 3 3 5 7 4+, $,,, )* 4 : 5 * 3 3 :$,,,: 4 : * 4 * 434 5 : 3 5 5 4 3 3 3 3 *34 $,,,A 3+;+ 5 : 5 4 3 3 < * 4!< # 4 ! 4 *# C 3 6 * 34 4 3 ! # 53 4 7 5 4 34 : " 5 *44 6 * = 3 =>4 $,,, 6 44 * 3 4 4 7 5 3 $,,, , + 4 * 335 7 5/ ++++++ & 5*4 *+, $,,, +:1? @4 A * > , + )'2()" <" 5 * !#6 4 *3 341 N < !9#5 3 * 5*B C999 4 4 "
20 .3* 45 # 2 $ 9 $,,,* * 5 3 6 @ 3 5 6 46 4 Load Correction : )* Measure Off On Primary Nominal 60.00000 pF 3 34 Secondary Nominal 4.000000 m 4 4 34 ? Measuring Correction A ( 222 %% Measured Primary 60.25518 pF Measured Secondary .0042580 3 34 D Freq 1.0000MHz Primary Cs 3 4 5 Range 49 Secondary DF HIT
Materials Measurement " @ )# I # δ$ & = ε = ε K r ' r
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21 εο ε 0 : ε = o 0.pFcm 08854 . # $ 0 # $: -= Hε ) / 0: J 0 J : H = + ,,,B = () Kx 10005. C =− DDxxma D
$ $ !# 999= J # $ ! , $ !' : : " J : 9K + 0 h = h : o o
5 J : " J : ! '7 $ !' 0 : 0: 6 " " " + ! " ! " ! -0)@
22 ! &' $ !. " 0: A # $ " 0:0 : :0 8 299 " 92 9 " : : 0:5 2=5 + 7 " =+ 0 : 0 =−() Mhhhoo C DDD=−() a xxaa − CMCaxa 0 : 0 : ()+ -− + ,,,B H = - -− ++
# *: 2$ 0 : 999= # $ : : 7 " : 0 : 0 : 9
23 $ $ !#%
h CC() C -C @ =−1 a f xf xa h CC() C-C o xa xf f a CC() C− C Cxser = xf xa f a 6 : ()− Ca CCCaxaf CC xfa #0:$ #0 : $ 0 : : J : - H = ()+ ,,,B CC()()−− C D D - DD=+axfxaxff xxf − CCxa f CC xf a =− : 5 ()()DDCCC−− D = xa a xf f a " x − CCxa f CC xf a " 7 :
h C 1.0005 K = xser x + 2 ho Ca 1Dx 999= # $ # *: 2$ 0
24 0
Typical Measurement Parameters for a Variety of Components and Materials
Component Type Frequency Voltage Equiv. Ckt. Quantity Capacitors Electrolytic, non- 60 Hz .1,.3,1 Series C, D polarized " Electrolytic, 120 Hz Low, dc Series C, D polarized bias " " " 100K-1MHz Series ESR, |Z| " plastic, ceramic > 1kHz .1 - 1vca Series C, D 1000pF " ceramic < 1000pF 1 MHz .1 to 1 vac Series/parallel C, D Inductors High-valued 50 1000 Hz varies Parallel L, Q, Rp " Low-valued (rf) 1k - 1MHz low Series L, Q, Rs Resistors Low values dc - 1kHz varies Series R, Q, L " High values dc - 100 Hz varies Parallel R, Q, Cp Materials Insulators dc, 1k, 1M 1, HVDC Parallel C, D, R, G, dielectric const, K " Semiconductors dc, low freq. varies Parallel C, G, C vs V " Conductors 100, 1k any Series R, Q, L " Magnetic 50-1 kHz varies Series/parallel L, Q, R Motors & Capacitance 1k, 1M 1 Parallel C, D transformer " Inductance 50 Hz to 1 Series L, Q 1MHz " Resistance DC, 100Hz 1 Series R, Q Cables Capacitance 1k, 1M 1 Series C " Inductance as required any Series L " Impedance 1k, 1M any Series/parallel Z Battery Impedance 100,1k 1 Series Z, R Circuit bd. Impedance 1k, 1M 1 Series C, Z, L, G Network Impedance as required any Series/parallel R. L, C, Q, G, Z, G, Y, θ Filters Impedance as required any Series/parallel R, L, C, Q, G, Z, G, Y, θ Transducer as required any Series/ parallel Z, C, L, R, θ s Sensors as required any Series/ parallel all
25