General Radio Instrument Notes #102 Series & Parallel Components of Impedance

8/9/2019 General Radio Instrument Notes #102 Series & Parallel Components of Impedance

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E N E R A L R A D I O COMPANY

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SERIES A N D PARALLEL COMPONENTS

OF IMPEDANCE

Users of General Radio Type 1608 A and 1650 A Impedance Bridges can choose tomeasure either series or parallel capacitance and either series or paralkl inductance. Thisfeature has apparently confounded some who regard the choice befween C and C or L and

L as an unnecessary complication in wh t should be a simple measurement.

The confusion r rather, lack of information bout series and parallel parametersi s nothing new. Writ ing in the January, 1946 General adio Experimenter, W. Norris Tuttle

noted that Discussion of . . series and parallel components, however, seldom appears in

the elementary textbooks. Dr. Tuttle, a member o f GR s engineering staff, went en to set

fhe record straight on series and parallel components of impedance. Sixteen years rater, for

the benefit of a new senerotion who may not be clear on the series-parallel distinction, we

offer the following reprint of Dr. Tuttle's excellent Experimenter article.

We have received lately a number of inquiries aboutthe meanings of such terms as series capacitance,

parallel capacitance, series resistance, ' parallel re-

sistance, etc. as thev are med in instnlction hooks forGeneral adio br id~es . Although ost en@neers thinkin terms of the series components of impedance, rnanvtypes of prohlems, pnrticnlarlv those involvinq vacuumtubes, are more dmplv handled in term? of the paraTlelcomponents. Certain h r i d g ~ ircuits give rlirectlv theseries companents of an impedance, while others canhe arranged to give the parallel the choicedepending on the intended application. Discussion ofthe relationship h~ ti vee n he series and parallel com-ponents, however, seldom appears in the elementarytextbooks.

That anv impedance can he represented both waysis clear from the fact that measurements on it at asingle frequency can determine only the relationshiphehveen the voltage across the impedance and the in-phase and quadrature cornponentq of the current Row-ing through it. Stated in terms of power enzineering, acircuit element draws a certain amount of power at n

particular value of power factor, and these two quantities compIetely define the effective impedance of theelement for the conditions applying. I t il sometimesconvenient t represent the impedance as a pure re-sistance in series with a pure reactance, but it is very

often more convenient to consider it as made up of adifferent value of resistance in parallel with a reactance.The two representations, however, are completely

equivalent and either pair of components can be simplydetermined in terms of the other pair.

For example it will he seen that. in three cases shownin Figure I, the first configuration of series elementswol~ld raw the same current, both in phase and mag-nitude aq the second confiptration. consisting of resis-tive and reactive elerncnts in The twoarrangements of each case are indistinguisha6Ie fromeach other by measurements ma d e at their terminalsat a xed frequency.

The general relationship between the elements of t heseries and parallel arrangements can be simply foundby equating the current drawn in the hvn cases.

I0 j20Figure 1

Exomples of

equivalent series

and parolld circvifs 10 15



where R and X are the series components and Rand X, are the parallel components, Rationalizing andequating the real and imaginary terms,

Given Rg a n d X

Given R p and X p

The quantity X /R is the fami liar Q or storage fac-tor nf an inductor or capacitor, and its reciprocal is thedissipation factor D, more f requency employed in de-scribing the losses in capacitors. Substituting thesequantities in Equations 2 ) and 35 ,

These equations give the parallel components ofimpedance directly in terms of the series components.The relationshjps, however, serve equally ~tlell whenthe series components are required and the parallelcomponents arc given, because the quantity or L

can he determinec? directly horn either the series orparallel components. Dividing 4 ) by 51,

so that Q can he determined immediately, whichevercomponents are given, and use in Equations 4 ) and( 5 ) to obtain the other components. A further simpli-fication is that only ane of the two Equations 4 ) and( 5 ) need be employed with 6 ) to make the complete

transformation. The three steps i n each case are as~ O ~ ~ O R I S :

If it is preferred to work in terms of dissipation factorthe corresponding steps are :

Given R nd X R

2 ) R = R s I +A

Gioen p and X p

XP1) =

P

It is seen that use of Q or D, which are associated156th equal simplicity wit11 either the series or parallelcomponents, greatly facilitates the transformation.Since 6 ) is reacfily borne in mind, the only relation

that need be remembered is that, as seen from 43, theratio between the parallel and series resistances is thequantity + Q 2 . It should be noted that the parallelresistance ancl parallel reactance are always greater thanthe corresponding series components. 2 s obvious thatfor large he series resistance must be small com-pared wit11 the series reactance, but the parallel resist-ance must be large compared with the parallel react-ance

One of the simplest examples of the utility of theparallel impedance components is in parallel resonant



The nomograph berow greatly slmplifies the process of converting from series to parallelvalues or rice versa) of inductance and capacitance, for values of dissFpation factor up to 10Q down to 0.13. To illustrate use of the nomograph, assume a parallel capacitance of 2 p f

and a of 7 A straight line connecting these two points s seen to cross t h center [C, bar ~ r t100. Therefore, the equivalent series capacitance is 100 f.



Figure 2. Series and parnilel resonant circuits. The copaci lance necessary

ro resonate with o given inductance wil l depend upon whether the

element3 ore t~n nec fs d n series or in parallel .

circuits where the coil losses are high I t will be seenin Figure 2 that parallel resonance occurs when thecapacitor reactance is exactly equal to the parallelreactance of the inductor regardless of the coil losses.If the tuning capacitance for parallel resonance is de-termined from the series components of the coil im-pedance on the other hand the required value dependsboth on the resistance and on the reactance. In theseries circuit the opposite applies nd resonance occurswhen the capacitor reactance is exactly equal to the

series reactance of the inductor. Where the Q of thecoil is high the diEerence between its series and paral-lel reactance is negligible in ordinary applications.Even with a Q of 10 the difference is only one per cent.But for lower vaIues of Q the difference rapidly in-creases. Tbe parallel reactance of an inductor with a

of s twice the series reactance so that only halfthe capacitance is required to tune it t o resonance ina paraIleI circuit as in a series circuit.

W N TUTTE

The foregoing article by Dr. Tuttle discusses seriesand parallel reectance not series and parallel capaci-tance and inductance as st . he formulas nd Ican be easily derived from those appearing in thearticle:

G E N E R L R D I O C O M P N YW ST CONCORD MASSACHUSETTS USA

Printed in U.S.A.



G E N E R A L R D I O COMPANY

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NSTRUMENT N O T S

OPERATING THE GENERAL RADIO TYPE 1133 A FREQUENCY

CONVERTER WITH COUNTERS O OTHER MAKES

The Type 1133-A Frequency Converter is de-signed for us e with the GR Type 1130-A Digital Timeand Frequency Meter and the Type 1153 A Digital

Frequency Meter. It can, however, be operated withother 10-Mc counters or a s a general-purpose f re-quency conve rter with other ac ces sor y equipment.

REFERENCE FREQUENCYINPUT

The conver ter ordinarily requires a 5-Mc reference-frequency hput (supplied by the counter7patched into a re a r connector. Qther reference-fre-quency sources can be used, however, as describedbelow.

Any source of 5 Mc/s capabIe of supplying 15m V o r mo re in to a 5 0 o h load (e.g. 30 mV behind50 ohm) can be used to drive the converter, Becauseof a narrow-band cry sta l filt er in the converter, alower-frequency sou rce with a s trong harmonic at 5Mc s can also be used.

The Type 1153 -P1 Frequency Multiplier, whichplugs into the r e a r of the converter, multiplies a 100-kc reference-frequency input of -volt r m s o r g r e a t er(1-volt peak-to-peak for a square wave to 5 Mc/s to

operate the converter. The multiplier require s asupp ly vo lh g e of 3-20 V a t 8 mA B will also operatewith other input frequen cies which are submultiples of

Mc/s, such a s 200 kc/s and 500 kc/s .

If a 1-Mc signal does not b v e sufficient 5-Mcharmonic voltage to drive the converter, a fast-switch-ing germanium or silicon diode can be connected inse ri es with the reference-frequency input connectorof the converter. Satisfactory diodes are the IN994 ndI-INDSCPOO types. The diode can be conveniently mountedin a Type 874-X Insertion Unit which can be pluggedinto the INPUT connector t the r e a r of the converter.

his scheme works well with Beckman Ins tmmentscounters.

10 MC/S @I-P SERIES 524 COUNTERS

The reference-frequency circuits of the con-v e n e r can b e o p erated f ro m a 10-Mc s ou rc e of 100mY o r greater in to 50 ohms if the fir st stage of theconverter is rewi red a s outlined below.

1 Remove instrument from cabinet (see I n s m c -tion Manual page 13).

2 Remove shiel d from 10 s Reference FrequencyGenerator section (remove 10 nuts , see page 14).

3. Clip out C401 and C4 4 (page 29 .

General Radio Instrument Notes #102 Series & Parallel Components of Impedance

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