• Peak, R.M.S. and Average A-C Values of E & I

    Numerical Comparison Table

    Peak


    R.M.S.


    Average


    1 0.707 0.637
    2 1.414 1.274
    3 2.121 1.911
    4 2.828 2.548
    5 3.535 3.185
    6 4.242 3.822
    7 4.949 4.459
    8 5.656 5.096
    9 6.363 5.733
    10 7.070 6.369
    11 7.777 7.006
    12 8.484 7.643
    13 9.191 8.280
    14 9.898 8.917
    15 10.605 9.554
    16 11.312 10.191
    17 12.019 10.828
    18 12.727 11.465
    19 13.433 12.102
    20 14.140 12.738
    21 14.847 13.375
    22 15.554 14.012
    23 16.261 14.643
    24 16.968 15.286
    25 17.675 15.923
    26 18.382 16.560
    27 19.089 17.197
    28 19.796 17.834
    29 20.503 18.471
    30 21.210 19.107
    31 21.917 19.744
    32 22.625 20.381
    33 23.332 21.018
    34 24.039 21.655
    35 24.746 22.292
    36 25.453 22.929
    37 26.160 23.566
    38 26.867 24.203
    39 27.574 24.840
    40 28.281 25.476
    41 28.988 26.113
    42 29.695 26.750
    43 30.402 27.387
    44 31.109 28.024
    45 31.816 28.661
    46 32.523 29.298
    47 33.230 29.935
    48 33.937 30.572
    49 34.644 31.209
    50 35.351 31.845
    Peak


    R.M.S.


    Average


    51 36.058 32.482
    52 36.765 33.119
    53 37.472 33.756
    54 38.179 34.393
    55 38.886 35.030
    56 39.593 35.667
    57 40.300 36.304
    58 41.007 36.941
    59 41.714 37.578
    60 42.421 38.214
    61 <43.128 38.851
    62 43.835 39.488
    63 44.542 40.125
    64 45.249 40.762
    65 45.956 41.399
    66 46.663 42.036
    67 47.370 42.673
    68 48.077 43.310
    69 48.784 43.947
    70 49.491 44.583
    71 50.198 45.220
    72 50.905 45.857
    73 51.612 46.494
    74 52.319 47.131
    75 53.026 47.768
    76 53.733 48.405
    77 54.440 49.042
    78 55.147 49.679
    79 55.854 50.316
    80 56.561 50.952
    81 57.268 51.589
    82 57.975 52.226
    83 58.682 52.863
    84 59.389 53.500
    85 60.096 54.137
    86 60.803 54.774
    87 61.510 55.411
    88 62.217 56.048
    89 62.924 56.685
    90 63.631 57.321
    91 64.338 57.958
    92 65.045 58.595
    93 65.752 59.232
    94 66.459 59.869
    95 67.166 60.506
    96 67.873 61.143
    97 68.580 61.780
    98 69.287 62.417
    99 69.994 63.054
    100 70.701 63.693

  • Vacuum Tube Formulas and Symbols

    Vacuum Tube Constants
    Amplication factor (Mu or u) is given by
    Dynamic plate resistance in ohms, is given by
    Mutual conductance in mhos, is given by

    Vacuum Tube Formulas
    Gain per stage is given by
    Voltage output appearing in RL, is given by
    Power output in RL, is given by
    Maximum power output in RL which results when
    RL = rp, is given by
    Maximum undistorted power output in RL which
    results when RL = 2rp,is given by
    Required cathode biasing resistor in ohms, for a
    single tube, is given by

    Vacuum Tube Symbols
    Mu or u
    rp
    gm
    Ep
    Eg
    Ip
    RL
    It
    Es
     
    =
    =
    =
    =
    =
    =
    =
    =
    =
    =
    Amplification factor,
    Dynamic plate resistance in ohms,
    Mutual conductance in mhos,
    Plate voltage in volts,
    Grid voltage in volts,
    Plate current in amperes,
    Plate load resistance in ohms,
    Total cathode current in amperes,
    Signal voltage in volts,
    change or variation in value, which may be either an increment (increase) or a decrement (decrease).

    Peak, R.M.S., and Average A-C Values of E & I

    Given Value To get . . .
    Peak R.M.S. Average
    Peak 0.707 * Peak 0.637 * Peak
    R.M.S. 1.41*RMS 0.9*RMS
    Average 1.57*Average 1.11*Average

     


  • Numerical Relations of Angle Functions

    Angle sin Cos Tan Angle o Sin Cos Tan
    0o .0000 1.0000 .0000 45o .7071 .7071 1.0000
    1o .0175 .9998 .0175 46o .7193 .6947 1.0355
    2o .0349 ..9994 .0349 47o .7314 .6820 1.0724
    3o .0523 .9986 .0524 48o .7431 .6691 1.1106
    4o .0698 .9976 .0699 49o .7547 .6561 1.1504
    5o .0872 .9962 .0875 50o .7660 .6428 1.1918
    6o .1045 .9945 .1051 51o .7771 .6293 1.2349
    7o .1219 .9925 .1228 52o .7880 .6157 1.2799
    8o .1392 .9903 .1405 53o .7986 .6018 1.3270
    9o .1564 .9877 .1584 54o .8090 .5878 1.3764
    10o .1736 .9848 .1763 55o .8192 .5736 1.4281
    11o .1908 .9816 .1944 56o .8290 .5592 1.4826
    12o .2079 .9781 .2126 57o .8387 .5446 1.5399
    13o .2250 .9744 .2309 57o .8480 .5299 1.6003
    14o .2419 .9703 .2493 59o .8572 .5150 1.6643
    15o .2588 .9659 .2679 60o .8660 .5000 1.7321
    16o .2756 .9613 .2867 61o .8746 .4848 1.8040
    17o .2924 .9563 .3057 62o .8829 .4695 1.8807
    18o .3090 .9511 .3249 63o .8910 .4540 1.9626
    19o .3256 .9455 .3443 64o .8988 .4384 2.0503
    20o .3420 .9397 .3640 65o .9063 .4226 2.1445
    21o .3584 .9336 .3839 66o .9135 .4067 2.2460
    22o .3746 .9272 .4040 67o .9205 .3907 2.3559
    23o .3907 .9205 .4245 68o .9272 .3746 2.4751
    24o .4067 .9135 .4452 69o .9336 .3584 2.6051
    25o .4226 .9063 .4663 70o .9397 .3420 2.7475
    26o .4384 .8988 .4877 71o .9455 .3256 2.9042
    27o .4540 .8910 .5095 72o .9511 .3090 3.0777
    28o .4695 .8829 .5317 73o .9563 .2924 3.2709
    29o .4848 .8746 .5543 74o .9613 .2756 3.4874
    30o .5000 .8660 .5774 75o .9659 .2588 3.7321
    31o .5150 .8572 .6009 76o .9703 .2419 4.0108
    32o .5299 .8480 .6249 77o .9744 .2250 4.3315
    33o .5446 .8387 .6494 78o .9781 .2079 4.7046
    34o .5592 .8290 .6745 79o .9816 .1908 5.1446
    35o .5736 .8192 .7002 80o .9848 .1736 5.6713
    36o .5878 .8090 .7265 81o .9877 .1564 6.3138
    37o .6018 .7986 .7536 82o .9903 .1392 7.1154
    38o .6157 .7880 .7813 83o .9925 .1219 8.1443
    39o .6293 .7771 .8098 84o .9945 .1045 9.5144
    40o .6428 .7660 .8391 85o .9962 .0872 11.43
    41o .6561 .7547 .8693 86o .9976 .0698 14.30
    42o .6691 .7431 .9004 87o .9986 .0523 19.08
    43o .6820 .7314 .9325 88o .99994 .0349 28.64
    44o .6947 .7193 .9657 89o .9998 .0175 57.29

  • Trigonometric Relationships

    In any right triangle, if we let
    Ø = the acute angle formed by the hypotenuse and
    the base leg,
    ø = the acute angle formed by the hypotenuse and
    the altitude leg,
    H = the hypotenuse,
    A = the side adjacent Ø and opposite ø,
    O = the side opposite Ø and adjacent ø,
    then sine of Ø = sin Ø = O/H
    cosine of Ø = cos Ø = A/H
    tangent of Ø = tan Ø = O/A
    cosecant of Ø = csc Ø = H/O
    secant of Ø = scc Ø = H/A
    cotangent of Ø = cot Ø = A/O
    also
    sin Ø = cos ø csc Ø = sec ø
    cos Ø = sin ø scc Ø = csc ø
    tan Ø = cot ø cot Ø = tan ø
    and
    1/sin Ø = csc Ø 1/csc Ø = sin Ø
    1/cos Ø = sec Ø 1/sec Ø = cos Ø
    1/tan Ø = cot Ø 1/cot Ø = tan Ø
       The expression “arc sin” indicates, “the angle whose sine is”…;
    like wise arc tan indicates, “the angle whose tangent is”…etc.
    See formulas in table below
    Known
    Values
    Formulas for determining Unknown Values of …
    A O H Ø ø
    A & O arc tan O/A arc tan A/O
    A & H arc cos A/H arc sin A/H
    A & Ø A tan Ø A/cos Ø 90° – Ø
    A & ø A/tan ø A/sin ø 90° – ø
    O & H arc sin O/H arc cos O/H
    O & Ø O/tan Ø O/sin Ø 90° – Ø
    O & ø O tan ø O/cos ø 90° – ø
    H & Ø 90° – Ø
    H & ø 90° – ø

  • Transmission Line Formulas

    Concentric Transmission Lines
    Characteristic impedance in ohms is given by

    R-f resistance in ohms per foot of copper line, is given by

    Attenuation in decibels per foot of line, is given by
    Where: Z
    ra
    d1

    d2

    f


    =
    ==
    =

    =

    =

    characteristic impedance in ohms,
    radio frequency resistance in ohms per foot of copper line,
    attenuation in decibels per foot of line,
    the inside diameter of the outer conductor, expressed in inches,
    the outside diameter of the inner conductor, expressed in inches,
    frequency in megacycles.
    Two-Wire Open Air Transmission Lines
    Characteristic impedance in ohms is given by

    Inductance in micro henrys per foot of line is given by

    Capacitance in micromicrofarads per foot of line is given by
    Attenuation in decibels per foot of wire is given by

    R-f resistance in Ohms per loop-foot of wire, is given by

    Where: Z
    D
    d
    L
    C
    db
    Rf
    f


    =
    =
    =
    =
    =
    =
    =
    =
    characteristic impedance in ohms,
    spacing between wire centers in inches,
    the diameter of the conductors in inches,
    inductance in microhenrys per foot of line,
    capacitance in micromicrofarads per foot of line,
    attenuation in decibels per foot of wire,
    r-f resistance in ohms per loop-foot of wire,
    frequency in megacycles.
    Vertical Antenna
       The capacitance of a vertical antenna, shorter than one-quarter
    wave length at its operating frequency, is given by
    Where: Ca
    l
    d
    f
    e
    =
    =
    =
    =
    =
    capacitance og the antenna in micromicrofarads,
    height of antenna in feet,
    diameter of antenna conductor in inches,
    frequency in megacycles.
    2.718 (the base of the natural system of logarithms).

  • Transient I and E in LCR Circuits

    The formulas whhich follow may be used to closely
    approximate the growth and decay of current and
    voltage in circuits involving L, C and Rwhere    i  =  instantaneous current in amperes at any
    given time (t),
    E  =  potential in volts as designated,
    R  =  circuit resistance in ohms,
    C  =  capacitance in farads,
    L  =  inductance in henrys,
    V  =  steady state potential in volts,
    VC =   reactive volts across C,
    VL =   reactive volts across L,
    VR =   voltage across R,
             RC  =  time constant of RC circuit in seconds,
    L/R  =  time constant of RL circuit in seconds,
    t   =  any given time in seconds after switch
    is thrown,
    e   =  a constant, 2.718 (base of the natural
    system of logarithms),
    Sw  =  switchThe time constant is defined as the time in seconds
    for current or voltage to fall to 1/e or 36.8% of its initial
    value or to rise to (1 – 1/e) or approximately 63.2%
    of its final value.
    Charging a de-energized Capacitive Circuit Discharging an Energized Capacitive Circuit
    E  =  applied potential. E  =  potential to which C is
    charged prior to closing Sw.
    Voltage is Applied to a De-energized Inductive Circuit An Energized Inductive Circuit is Short Circuited
    E  =  applied potential. E  =  counter potential induced in coil when
    Sw is closed.


    Steady State Current Flow

    In a Capacitive Circuit In an Inductive Circuit
       In a capacitive circuit, where resistance loss components
    may be considered as negligible, the flow of current at a given
    alternating potential of constant frequency, is expressed by
       In an Inductive circuit, where inherent resistance and
    capacitance components may be so low as to be negligible,
    the flow of current at a given alternating potential of a
    constant frequency, is expressed by
    where I
    XC
    E
    =
    =
    =
    current in amperes,
    capacitive reactance of the circuit in ohms,
    applied potential in volts.
    where I
    XL
    E
    =
    =
    =
    current in amperes,
    Inductive reactance of the circuit in ohms,
    applied potential in volts.

     


  • Most Used Formulas

    Resistance Formulas DIELECTRIC CONSTANTS
    In series 
    In parallel 

    Two resistors
    in parallel
    Kind of                                                Approximate*
    Dielectric                                                K Value

    Air (at atmospheric pressure) ………………..
    Bakelite …………………………………………….
    Beeswax …………………………………………..
    Cambric (varnished) ……………………………
    Fibre (Red) ……………………………………….
    Glass (window or flint) …………………………
    Gutta Percha ……………………………………..
    Mica ………………………………………………..
    Paraffin (solid) ……………………………………
    Paraffin Coated Paper …………………………
    Porcelain …………………………………………..
    Pyrex ……………………………………………….
    Quartz ………………………………………………
    Rubber ……………………………………………..
    Slate …………………………………………………
    Wood (very dry) …………………………………
    1.0
    5.0
    3.0
    4.0
    5.0
    8.0
    4.0
    6.0
    2.5
    3.5
    6.0
    4.5
    5.0
    3.0
    7.0
    5.0

    * These values are approximate, since true values
    depend upon quality or grade of material used, as
    well as moisture content, temperature and frequency
    characteristics of each

    Capacitance
    In parallel
    In series

    Two capacitors
    in series
    The Quantity of Electricity Stored Within a Capacitor is Given by Self-Inductance
               
    where   Q = the quantity stored in coulombs,
    E = the potential impressed across the
    capacitor in volts,
    C = capacitance in farads.
    In series 
    In parallel 

    Two inductors
    in parallel
    The Capacitance of a Parallel Plate
    Capacitor is Given by
    Coupled Inductance
               
    where   C = capacitance in mmfd.,
    K = dielectric constant,
    *S = area of one plate in square centimeters,
    N = number of plates,
    *d = thickness of the dielectric in centimeters
    (same as the distance between plates).*When S and d are given in inches, change
    constant 0.0885 to 0.224. Answer will still be
    in micromicrofarads.
    In series with fields aiding

    In series with fields opposing

    In parallel with fields aiding

    In parallel with fields opposing
    Coupled Inductance (cont.) Resonance (cont.)
      where      L<subt< sub=””></subt<> = the total inductance,

    M = the mutual inductance,

    L1 and L2 = the self inductance of                                                        the individual coils.

      where      F<subr< sub=””></subr<> = the resonant frequency
    in cycles per second,
    L = inductance in henrys,
    C = capacitance in farads,
    = 6.28,
    = 39.5,
    Mutual Inductance Reactance
       The mutual inductance of two r-f coils with fields
    interacting, is given by

    where    M = mutual inductance, expressed
    in same units as LA and LO,
    LA = Total inductance of coils
    L1 and L2 with fields aiding,
    LO = Total inductance of coils
    L1 and L2 with fields opposing,
      of an inductance is expressed by

    of an capacitance is expressed by

    where   XL =  inductive reactance in ohms,
    (known as positive reactance),
    XC =  capacitive reactance in ohms,
    (known as negative reactance),
    F  =  frequency in cycles per second,
    L  =  inductance in henrys,
    C  =  capacitance in farads,
    =  6.28,
    Coupling Coefficient Frequency from Wavelength
       When two r-f coils are inductively coupled so as to
    give transformer action, the coupling coefficient is
    expressed by,

    where    K  =  the coupling coefficient;
    (K x 102 = coupling
    coefficient in %),
    M  =  the mutual inductance value,
    L1 and L2  =  the self-inductance of the two coils
    respectively, both being expressed
    in the same units.

    where    =  wavelength in meters.
    where    =  wavelength in centimeters.
    Resonance Wavelength from Frequency
        The resonant frequency, or frequency at which
    inductive reactance XL equals capacitive reactance
    XC is expressed by

    also  
    and  

    where    F =  frequency in kilocycles.
    where    F =  frequency in megacycles.
    Q or Figure of Merit Impedance (cont.)
     of a simple reactor    In series circuits where phase angle and any two of the
    Z, R and X components are known, the unknown
    component may be determined from the expressions:
     of a single capacitor                 

                

      where Q  =  a ratio expressing the figure of
    merit,
      where Z  =  magnitude of impedance in ohms,
    XL  =  inductive reactance in ohms, R  =  resistance in ohms,
    XC  =  capacitive reactance in ohms, X  =  reactance (inductive or
    capacitive) in ohms,
    RL  =  resistance in ohms acting in series
    with inductance,
    Nomenclature
    RC  =  resistance in ohms acting in series
    with capacitance,
    Z  =  absolute or numerical value of
    impedance magnitude in ohms,
    Impedance R  =  resistance in ohms,
       In any a-c circuit where resistance and reactance
    values of the R, L and C components are given, the
    absolute or numerical magnitude of impedance and
    phase angle can be computed from the formulas
    which follow:
    XL  =  inductive reactance in ohms,
    XC  =  capacitive reactance in ohms,
    L  =  inductance in henrys,
    C  =  capacitance in farads,
       In general the basic formulas expressing total
    impedance are:
    RL  =  resistance in ohms acting in
    series with inductance,
     for series circuits, RC  =  resistance in ohms acting in
    series with capacitance,
           =  phase angle in degrees by which
    current leads voltage in a capacitive
    circuit, or lags voltage in an inductive
    circuit.  In a resonant circuit , where
    XL equals XC, equals 0o.


    for parallel circuits,
        
     Degrees X 0.0175 = radians.
    1 radian = 57.3o
       See page 17 for formulas involving impedance, conductance, susceptance and admittance. Numerical Magnitude of Impedance . . .
     of resistance alone
                               Z = R
                              = 0o
       
    of resistance in series,
    Z = R1 + R2 + R3etc.
    = 0o
       
    of inductance and capacitance in series,Z = XLXC
    = -90o when XL < XC
    = 0o when XL = XC
    = +90o when XL > XC
       
    of inductance alone
    Z = XL
    = +90o
       
    of inductance in series,
    Z = XL1 + XL2 + XL3etc.
    = +90o
       
    of resistance, inductance and capacitance in series

       
    of capacitance alone,
    Z = XC
    = -90o
       
    of resistance in parallel,

    = 0o
       
    of capacitance and resistance in parallel,
    Z = XC1 + XC2 + XC3etc.
    = -90o
       
    or where only 2 capacitances C1 and C2 are
    involved,

    = -90o
       
    of resistance in parallel,

    = 0o
       
    of resistance and inductance in series,

       
    of inductance in parallel,

    = +90o
       
    of resistance and capacitance in series,


    or where only 2 inductances L1 and L2 are involved,

       
    of inductance and capacitance in parallel,

       
    of capacitance in parallel


    or where only 2 capacitances C1 and C2 are involved,

       
    of inductance, resistance and capacitance in parallel,

       
    of resistance and inductance in parallel,

       
    of inductance and series resistance in parallel with
    capacitance,

       
    of capacitance and resistance in parallel,

       
    of capacitance and series resistance in parallel with
    inductance and series resistance,

    Conductance Admittance
      In direct current circuits, conductance is expressed
    by,

    where      G = conductance in mhos,
    R = resistance in ohms,
    In d-c circuits involving resistances R1, R2, R3, etc.
    in parallel,
    the total conductance is expressed by
    Gtotal = G1 + G2 + G3 …etc.
    and the total current by
    Itotal = E Gtotal
    and the amount of current in any single
    resistor,
    R2 for example, in a parallel group,
    by


    R, E and I in Ohm’s law formulas for d-c circuits
    may be expressed in terms of conductance as follows:
                     
    where      G  =  conductance in mhos,
    R  =  resistance in ohms,
    E  =  potential in volts,
    I  =  current in amperes,
      In an alternating current circuit, the admittance of a series
    circuit is expressed by,

    Admittance is also expressed as the reciprocal of
    impedance, or

    where      Y = admittance in mhos,
    R = resistance in ohms,
    X = reactance in ohms,
    Z = impedance in ohms,
    R and X in Terms of G and B
      Resistance and reactance may be expressed in terms
    of conductance and susceptance as follows:
               
    G, B, Y and Z in Parallel Circuits
    Susceptance   In any given a-c circuit containing a number of smaller
    parallel circuits only,

    the effective conductance GT is expressed by
    GT = G1 + G2 + G3etc.

    and the effective susceptance BT by
    BT = B1 + B2 + B3etc.

    and the effective admittance YT by

    and the effective impedance ZT by

    where       R  =  resistance in ohms,
    X  =  reactance (capacitive or induc-
    tive) in ohms,
    G  =  conductance in mhos,
    B  =  susceptance in mhos,
    Y  =  admittance in mhos,
    Z  =  impedance in ohms,


  • 70-Volt Loud-Speaker Matching Systems

    The EIA 70.7 volt constant voltage system of power
    distribution provides the engineer and technician with
    a simple means of matching a number of loudspeakers
    to an amplifier.   To use this method:
     Since the voltage at rated amplifier power is 70.7,
    this reduces to:
                         (2)
    1.  Determine the power required at each                 loudspeaker.

    2.  Add the power required for the individual
    speakers and select and amplifier with a rated
    power output equal to or greater than this total.

    3.  Select 70.7-volt transformer having primary
    wattage taps as determined in step 1.*

    4.  Wire the selected primaries in parallel across
    the 70.7-volt line

    From formula (2) these relationships are:
    1 watt requires 5000 ohm primary
    2 watt requires 2500 ohm primary
    5 watt requires 1000 ohm primary
    10 watt requires 500 ohm primary

    Once the primary taps have been determined,
    continue on through step 4 and 5 as outlined above.
    When selecting transformer primary taps, use the next
    highest available value above the computed value.  A
    mismatch of 25% is generally considered permissible.

    Example:  Required

    5. Connect each secondary to its speaker; selecting
    the tap which matches the voice coil inpedance.
      One 6 watt speaker with 4 ohm voice coil.
    Two 10 watt speakers with 8 ohm voice coils
    (use one transformer at this location).
         For transformers rated in impedance, the
    following formulas may be used to determine
    the proper taps in step 3.
      (1-2) Total power = 6 + 10 + 10 = 26 watts
    (use 30-watt amplifier or other amplifier
    capable of handling at least 26 watts).
      Primary         (Amplifier output voltage)2
    Impedance.       Desired speaker power
         (3)  ohms
    (use 1000 ohm transformer)
    ohms
    or                                             (1)
     *These transformers have the primary taps marked
    in watts and the secondaries marked in ohms.
       (4-5) See sketch below.

  • Minimum Loss Pads

    For Matching Two Impedances where Z1 > Z2 Matched, use a resistor , RL in series with the smaller
    impedance such that
    If the smaller impedance only is to be matched, use a
    resistor RS in shunt across the larger impedance such that
    Where Only One Impedance is to be Matched
    If the larger impedance only is to be Here also

    Tables of R1 and R 2 Values

    When Z1 is 600 ohms
    and Z2 is less than 600 ohms.

    Z2 500 400 300 250 200 150 100 75 50 40 30 25
    R1 245 346 424 458 490 520 548 561 575 580 585 587
    R2 1,225 694 425 328 245 173 110 80.2 52.2 41.4 30.8 25.6
    db
    Loss
    3.8 5.7 7.6 8.7 10.0 11.4 13.4 14.8 16.6 17.6 18.9 19.7

    When Z2 is less than 25 ohms,


    and      R2 = Z2         
    Where Z2 is 600 ohms
    and Z1 is greater than 600 ohms.

    Z1 800 1,000 1,200 1,500 2,000 2,500 3,000 3,500 4,000 5,000 6,000 8,000 10,000
    R1 400 632 849 1,162 1,673 2,180 2,683 3,186 3,688 4,690 5,692 7,694 9,695
    R2 1,200 949 849 775 717 688 671 659 651 638 633 624 619
    db
    Loss
    4.8 6.5 7.6 9.0 10.5 11.6 12.5 13.3 13.9 15.0 15.8 17.1 18.1

    When Z1 is greater than 10,000 ohms,

    let  R1 = Z 1 – 300
    and  R1 = 600