RCL (1943)

from the study of capacitance you will recall that a capacitor stores electricity from the study of inductance you know that an inductor reacts against any change of current flow resistance is always present in both capacitors and inductors combinations of resistance capacitance and inductance are very widely used in radio apparatus to understand how capacitors work in combination with inductors it is important first to understand the effect of a capacitor upon an alternating current you will remember that when an EMF is impressed across a condenser and resistance in series a large current flows at first gradually diminishing toward zero value as the full charge is reached you will also remember that when a fully charged condenser is short circuited through a small resistance a very heavy current flows immediately the amount gradually decreasing as the condenser is drained of its charge you can see as shown by these graphs that at a given time after the beginning of a charge or a discharge an equal current will be flowing in each case if the polarity of a charging Source perhaps a battery were reversed instead of merely discharging a Charged condenser by a short circuit the so-called charge and discharge currents would be identical in value but exactly opposite in direction they can be thought of conveniently as recharge currents it is convenient to think of each curve as a recharge curve when dealing with any alternating current and to represent a recharge current in One Direction in the normal manner the recharge curve in the opposite Direction can then be represented by turning the graph upside down direction of the current can be further distinguished by labeling the one way plus and the other way minus the time axis can be designated as zero since it is halfway between the largest values of plus and minus then a graph depicting a series of reversals would develop alternately above the time AIS and below the time a axis such a graph shows both intensity of current and direction of current flow current in relation to time can thus be pictured the plus and minus signs can also indicate voltage changes in which case the terms plus and minus correspond to positive and negative for instance on this graph is shown the battery voltage which has caused this Square shaped Wave by its periodic reversal voltage in relation to time can thus be pictured by Design or accident all AC circuits contain resistance you will remember that because of this resistance the voltage as measured across the terminals of a condenser increases as the current decreases pieces also that when the condenser voltage reaches a value equal to that of the battery EMF no further current flows this condition can be pictured on a graph as the voltage increases the current decreases until it finally no longer flows on a graph of the battery voltage at a point marked x a condition may be assumed when the voltage is still positive but about to reverse on the current curve the current is then reached practically zero on the curve of the condenser voltage the voltage has reached a value substantially equal to the battery EMF then when the battery voltage is reversed a heavy current immediately flows in the opposite direction the voltage across the condenser however requires time to adjust itself to a negative value because of the resistance in the circuit thus for a period of time following each reversal a condition exists where the current has changed Direction and is in the negative or minus section while the voltage is still in the positive or plus section at point Y the opposite condition exists and the current is positive while the voltage is negative you can say therefore that in a circuit consisting of capacitance and resistance the current and voltage are not in step and that the current leads the voltage also in a circuit containing inductance and resistance the current and voltage are not quite in step over the graph of alternating battery voltage there is superimposed the curve which the current takes due to the reactance of the inductor again periods are found when the voltage and current are on opposite sides of the zero line in this case however the current lags behind the voltage which produces it a quickly reversible battery has been used to illustrate the generating of an alternating current and voltage on a graph this produces a square shaped wave waves of such abrupt characteristics are not common the voltage produced by a simple AC generator or a vacuum tube oscillator follows this curve as shown on an oscilloscope it's called a sine wave one complete swing above and below the time axis is called a cycle such a wave generated along the timeline by the gradual alternation of a point between positive and negative is a curve which in its proportions is as common as the seasons of the year or the movement of ocean waves it's the type of curve which describes the motion of a pendulum it also describes the motion of the balance wheel of a watch since A S Wave involves such a gradual change of current the sound made by a S Wave is reproduced through a loudspeaker has a musical quality waves of certain shapes are used in special applications they include square waves and other odd shapes although they are seldom used to produce audible sounds it's interesting to note how their sounds differ from those of sine waves the normal sine wave representing a voltage applied to across an inductance will be of the type shown in this graph the current through an inductance in negligible resistance can also be pictured as a sine wave and since the current in an inductance comes later than or lags the voltage producing it the current sine wave reaches its peak values later than the voltage sine wave does in a capacitive circuit the voltage and current can be pictured as sine waves but in this case the current comes before or leads the voltage this graph shows the sine waves representing the voltage and current in a circuit having resistance only notice that when there is zero voltage there is zero current when the voltage increases so does the current they keep right in step or as it's called in Phase now if a small part of the resistance is removed and a small inductive reactants is substituted in its place the entire current Curve will move slightly to the right showing that the current now lags the voltage and the current and voltage are no longer in Phase the points of zero voltage and zero current no longer coincide more resistance is now removed and a comparable amount of inductive reactance is substituted finally a point is reached where there is no resistance left an inductive reactance alone displaces the current curve to its maximum distance to the right starting again with a purely resistive circuit capacitive reactant is substituted for resistance this displac the current curve to the left showing that the current now leads the voltage so that they're no longer in Phase when the total opposition to current flow becomes capacitive reactants only the curve reaches its maximum displacement to the left notice that when this occurs the current is zero at the time the voltage is maximum when the maximum inductive reactant is placed in the circuit and displaces a current curve all the way to the right the current is also zero what the time the voltage is maximum the curve has now been shifted toward the left it indicates a circuit with capacitance and some resistance if you wish to bring the current curve back in phase one way to do it is to substitute more resistance for the capacitive reactants there is also another method since inductive reactant tends to shift the curve in the opposite direction the assertion of inductance in series with the capacitance will tend to balance the phase shift a point of complete balance occurs when the amount of inductive reactants exactly equals the amount of capacitive reactants at this point the circuit behaves as though neither inductance nor capacitance were present and as though there were only resistance in the circuit this point is called resonance e suppose you are working with this circuit and that you wish to calculate the voltage at the AC Source this coil which is wound with heavy copper wire has an inductance quality such as to produce 4 Ohms of reactants at the AC frequency with which you are working this resistance of 3 ohms is connected in series with it measuring the current with a special ammeter which reads maximum values you find it to be exactly 1.0 amp If This Were a DC circuit the first step in calculating the voltage at the source would be to add together all of the series resistances to get a total value of R now suppose that in this RL circuit you add arithmetically the resistance and reactance 4 Ohms plus 3 ohms is 7 ohms substituting these values in the ohms Law formula you have e = i * r or e = i * 7 or e = 7 vol then on checking the maximum value of the voltage drop across the inductance you find a drop of 4 volts checking the voltage drop across the resistor you find a drop of 3 volts this makes a total drop of 7 volts and checks with your calculations now you measure the source voltage expecting it to read 7 volts but it reads 5 volts obviously there must be a characteristic concerning AC to which DC formulas do not apply an explanation of this will be given with the help of a drawing first consider two voltage curves or sine waves which are in Phase to add these sine waves two corresponding parts of the curves may be taken the voltage of this curve as measured by the height is 3 volts the height as shown by the dividers indicates the instantaneous values of the curve the second curve represents a value of 4 volts the two curves can be added together by adding all their instantaneous values and then plotting graphically the resulting values by connecting these points a new curve is derived which is the sum of the first two this is a graphic method of adding inphase voltages and is as simple as adding DC out of phase voltages can be added by the same method here again is a 3vt sine wave and a 4volt wave since in this case the 4volt wave represents a voltage drop across a pure inductance it is displaced to the left or leads the 3vt wave which is due to resistance alone here is a 3vt peak the 4volt peak is of purely reactive nature and therefore is displaced as you see it note that these values are the same as the voltage drops which were measured a few moments ago across the inductance 4 volts across the Resistance 3 volts now the instantaneous values of the two curves are added as in the case of the inphase components and the resultant values are plotted on the graph when these points are connected a new curve is formed between the first two notice that the value of the new curve is not the direct sum of the other two or 7 volts but is 5 volts this corresponds with the reading on the meter when the voltage source was measured this graphic method is one way of determining Source voltage when combining resistive and reactive voltages it is possible to compare a complete voltage cycle with the complete revolution of a wheel in other words 360° this voltage curve which may be due to the drop across a resistance consists of one complete cycle it occupies 360° in the graph now a voltage due to an inductive reactance in series with the resistance is introduced you will expect it to be displaced to the left the maximum amount due to the reactant or to the point at which the voltage due to the resistance alone is maximum at the time when the voltage due to the reactants is zero as can be measured on the scale this displacement amounts to exactly 90° or you can say that the resistance voltage passes through zero 90° after the inductance voltage goes through zero in that respect the inductive voltage leads the resistance voltage 90° because of this 90° relation you can use a geometric method of adding voltages in a circuit containing resistance and reactant first draw a straight horizontal line of a length representing one of the voltages to be added say the voltage across the resistor you may use any unit of measurement centim in or feet this line is 3 units long and is labeled e subr it corresponds to the 3vt drop across the resistor the 4vt drop across the reactants must also be considered to represent the four volts draw another line four units long notice that this is drawn at right angles or 90° from one end of the first line it is labeled E subx A 90° angle is used in this case because it corresponds to the 90° displacement between the 3vt curve representing the dropic across the resistor and the 4vt curve representing the drop across the reactants now for the next step draw a straight line connecting the extremities of the two lines already drawn this line is labeled e subtotal finally measure the line just drawn its length 5 units corresponds to the measured voltage at the source as before the voltmeter indicates this voltage as being 5 volts this is true even though the drop across the resistor is 3 volts and though the drop across the reactants is 4 volts you will recognize that the method of calculation used is that of Pythagoras who many centuries ago proved that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides in the Triangle just drawn there have been represented the voltages across the resistance the inductance and the total of the two together in a series circuit of course the same current must pass through all the components you know from Ohm's law that the voltage across the resistance must be the current I times the resistance r likewise the voltage across the inductance is the same current I times the reactance X the voltage across the entire circuit is the current I times the total opposition of the circuit this total opposition is called impedance and is designated by the letter z now the hypotenuse of the triangle can be found as it was before but using the new terms which have been substituted and which contain the circuit values i z^ s equal iir 2 plus ix2 then the eyes can be taken out and you have a triangle relating the resistance reactance and impedance therefore in a series circuit of resistance and reactants the impedance to alternating current may be found by the theorem of Pythagoras just as the total voltage across the circuit can be found from the voltage es across the parts of the circuit in this impedance triangle X is always drawn at right angles to R to represent the difference in Phase between the voltages across the reactants and resistance since the effect of the inductive reactant is opposite to that of capacitive reactant the impedance triangle for the latter may be drawn inverted you will remember from the study of inductance that as the frequency is increased the reactance increases as the frequency increases the reactant increases from the study of capacitance you will also remember that the greater the frequency the less the capacitive reactants note that this graph is drawn below the horizontal axis this is to indicate that the effect of capacitive reactants is opposite to that of inductive reactants the graph of which was drawn above the axis by combining the two curves a frequency will be found at which the inductive reactance exactly equals the capacitive reactance when this occurs with the inductance and capacitance in series with each other the two balance each other and a condition of resonance results series resonance in this case simply means that the reactive component of the circuit ceases to exist and the total impedance becomes equal to the resistance alone if the frequency has changed ever so slightly higher the inductive reactant component begins to predominate and the impedance Rises if the frequency is changed slightly lower the capacitive reactant component begins to dominate and again the impedance Rises the farther from resonance the frequency is moved the more one or the other of the reactive components predominates and the higher the impedance becomes because it's the nature of capacitive reactants to increase at lower frequencies capacity is the reactance which increases impedance at those frequencies and since inductive reactance increases as the frequency increases it is responsible for increase in impedance at higher frequencies a graph can be drawn showing the series resonance in terms of impedance alone note that the value of impedance from A to B is lowest at resonance impedance never can be quite zero however because of small amounts of resistance always present in the best of circuit components in the case of parallel circuits of induct and capacitance there are two possible paths for the alternating current to travel either through the inductance or through the capacitance here again at low frequencies the reactance of the inductor is very low since the inductor alone constitutes a complete path the impedance from A to B also will be low at these frequencies as the frequency is increased the inductive reactance becomes greater and the the circuit impedance increases to its maximum amount at resonance after the point of resonance of L and C however the bypass effect of the capacitor becomes evident and as the frequency is further increased the reactance of the capacitor decreases and the circuit impedance drops lower and lower resonance may be defined in two ways first resonance is the frequency at which inductive and capacitive reactances are equal second the resonant frequency is a point in a series combination of L and C where the impedance due to their combined reactants is minimum and in a parallel combination it is a point where the impedance is maximum the effect of high or low impedance at resonance may be Illustrated in many ways for instance my voice has been recorded on film in such a manner that all frequencies from 30 to 10,000 Cycles are given equal emphasis or as it is commonly expressed response but if somewhere across the recording circuit there is inserted a series L and C combination resonant at 1,000 Cycles the impedance at that frequency will be low enough to partially shortcircuit my voice at frequencies in the vicinity of 1,000 Cycles when the switch is closed there is a noticeable change in the quality of my voice the response curve shows why this happens it is due to the filtering out of frequencies in the vicinity of 1,000 Cycles when the switch is open my voice sounds normal again in a similar way there might be in Ed across the circuit a parallel combination of CNL also tuned to 1,000 Cycles this combination presents a high impedance near resonance and a low impedance at all other frequencies when the switch is closed which controls this combination of CNL a totally different quality of voice reproduction is evident this is because all frequencies except those in the vicinity of residence are dissipated the net response then is only a very narrow band of frequencies around a thousand Cycles producing a quality not unlike the early telephone it is in Mark contrast to the response curve normally used for modern reproduction of sound with Motion Pictures combinations of R C and L used in such a manner are called filters and may take the form of equalizers compensators tone controls and noise filters a very important use for parallel resonant RCL combinations is the so-called tune circuit used in radio transmission and reception since the reactants values of c andl determine the resonant frequency it is obvious that by varying the values of either C or L the resonant frequency will be changed in tuning a radio receiver L usually remains constant and the value of C is changed in general the maximum practical change in capacitance with a given inductance will allow a change in resonant frequency in a ratio of 3:1 or in the case of a broadcast receiver from 550 kilocycles to 1500 kilocycles for for