A differentiating circuit is a circuit whose output voltage is proportional to the first time derivative of the input voltage:
Rice. 3.7.1. Diagram of the differentiating circuit
The differentiating circuit (Fig. 3.7.1) consists of a resistor R and capacitor WITH, whose parameters are chosen in such a way that the active resistance is many times less than the capacitive resistance.
The voltages at the input and output of the circuit are related by the relation:
u in = u out + u C;
u out = i· R
u C =u in - u out = u in - iR ;
If the value i R significantly less than u vh, then u in ≈ u C.
The value τ = RC called time constant of the differentiating circuit.
The smaller the time constant compared to the input pulse width, the higher the differentiation accuracy.
If a sinusoidal voltage is applied to the input of the differentiating circuit, then the output voltage will also be sinusoidal, however, it will be phase-shifted relative to the input voltage, and its amplitude will be less than that of the input. Thus, the differentiating circuit, which is a linear system, does not change the spectral composition of the voltage supplied to it.
Applying a rectangular pulse to the input of the differentiating circuit, which, as is known, consists of an infinite number of sinusoidal components, changes the amplitude and phase of these components, which leads to a change in the shape of the output voltage compared to the shape of the input.
When a rectangular pulse is applied to the input of the differentiating circuit, the capacitor begins to charge WITH through resistance R.
At the initial moment of time, the voltage across the capacitor is zero, so the output voltage is equal to the input. As the capacitor charges, the voltage across it begins to increase exponentially:
u c= u in (1 – e– t/τ) ;
where τ = RC is the time constant of the circuit.
Voltage at the output of the differentiating circuit:
u out = u in - u c= u in - u in (1 – e– t / τ) = u in · e– t / τ) ;
Thus, as the capacitor charges, the voltage at the output of the circuit decreases exponentially. When the capacitor is fully charged, the voltage at the output of the differentiating circuit will become zero.
At the end of the rectangular pulse, the voltage at the input of the circuit will abruptly decrease to zero. Since the capacitor at this time remains fully charged, from that moment it will begin to discharge through the resistance R. At the beginning of the discharge of the capacitor, the voltage at the output of the circuit is approximately equal in magnitude to the voltage across the capacitor, but with the opposite sign, since the direction of the discharge current is opposite to the charge current. As the capacitor discharges, the voltage at the output of the circuit decreases exponentially.
Consider the RC circuit shown in fig. 3.20 a. Let voltage u1(t) act at the input of this circuit.
Rice. 3.20. Differentiating RC-(a) and RL-(b) chains.
Then the following relation holds for this chain:
and taking into account the transformations we will have
If for a given signal we choose the time constant of the circuit τ=RC so large that the contribution of the second term on the right side of (3.114) can be neglected, then the variable voltage component uR≈u1. This means that at large time constants, the voltage across the resistance R repeats the input voltage. Such a circuit is used when it is necessary to transmit signal changes without transmitting a constant component.
For very small values of τ in (3.114), the first term can be neglected. Then
i.e., for small time constants τ, the RC circuit (Fig. 3.20, a) differentiates the input signal, therefore such a circuit is called a differentiating RC circuit.
The RL chain also has similar properties (Fig. 3.20, b).
Rice. 3.21. Frequency (a) and transient (b) characteristics of differentiating circuits.
Signals passing through RC and RL circuits are called fast if
or slow if
It follows that the considered RC circuit differentiates slow signals and passes fast signals without distortion.
For harmonic e. d.s. a similar result can be easily obtained by calculating the transfer coefficient of the circuit (Fig. 3.20, a) as the transfer coefficient of a voltage divider with stationary resistances R and XC \u003d 1 / ωC:
For small τ, namely, when τ<<1/ω, выражение (3.116) преобразуется в
In this case, the phase of the output voltage (argument K) is equal to π/2. The phase shift of a harmonic signal by π/2 is equivalent to its differentiation. When τ>>1/ω, the transfer coefficient is K≈1.
In the general case, the transmission coefficient module (3.116), or the frequency response of the circuit (Fig. 3.20, a):
and the argument K, or the phase characteristic of this circuit:
These dependencies are shown in Fig. 3.21 a.
The RL circuit in Fig. 1 has the same characteristics. 3.20,b with time constant τ=L/R.
If we take a single voltage jump as the output signal, then by integrating equation (3.114) we can obtain the transient response of the differentiating circuit, or the time dependence of the output signal with a single voltage jump at the input:
The transient response graph is shown in fig. 3.21b.
Rice. 3.22. Integrating RC-(a) and LC-(b) chains.
Consider the RC circuit shown in fig. 3.22, a. It is described by the equation
For small τ=RC (for "slow" signals) uC≈u1. For "fast" signals, the voltage u1 is integrated:
Therefore, an RC circuit whose output voltage is taken from the capacitance C is called an integrating circuit.
The transfer coefficient of the integrating circuit is determined by the expression
At ω<<1/τ K≈1.
The frequency and phase characteristics are described respectively by the expressions
Rice. 3.23. Frequency (a) and transient (b) characteristics of integrating circuits.
and are shown in Fig. 3.23, a. The transient response (Fig. 3.23, b) is obtained by integrating (3.121) with:
For equal time constants, the RL circuit shown in Fig. 1 has the same properties. 3.22b.
An electrical circuit in which the output voltage U out (t) (or current) is proportional to the time integral of the input voltage U in (t) (or current):
Rice. 1 .
Operational amplifier integrator.<В основе действия И. ц. лежит накопление заряда на конденсаторе с ёмкостью WITH under the influence of the applied current or the accumulation of magnetic. flux in an inductor L under the action of the applied voltage I. c. with a condenser.<С наиб, точностью указанный принцип реализуется в интеграторе на операц. усилителе (ОУ) (рис. 1). Для идеального ОУ разность напряжений между его входами и входные токи равны нулю, поэтому ток, протекающий через сопротивление R, equal to charge current
capacitor WITH, and the voltage at the point of their connection is zero. As a result, the Product RC=t, which characterizes the charge rate of the capacitor, called. time constant I. c.<Широко используется простейшая RC-I. c. (Fig. 2, a). In this circuit, the capacitor charge current is determined by the difference between the input and output voltages; therefore, the integration of the input voltage is performed approximately and the more accurately, the lower the output voltage compared to the input. The last condition is satisfied if the time constant t is much greater than the time interval over which the integration takes place. For correct integration of the pulsed input signal, it is necessary that t be much larger than the pulse duration T (Fig. 3). RL-I has similar properties. c., shown in Fig. 2b, for which the time constant is equal to L/R.
Rice. 3.1 -
input rectangular pulse; 2 -
output voltage of the integrating circuit at tdT.
I. c. are used to convert pulses modulated in duration into pulses modulated in amplitude, to lengthen the pulses, obtain a sawtooth voltage, highlight the low-frequency components of the signal, etc. I. c. for operations. amplifiers are used in automation devices and analog computers to implement the integration operation.
53. Transient processes. Commutation laws and their application.
Transitional processes- processes that occur in electrical circuits under various influences, leading them from a stationary state to a new stationary state, that is, under the action of various kinds of switching equipment, for example, keys, switches to turn on or off a source or receiver of energy, in case of breaks in the circuit , in case of short circuits of individual sections of the circuit, etc.
The physical reason for the occurrence of transient processes in circuits is the presence of inductors and capacitors in them, that is, inductive and capacitive elements in the corresponding equivalent circuits. This is explained by the fact that the energy of the magnetic and electric fields of these elements cannot change abruptly at switching(the process of closing or opening switches) in a circuit.
The transient process in the circuit is described mathematically by the differential equation
- heterogeneous (homogeneous), if the equivalent circuit of the circuit contains (does not contain) sources of EMF and current,
- linear (non-linear) for a linear (non-linear) circuit.
The duration of the transient process lasts from fractions of nanoseconds to years. Depends on the specific circuit. For example, the self-discharge time constant of a capacitor with a polymer dielectric can reach a millennium. The duration of the transition process is determined time constant chains.
The switching laws apply to energy-intensive (reactive) elements, i.e., to capacitance and inductance. They say: the voltage on the capacitance and the current in the inductance with finite impacts are continuous functions of time, that is, they cannot change abruptly.
Mathematically, this formulation can be written as follows
For capacity;
for inductance.
The switching laws are a consequence of the definitions of the elements of capacitance and inductance.
Physically, the switching law for inductance is explained by the resistance of the self-induction EMF to a change in current, and the switching law for a capacitance is explained by the resistance of the electric field strength of a capacitor to a change in external voltage.
54. Eddy currents, their manifestations and use.
Eddy currents or Foucault currents(in honor of J. B. L. Foucault) - eddy induction currents that arise in conductors when the magnetic field penetrating them changes.
Eddy currents were first discovered by the French scientist D. F. Arago (1786-1853) in 1824 in a copper disk located on an axis under a rotating magnetic needle. Due to eddy currents, the disk came into rotation. This phenomenon, called the Arago phenomenon, was explained a few years later by M. Faraday from the standpoint of the law of electromagnetic induction he discovered: a rotating magnetic field induces eddy currents in a copper disk that interact with a magnetic needle. Eddy currents were studied in detail by the French physicist Foucault (1819-1868) and named after him. He discovered the phenomenon of heating of metal bodies rotating in a magnetic field by eddy currents.
Foucault currents arise under the influence of an alternating electromagnetic field and, by their physical nature, do not differ in any way from induction currents arising in linear wires. They are vortex, that is, they are closed in a ring.
The electrical resistance of a massive conductor is small, so the Foucault currents reach a very large force.
The thermal effect of Foucault currents is used in induction furnaces - a conducting body is placed in a coil fed by a high-frequency high-power generator, eddy currents arise in it, heating it to melting.
With the help of Foucault currents, the metal parts of vacuum installations are heated for their degassing.
In many cases, Foucault currents may be undesirable. To combat them, special measures are taken: in order to prevent energy losses for heating the cores of transformers, these cores are recruited from thin plates separated by insulating layers. The advent of ferrites made it possible to manufacture these cores as solid cores.
Eddy current testing is one of the methods of non-destructive testing of products made of conductive materials.
55. Transformer, basic properties and types of construction.
DIFFERENTIATION CIRCUIT- a device designed to differentiate in time electric. signals. Output reaction D. c. u out ( t) is related to the input action u in ( t) ratio , where - post. a quantity that has the dimension of time. There are passive and active D. c. Passive D. c. used in pulse and digital devices to shorten pulses. Active D. c. used as differentiators in analog computing. devices. The simplest passive D. c. shown in fig. 1, A. The current through the capacitance is proportional to the derivative of the voltage applied to it. If the parameters of D. c. are chosen thus,
What u c =u vh, then 
, a . Condition u c =u input is performed if at the highest frequency of the spectrum of the input signal Option passive D. c. shown in fig. 1, b. Under the condition we have and
Rice. 1. Schemes of passive differentiating circuits: A- capacitive RC; b- inductive RL.
Therefore, at the given parameters D. of c. differentiation is the more accurate, the lower the frequencies on which the energy of the input signal is concentrated. However, the more accurate the differentiation, the lower the coefficient. transfer circuit and hence the output level. This contradiction is eliminated in active D. c., where the process of differentiation is combined with the process of amplification. In active D. c. use operational amplifiers(OS) covered by negative feedback (Fig. 2). Input voltage u in ( t) is differentiated by a chain formed by a succession. container connection WITH And R eq - the equivalent resistance of the circuit between the terminals 2-2 ", and then the op-amp is amplified. If you apply voltage to the inverting input of the op-amp, then, provided that its gain, , we get
Rice. 2. Scheme of an active differentiating circuit.
Rice. 3. Passage of an impulse through a differentiating circuit RC: A- input impulse, u in = E at ; b- voltage on the capacitance u c (t); V- output voltage .
For compare. assessments of active and passive D. c. ceteris paribus, you can use the ratio . When passing through D. c. pulse signals there is a decrease in their duration, hence the concept of D. c. as about shortening. Time diagrams illustrating the passage of a rectangular pulse through a passive D. c. are shown in fig. 3. It is assumed that the input voltage source is characterized by zero ext. resistance, and D. c. - the absence of parasitic capacitances. The presence of internal resistance leads to a decrease in the voltage amplitude at the input terminals and, consequently, to a decrease in the amplitudes of the output pulses; the presence of parasitic capacitances - to delaying the processes of rise and fall of output pulses. Active D. of c also have a similar shortening effect.
We have every right to move on to the consideration of chains consisting of these elements 🙂 This is what we will do today.
And the first circuit, the work of which we will consider - differentiating RC circuit.
Differentiating RC circuit.
From the name of the circuit, in principle, it is already clear what elements are included in its composition - this is a capacitor and a resistor 🙂 And it looks like this:
This scheme is based on the fact that current flowing through a capacitor, is directly proportional to the rate of change of the voltage applied to it:
The voltages in the circuit are related as follows (according to the Kirchhoff law):
At the same time, according to Ohm's law, we can write:
We express from the first expression and substitute into the second:
Provided that (that is, the rate of change of voltage is low), we get an approximate dependence for the output voltage:
Thus, the circuit fully justifies its name, because the output voltage is differential input signal.
But another case is also possible, when title="Rendered by QuickLaTeX.com" height="22" width="134" style="vertical-align: -6px;"> (быстрое изменение напряжения). При выполнении этого равенства мы получаем такую ситуацию:!}
That is: .
It can be seen that the condition will be better satisfied for small values of the product , which is called circuit time constant:
Let's see what meaning this characteristic of the chain carries 🙂
The charge and discharge of the capacitor occurs according to the exponential law:
Here, is the voltage across the charged capacitor at the initial moment of time. Let's see what the voltage value will be after time:
The voltage on the capacitor will decrease to 37% of the original.
It turns out that is the time for which the capacitor:
- when charged - will charge up to 63%
- when discharged - discharged by 63% (discharged up to 37%)
With the time constant of the circuit we figured out, let's get back to differentiating RC circuit 🙂
We have analyzed the theoretical aspects of the functioning of the circuit, so let's see how it works in practice. And for this, let's try to apply some kind of signal to the input and see what happens at the output. As an example, let's apply a sequence of rectangular pulses to the input:
And here is how the waveform of the output signal looks like (the second channel is blue):
What do we see here?
Most of the time, the input voltage is constant, which means its differential is 0 (derivative of a constant = 0). This is exactly what we see on the graph, which means that the chain performs its differentiating function. And what are the bursts on the output oscillogram connected with? It's simple - when the input signal is “turned on”, the capacitor is charging, that is, the charging current passes through the circuit and the output voltage is maximum. And then, as the charging process proceeds, the current decreases exponentially to zero, and with it the output voltage decreases, because it is equal to . Let's zoom in on the waveform and then we will get a clear illustration of the charging process:
When the signal is “turned off” at the input of the differentiating circuit, a similar transient occurs, but it is caused not by charging, but by discharging the capacitor:
In this case, we have a small circuit time constant, so the circuit differentiates the input signal well. According to our theoretical calculations, the more we increase the time constant, the more the output signal will be similar to the input. Let's put it to the test 🙂
We will increase the resistance of the resistor, which will lead to an increase:
You don’t even need to comment on anything here - the result is obvious 🙂 We confirmed the theoretical calculations by conducting practical experiments, so let's move on to the next question - to integrating RC circuits.
We write expressions for calculating the current and voltage of this circuit:
At the same time, we can determine the current from Ohm's Law:
Equate these expressions and get:
We integrate the right and left sides of the equality:
As in the case with differentiating RC chain two cases are possible here:
In order to make sure that the circuit is working, let's apply to its input exactly the same signal that we used when analyzing the operation of the differentiating circuit, that is, a sequence of rectangular pulses. For small values, the output signal will be very similar to the input signal, and for large values of the circuit time constant, we will see a signal at the output that is approximately equal to the integral of the input. What will be the signal? The sequence of pulses is a section of equal voltage, and the integral of the constant is a linear function (). Thus, at the output, we should see a sawtooth voltage. Let's check the theoretical calculations in practice:
The yellow color here shows the signal at the input, and the blue color, respectively, the output signals at different values of the circuit time constant. As you can see, we got exactly the result that we expected to see 🙂
This is where we end today's article, but we do not finish studying electronics, so see you in new articles! 🙂
RC circuit time constant
RC electric circuit
Consider the current in an electrical circuit consisting of a capacitor with a capacity C and a resistor R connected in parallel.
The value of the charge or discharge current of the capacitor is determined by the expression I = C(dU/dt), and the value of the current in the resistor, according to Ohm's law, will be U/R, Where U is the charge voltage of the capacitor.
It can be seen from the figure that the electric current I in elements C And R chains will have the same value and opposite direction, according to Kirchhoff's law. Therefore, it can be expressed as follows:
Solve the differential equation C(dU/dt)= -U/R
We integrate: 
From the table of integrals here we use the transformation 
We get the general integral of the equation: log|U| = - t/RC + Const.
Let's express the tension U potentiation: U=e-t/RC * e Const.
The solution will take the form:
U=e-t/RC * Const.
Here Const- a constant, a value determined by the initial conditions.
Therefore, the voltage U charge or discharge of the capacitor will change in time according to the exponential law e-t/RC .
Exponent - function exp(x) = e x
e– Mathematical constant approximately equal to 2.718281828...
Time constant τ
If the capacitor has a capacity C in series with a resistor R connect to a constant voltage source U, a current will flow in the circuit, which at any time t charge the capacitor up to U C and is defined by the expression:
Then the voltage U C at the capacitor terminals will increase from zero to the value U by exponent:
U C = U( 1 - e-t/RC )
At t=RC, the voltage across the capacitor will be U C = U( 1 - e -1 ) = U( 1 - 1/e).
Time numerically equal to the product RC, is called the circuit time constant RC and is denoted by the Greek letter τ
.
Time constant τ=RC
During τ
the capacitor will charge up to (1 - 1 /e)*100% ≈ 63.2% of the value U.
For time 3 τ
voltage will be (1 - 1 /e 3)*100% ≈ 95% value U.
Over time 5 τ
voltage will rise to (1 - 1 /e 5)*100% ≈ 99% value U.
If to a capacitor with a capacity C, charged to voltage U, connect a resistor in parallel with the resistance R, then the capacitor discharge current will flow in the circuit.
The voltage across the capacitor during discharge will be U C = Ue-t/τ = U/e t/τ .
During τ
the voltage across the capacitor will decrease to a value U/e, which will be 1 /e*100% ≈ 36.8% of value U.
For time 3 τ
the capacitor will discharge to (1 /e 3)*100% ≈ 5% of value U.
Over time 5 τ
before (1 /e 5)*100% ≈ 1% value U.
Parameter τ widely used in calculations RC- filters for various electronic circuits and assemblies.
Connection of instantaneous values of voltages and currents on the elements
electrical circuit
For a series circuit containing a linear resistor R, an inductor L and a capacitor C, when it is connected to a source with a voltage u (see Fig. 1), we can write
where x is the desired function of time (voltage, current, flux linkage, etc.); - known disturbing effect (voltage and (or) current of the source of electrical energy); - k-th constant coefficient determined by the parameters of the circuit.
The order of this equation is equal to the number of independent energy storage devices in the circuit, which are inductors and capacitors in a simplified circuit obtained from the original circuit by combining the inductances and, accordingly, the capacitances of the elements, the connections between which are serial or parallel.
In the general case, the order of a differential equation is determined by the relation
| , | (3) |
where and - respectively, the number of inductors and capacitors after the specified simplification of the original circuit; - the number of nodes at which only branches containing inductors converge (in accordance with Kirchhoff's first law, the current through any inductor in this case is determined by the currents through the remaining coils); - the number of circuit circuits, the branches of which contain only capacitors (in accordance with the second Kirchhoff law, the voltage on any of the capacitors in this case is determined by the voltages on the others).
The presence of inductive connections does not affect the order of the differential equation.
As is known from mathematics, the general solution of equation (2) is the sum of the particular solution of the original non-homogeneous equation and the general solution of the homogeneous equation obtained from the original one by equating its left side to zero. Since there are no restrictions on the choice of a particular solution (2) from the mathematical side, in relation to electrical engineering, it is convenient to take as the latter the solution corresponding to the desired variable x in the steady post-switching mode (theoretically for ).
A particular solution of equation (2) is determined by the form of the function on its right side, and therefore is called forced component. For circuits with given constant or periodic voltages (currents) of sources, the forced component is determined by calculating the stationary mode of operation of the circuit after switching by any of the previously considered methods for calculating linear electrical circuits.
The second component of the general solution x of equation (2) - solution (2) with a zero right side - corresponds to the regime when external (forcing) forces (energy sources) do not directly affect the circuit. The influence of sources is manifested here through the energy stored in the fields of inductors and capacitors. This mode of operation of the circuit is called free, and the variable is called free component.
In accordance with the above, . the general solution of equation (2) has the form
| (4) |
Relation (4) shows that, in the classical method of calculation, the post-switching process is considered as a superposition of two modes on each other - forced, as it were, occurring immediately after switching, and free, occurring only during the transient process.
It must be emphasized that, since the principle of superposition is valid only for linear systems, the solution method based on the specified decomposition of the required variable x is valid only for linear circuits.
Initial conditions. Switching laws
In accordance with the definition of the free component in its expression, integration constants take place, the number of which is equal to the order of the differential equation. The constants of integration are found from the initial conditions, which are usually divided into independent and dependent ones. The independent initial conditions include the flux linkage (current) for the inductor and the charge (voltage) on the capacitor at the moment of time (switching moment). Independent initial conditions are determined based on the laws of commutation (see Table 2).
Table 2. Switching laws
See more at: http://www.toehelp.ru/theory/toe/lecture24/lecture24.html#sthash.jqyFZ18C.dpuf
Integrating RC circuit
Consider an electrical circuit of a resistor with resistance R and a capacitor C shown in the figure.
Elements R And C are connected in series, which means that the current in their circuit can be expressed based on the derivative of the capacitor charge voltage dQ/dt = C(dU/dt) and Ohm's law U/R. Let us denote the voltage at the resistor terminals U R.
Then the equality will take place:
We integrate the last expression 
. The integral of the left side of the equation will be equal to Uout + Const. Let's move the constant component Const to the right side with the same sign.
On the right side of the time constant RC take it out of the integral sign:
As a result, it turned out that the output voltage U out directly proportional to the integral of the voltage at the terminals of the resistor, and therefore to the input current I in.
DC Const does not depend on the ratings of the circuit elements.
To provide a direct proportional dependence of the output voltage U out from the integral of the input U in, the input voltage must be proportional to the input current.
Nonlinear relationship Uin/Iin in the input circuit is caused by the fact that the charge and discharge of the capacitor occurs exponentially e-t/τ , which is most non-linear at t/τ≥ 1, that is, when the value t equal or more τ
.
Here t- the time of charge or discharge of the capacitor within the period.
τ
= RC- time constant - product of quantities R And C.
If we take denominations RC chains when τ
will be much more t, then the initial section of the exponent for a short period (relative to τ
) can be linear enough to provide the necessary proportionality between the input voltage and current.
For a simple chain RC the time constant is usually taken 1-2 orders of magnitude greater than the period of the alternating input signal, then the main and significant part of the input voltage will drop at the resistor terminals, providing a sufficiently linear dependence U in / I in ≈ R.
In this case, the output voltage U out will be with an acceptable error proportional to the integral of the input U in.
The larger the denominations RC, the smaller the variable component at the output, the more accurate the function curve will be.
In most cases, the variable component of the integral is not required when using such circuits, only a constant is needed. Const, then the denominations RC you can choose as large as possible, but taking into account the input resistance of the next stage.
As an example, the signal from the generator - a positive meander 1V with a period of 2 mS, will be fed to the input of a simple integrating circuit RC with denominations:
R= 10 kOhm, WITH= 1uF. Then τ
= RC= 10 mS.
In this case, the time constant is only five times the time of the period, but visually the integration can be traced quite accurately.
The graph shows that the output voltage at the level of a constant component of 0.5V will be triangular, because the sections that do not change in time will be a constant for the integral (we denote it a), and the integral of the constant will be a linear function. ∫adx = ax + Const. Constant value a determines the tangent of the slope of the linear function.
We integrate the sinusoid, we get the cosine with the opposite sign ∫sinxdx = -cosx + Const.
In this case, the constant component Const = 0.
If you apply a triangular waveform to the input, the output will be a sinusoidal voltage.
The integral of the linear section of the function is a parabola. In the simplest version ∫xdx = x 2 /2 + Const.
The sign of the multiplier will determine the direction of the parabola.
The disadvantage of the simplest circuit is that the variable component at the output is very small relative to the input voltage.
Consider an Operational Amplifier (OA) as an integrator according to the circuit shown in the figure.
Taking into account the infinitely large resistance of the op-amp and the Kirchhoff rule, the equality will be true here:
I in \u003d I R \u003d U in / R \u003d - I C.
The voltage at the inputs of an ideal op-amp is zero here, then at the terminals of the capacitor U C = U out = - U in .
Hence, U out determined based on the current of the common circuit.
With element values RC, When τ = 1 Sec, the output AC voltage will be equal in value to the integral of the input. But opposite in sign. Ideal integrator-inverter with ideal circuit elements.
RC Differential Circuit
Consider a differentiator using an Operational Amplifier.
An ideal op-amp here will provide equality of currents I R = - I C according to Kirchhoff's rule.
The voltage at the inputs of the op-amp is zero, therefore, the output voltage U out = U R = - U in = - U C .
Based on the derivative of the capacitor charge, Ohm's law and the equality of the currents in the capacitor and resistor, we write the expression:
U out = RI R = - RI C = - RC(dU C /dt) = - RC(dU in /dt)
From this we see that the output voltage U out proportional to the derivative of the charge of the capacitor dU in /dt, as the rate of change of the input voltage.
With the value of the time constant RC equal to one, the output voltage will be equal in value to the derivative of the input voltage, but opposite in sign. Therefore, the considered circuit differentiates and inverts the input signal.
The derivative of a constant is zero, so there will be no constant in the output when differentiating.
As an example, let's apply a triangular signal to the input of the differentiator. The output is a rectangular signal.
The derivative of the linear section of the function will be a constant, the sign and value of which will be determined by the slope of the linear function.
For the simplest two-element RC differentiating circuit, we use the proportional dependence of the output voltage on the derivative of the voltage at the capacitor terminals.
U out = RI R = RI C = RC(dU C /dt)
If we take the values of the RC elements so that the time constant is 1-2 orders of magnitude less than the length of the period, then the ratio of the input voltage increment to the time increment within the period can determine the rate of change of the input voltage to a certain extent accurately. Ideally, this increment should tend to zero. In this case, the main part of the input voltage will drop at the terminals of the capacitor, and the output will be an insignificant part of the input, so such circuits are practically not used to calculate the derivative.
The most common RC differentiating and integrating circuits are used to change the pulse length in logical and digital devices.
In such cases, the RC values are calculated exponentially e-t/RC based on the length of the pulse in the period and the required changes.
For example, the figure below shows that the pulse length T i at the output of the integrating chain will increase by time 3 τ
. This is the time for the capacitor to discharge to 5% of the amplitude value.
At the output of the differentiating circuit, the amplitude voltage appears instantly after the pulse is applied, since it is zero at the terminals of the discharged capacitor.
This is followed by the charging process and the voltage at the terminals of the resistor decreases. For time 3 τ
it will decrease to 5% of the amplitude value.
Here 5% is a significant value. In practical calculations, this threshold will be determined by the input parameters of the logic elements used.
