A differentiating circuit is a circuit whose output signal is proportional to the derivative of the input signal.
A signal is a physical quantity that carries information. Below we will consider impulsive voltage signals - voltage pulses.
The diagram of real differentiating circuits is shown in Figures 13-33 a and 13-33 b.
The proportionality factor M represents the time constant of the circuit 
.
For RC circuit 
=RC, for chainRL 
=L/R.
Figure 13-33. Scheme of differentiating circuits.
Differentiating RC circuit. (low pass filter)
This circuit is also a four-terminal network. In a differentiating RC circuit, the signal is removed from resistor R, that is, 
(see Fig. 13-33 a). The differentiating (input) signal has a rectangular shape (see Figure 13-33a below).
Let us consider the effect of such a signal (voltage pulse) on a differentiating RC circuit.
Figure 13-34. Differentiated signal (a) and signal at the output of the differentiating RC circuit (b), 
In the moment 
(circuit switching on) output voltage 
. This follows from the fact that at the moment of switching on in the circuit according to the second law of commutation, the voltage on the capacitor retains its value that was before commutation, that is, equal to 0, therefore, the entire voltage will be applied to the resistor R( 
).
Then 
will decrease exponentially
(13.29)
If 
,during the action of the input pulse ( 
) the capacitor will be almost completely charged and at the moment 
when the impulse ends 
0, capacitor voltage 
will become equal 
(in Fig. 13-34 b 
shown by the dotted line), and the voltage across the resistor R 
will drop to 0. Since the circuit is now disconnected from the input voltage ( 
=0,
), the capacitor will begin to discharge and after a while 
the voltage across it will become equal to 0. The current in the circuit from the moment 
will change direction, and the voltage across the resistor R at the moment 
the jump becomes equal 
and will begin to decrease exponentially 
, and after a while 
will become equal to 0.
Thus, at the output of the circuit, two pointed pulses of positive and negative polarities are formed, the areas of which are equal and the amplitude is equal 
.
If 
output pulse shape 
will have a different appearance than in Fig.
Let's consider two extreme cases: 
And 
(see Fig. 13-35 b and 13-35 c)
Figure 13-35. Changing the pulse shape at the output of the differentiating circuit depending on the ratio between 
And 
.
A. 
(see Fig. 13-35 b)
In this case, during the duration of the pulse, the capacitor manages to be fully charged even before the pulse ends. At the moment of switching on, a voltage jump of positive polarity is obtained across the resistor, equal to the amplitude of the rectangular pulse 
, and then the voltage decreases steeply exponentially and, as the capacitor charges, drops to zero until the end of the pulse. At the end of the pulse (at the moment 
) the capacitor will begin to discharge, and due to the passage of current through the resistor R at the input, a pulse of negative amplitude polarity is formed - 
. The area of this impulse will be equal to the area of the positive impulse. Such chains are called differentiating shortening chains.
B. 
(see Figure 13-35).
Since the charging time of the capacitor is approximately equal 
, the capacitor will have time to charge no earlier than after 
. Therefore, the voltage across the resistor 
, equal at the moment 
, will decrease exponentially and become equal to zero in 
. Therefore, in time 
pulse 
to resistance R is practically not distorted and repeats the shape of the input pulse.
Such a circuit is used as a transition circuit between amplifier stages and is intended to eliminate the influence of the constant voltage component from the collector of the transistor of the previous stage to the subsequent one.
From formulas and Figures 13-34 and 13-35 we can conclude that the amplitude of the output pulses at different ratios between 
And 
remains unchanged and equal 
, and their duration decreases 
decreases. The accuracy of differentiation will be higher, the smaller 
compared with 
.
The most accurate differentiation can be achieved using operational amplifiers.
Let's consider the frequency response of the differentiating RC circuit shown in Fig. 13-35a.
Rice. 13-35 a. Frequency response of the differentiating circuit of the RC circuit.
The frequency transfer coefficient of the differentiating RC circuit is equal to:
If we equate 
to 1/ 
, then we obtain the lower limit of the bandwidth of the differentiating RC circuit 
.
From graph 2-35a it can be seen that the bandwidth of the differentiating RC circuit is limited only at low frequencies.
Differentiating chains
- these are circuits in which the output voltage is proportional to the derivative of the input voltage. These circuits solve two main problems of signal conversion: obtaining pulses of very short duration (pulse shortening), which are used to trigger controlled electrical energy converters, triggers, monovibrators and other devices; performing a mathematical operation of differentiation (obtaining a derivative with respect to time) of complex functions specified in the form of electrical signals, which is often found in computer technology, automatic control equipment, etc.
The circuit diagram of the capacitive differentiating circuit is shown in Fig. 1. The input voltage is applied to the entire circuit, and the output voltage is removed from the resistor R. The current flowing through the capacitor is related to the voltage across it by the known relation i C = C (dU C / dt). Considering that the same current flows through resistor R, we write the output voltage
If U OUT<< U ВХ, что справедливо, когда падение напряжения на резисторе много меньше напряжения U С, то уравнение можно записать в приближенном виде U ВЫХ . Соотношение U ВЫХ << U ВХ » U C выполняется, если величина сопротивления R много меньше величины реактивного сопротивления конденсатора, т.е. R << 1/wC (для сигнала синусоидальной формы) и R << 1/w в C, где w в – частоты высшей гармоники импульсного сигнала.
The quantity t = RC is called the time constant of the circuit. From the course on electricity we know that a capacitor is charged (discharged) through a resistor according to an exponential law. After a period of time t = t = RC the capacitor is charged to 63% of the applied input voltage, after t = 2.3 t - to 90% of U IN and after 4.6 t - to 99% of U IN.
Let a rectangular pulse of duration t I be applied to the input of the differentiating circuit (Fig. 1) (Fig. 2, a). Let t И = 10 t. Then the output signal will have the form shown in Fig. 2, d. Indeed, at the initial moment of time, the voltage on the capacitor is zero, and it cannot change instantly. Therefore, the entire input voltage is applied to the resistor. Subsequently, the capacitor is charged with an exponentially decreasing current. In this case, the voltage on the capacitor increases, and the voltage on the resistor decreases so that at each moment of time the equality U BX = U C + U OUT is satisfied. After a period of time t ³ 3 t, the capacitor is charged almost to the input voltage, the charging current will stop and the output voltage will become zero.
When the input pulse ends (U BX = 0), the capacitor will begin to discharge through resistor R and the input circuit. The direction of the discharge current is opposite to the direction of the charging current, so the polarity of the voltage across the resistor changes. As the capacitor discharges, the voltage across it decreases, and along with it, the voltage across resistor R decreases. The result is shortened pulses (at t И > 4¸5 RC). The change in pulse shape for other ratios of pulse duration and time constant is shown in Fig. 2,b,c.
Integrating circuit is a circuit in which the output voltage is proportional to the time integral of the input voltage. Integrating circuits (Fig. 3) differ from differentiating ones (Fig. 1) in that the output voltage is removed from the capacitor. When the voltage across capacitor C is negligible compared to the voltage across resistor R, i.e. U OUT = U C<< U R , то ток i в цепи пропорционален входному напряжению, которое прикладывается ко всей цепи. Поэтому
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 with resistance R connected in parallel.
The value of the capacitor charge or discharge current 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- capacitor charge voltage.
From the figure it can be seen 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:
Solving the differential equation C(dU/dt)= -U/R
Let's integrate: 
From the table of integrals here we use the transformation 
We obtain the general integral of the equation: ln|U| = - t/RC + Const.
Let us express the tension from it U potentiation: U = e-t/RC * e Const.
The solution will look like:
U = e-t/RC * Const.
Here Const- constant, value determined by the initial conditions.
Therefore, the voltage U the charge or discharge of the capacitor will change over 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 a capacitor with a capacity C in series with a resistor R connect to a constant voltage source U, a current will flow in the circuit, which for any time t will charge the capacitor to the value U C and is determined by the expression:
Then the tension U C at the capacitor terminals will increase from zero to the value U exponentially:
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 R.C., is called the time constant of the circuit R.C. and is denoted by the Greek letter τ
.
Time constant τ = RC
During τ
the capacitor will charge to (1 - 1 /e)*100% ≈ 63.2% of the value U.
In time 3 τ
the voltage will be (1 - 1 /e 3)*100% ≈ 95% of the value U.
In time 5 τ
the voltage will increase 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 resistance R, then the capacitor discharge current will flow through 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 the value U/e, which will be 1 /e*100% ≈ 36.8% value U.
In time 3 τ
the capacitor will discharge to (1 /e 3)*100% ≈ 5% of value U.
In time 5 τ
to (1 /e 5)*100% ≈ 1% value U.
Parameter τ widely used in calculations R.C.-filters of various electronic circuits and components.
Relationship between instantaneous values of voltages and currents on elements
Electric circuit
For a series circuit containing a linear resistor R, an inductor L and a capacitor C, when connected to a source with voltage u (see Fig. 1), we can write
where x is the desired function of time (voltage, current, flux linkage, etc.); - known disturbing influence (voltage and (or) current of the electrical energy source); - kth 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 understood as inductors and capacitors in a simplified circuit obtained from the original one 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 the differential equation is determined by the relation
| , | (3) |
where and are, 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 Kirchhoff’s second law, the voltage on any of the capacitors in this case is determined by the voltages on the others).
The presence of inductive couplings does not affect the order of the differential equation.
As is known from mathematics, the general solution of equation (2) is the sum of a particular solution of the original inhomogeneous equation and a general solution of the homogeneous equation obtained from the original one by equating its left side to zero. Since from the mathematical side no restrictions are imposed on the choice of a particular solution (2), in relation to electrical engineering, it is convenient to take as the latter the solution corresponding to the desired variable x in the steady-state post-commutation mode (theoretically for ).
A particular solution to equation (2) is determined by the type of function on its right side, and is therefore called forced component. For circuits with given constant or periodic source voltages (currents), the forced component is determined by calculating the stationary operating mode of the circuit after switching by any of the previously discussed methods for calculating linear electrical circuits.
The second component of the general solution x of equation (2) - solution (2) with a zero right-hand 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 free component.
In accordance with the above, . the general solution to equation (2) has the form
| (4) |
Relation (4) shows that with the classical calculation method, the post-commutation process is considered as the superposition of two modes - forced, which occurs immediately after switching, and free, which occurs only during the transition process.
It must be emphasized that since the superposition principle is valid only for linear systems, the solution method based on the specified expansion of the desired variable x is valid only for linear circuits.
Initial conditions. Commutation 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. Constant integrations are found from initial conditions, which are usually divided into independent and dependent. Independent initial conditions include flux linkage (current) for the inductor and charge (voltage) on the capacitor at an instant in time (commutation instant). Independent initial conditions are determined based on the commutation laws (see Table 2).
Table 2. Commutation laws
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RC integrating circuit
Consider an electrical circuit consisting of a resistor with a resistance R and a capacitor with a capacity 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. We denote the voltage at the resistor terminals U R.
Then the equality will take place:
Let's integrate the last expression 
. The integral of the left side of the equation will be equal to U out + Const. Let's move the constant component Const to the right side with the same sign.
On the right side the time constant R.C. Let's 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 resistor terminals, and therefore to the input current I in.
Constant component Const does not depend on the ratings of the circuit elements.
To ensure a directly proportional dependence of the output voltage U out from the input integral U in, the input voltage must be proportional to the input current.
Nonlinear relation U in /I in 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 nonlinear at t/τ≥ 1, that is, when the value t comparable or more τ
.
Here t- time of charging or discharging the capacitor within the period.
τ
= R.C.- time constant - product of quantities R And C.
If we take the denominations R.C. chains when τ
will be much more t, then the initial portion of the exponential for a short period (relative to τ
) can be quite linear, which will provide the necessary proportionality between the input voltage and current.
For a simple circuit R.C. the time constant is usually taken 1-2 orders of magnitude larger 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 fairly 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 higher the denominations R.C., 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 the constant one is needed Const, then the denominations R.C. you can choose as large as possible, but taking into account the input impedance of the next stage.
As an example, a signal from a generator - a positive square wave of 1V with a period of 2 mS - will be fed to the input of a simple integrating circuit R.C. with denominations:
R= 10 kOhm, WITH= 1 uF. Then τ
= R.C.= 10 mS.
In this case, the time constant is only five times longer than the period time, but visual 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 in shape, because the sections that do not change over 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. Value of the constant a will determine the slope of the linear function.
Let's integrate the sine wave and get a 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 portion of a function is a parabola. In its simplest form ∫xdx = x 2 /2 + Const.
The sign of the multiplier will determine the direction of the parabola.
The disadvantage of the simplest chain is that the alternating component at the output is very small relative to the input voltage.
Let us consider an Operational Amplifier (O-Amp) as an integrator according to the circuit shown in the figure.
Taking into account the infinitely large resistance of the op-amp and Kirchhoff’s rule, the equality will be valid here:
I in = I R = U in /R = - 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 will be determined based on the current of the common circuit.
At element values R.C., When τ = 1 Sec, the output alternating voltage will be equal in value to the integral of the input. But, opposite in sign. An ideal integrator-inverter with ideal circuit elements.
RC Differentiation Circuit
Let's consider a differentiator using an Operational Amplifier.
An ideal op-amp here will ensure equal 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 current values 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 capacitor charge dU in /dt, as the rate of change of input voltage.
For a time constant R.C., equal to unity, the output voltage will be equal in value to the derivative of the input voltage, but opposite in sign. Consequently, the considered circuit differentiates and inverts the input signal.
The derivative of a constant is zero, so there will be no constant component at the output when differentiating.
As an example, let's apply a triangular signal to the differentiator input. The output will be a rectangular signal.
The derivative of the linear portion of the function will be a constant, the sign and magnitude of which is determined by the slope of the linear function.
For the simplest differentiating RC chain of two elements, 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, therefore such circuits are practically not used for calculating the derivative.
The most common use of RC differentiating and integrating circuits is to change the pulse length in logic and digital devices.
In such cases, RC denominations are calculated exponentially e-t/RC based on the pulse length 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 it takes for the capacitor to discharge to 5% of the amplitude value.
At the output of the differentiating circuit, the amplitude voltage appears instantly after applying a pulse, since it is equal to zero at the terminals of the discharged capacitor.
This is followed by the charging process and the voltage at the resistor terminals decreases. In time 3 τ
it will decrease to 5% of the amplitude value.
Here 5% is an indicative value. In practical calculations, this threshold is determined by the input parameters of the logic elements used.
Complex electronic devices consist of simple circuits. Consider a circuit consisting of a resistor and a capacitor connected in series with an ideal voltage generator, shown in Fig. 3.3.
Fig.3.3. Differentiation chain
If the output voltage is removed from a resistor, then the circuit is called differentiating, if from a capacitor, it is called integrating. These linear circuits are characterized by steady-state and transient characteristics. This is due to the fact that a change in the voltage acting in the circuit leads to the fact that currents and voltages in different parts of the circuit acquire new values. The change in the state of the circuit does not occur instantly, but over a certain period of time. Therefore, a distinction is made between steady-state and transition states of an electrical circuit.
Electrical processes are considered steady (stationary) if the law of change of all voltages and currents coincides, within constant values, with the law of change of the voltage acting in the circuit from an external source. Otherwise, the circuit is considered to be in a transitional (non-stationary) state.
Stationary characteristics include the amplitude-frequency and phase characteristics of a linear circuit.
The unsteady state of a linear circuit is described by a transient characteristic.
Let us assume that an ideal voltage generator is connected to the input of the circuit. Based on Kirchhoff’s second law for a differentiating circuit, we can write a differential equation relating voltage and current in the branches of the circuit:
(3.2)
Since the voltage at the output of the circuit is:
(3.3)
Substituting the current value into the integral, we obtain:
(3.4)
Let's differentiate the left and right sides of the last equation with respect to time:
(3.5)
Let's rewrite this equation in the following form:
, (3.6)
Where = is a circuit parameter called the circuit time constant.
Depending on the value of the time constant, two different relationships are possible between the first and second terms on the right side of the equation.
If the time constant is large compared with the period of the harmonic signals >>Or with the duration of the pulses >> that can be applied to the input of this circuit, then
And the voltage at the output of the circuit repeats the input voltage with slight distortion:
If the time constant is small compared to the period of the harmonic signals<<Или с длительностью импульсов <<, то
Hence the output voltage is:
Thus, depending on the value of the time constant, such a circuit can either transmit the input signal to the output with certain distortions, or differentiate it with a certain degree of accuracy. In this case, the shape of the output signal will be different. Below in Fig. Figure 3.4 shows the input voltage, resistor and capacitor voltages for cases where the time constant is large and the time constant is small.
A B
Rice. 3.4. Voltages on the elements of the differentiating circuit at ( A) And ( B)
At the initial moment of time, a voltage jump appears across the resistor equal to the amplitude of the input signal, and then the capacitor begins charging, during which the voltage across the resistor will decrease.
When the time constant is , the capacitor does not have time to charge up to the amplitude of the input pulse and the circuit transmits the input signal to the output with slight distortion. At<< конденсатор успеет полностью зарядиться до амплитуды входного напряжения за время действия первого импульса, а за время паузы между импульсами – полностью разрядиться. При этом на выходе цепи появляются укороченные импульсы, приблизительно соответствующие производной от входного сигнала. Считается, что когда Цепочка дифференцирует входной сигнал.
Now let's determine the transmission coefficient of the differentiating circuit. The complex transmission coefficient of the differentiating circuit when a harmonic signal is applied to the input is equal to:
. (3.11)
Let us denote the relation 
, where is the cutoff frequency of the differentiating circuit passband.
The expression for the transmission coefficient will take the form:
The module of the transmission coefficient is equal to:
. (3.13)
- the cutoff frequency of the passband at which the reactance module becomes equal to the value of the active resistance, and the circuit transmission coefficient is equal to . The dependence of the modulus of the transmission coefficient on frequency is called the amplitude-frequency response (AFC).
The dependence of the phase angle between the output and input voltages on frequency is called the phase characteristic (PFC). Phase characteristic:
Below in Fig. 3.5 shows the frequency response and phase response of the differentiating circuit:
Rice. 3.5. Amplitude-frequency and phase characteristics
Differentiating chain
From the amplitude-frequency characteristic it is clear that the passage of signals through the differentiating circuit is accompanied by a decrease in the amplitudes of the low-frequency components of its spectrum. The differentiating circuit is a high-pass filter.
From the phase characteristic it is clear that the phases of low-frequency components are shifted by a larger angle than the phases of high-frequency components.
The transient response of the differentiating circuit can be obtained if voltage is applied to the input in the form of a single step. The complex transmission coefficient is equal to
Consider an electrical circuit consisting of a resistor with a resistance R and a capacitor with a capacity 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. We denote the voltage at the resistor terminals U R.
Then the equality will take place:
Let's integrate the last expression 
. The integral of the left side of the equation will be equal to U out + Const. Let's move the constant component Const to the right side with the same sign.
On the right side the time constant R.C. Let's 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 resistor terminals, and therefore to the input current I in.
Constant component Const does not depend on the ratings of the circuit elements.
To ensure a directly proportional dependence of the output voltage U out from the input integral U in, the input voltage must be proportional to the input current.
Nonlinear relation U in /I in 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 nonlinear at t/τ≥ 1, that is, when the value t comparable or more τ
.
Here t- time of charging or discharging the capacitor within the period.
τ
= R.C.- time constant - product of quantities R And C.
If we take the denominations R.C. chains when τ
will be much more t, then the initial portion of the exponential for a short period (relative to τ
) can be quite linear, which will provide the necessary proportionality between the input voltage and current.
For a simple circuit R.C. the time constant is usually taken 1-2 orders of magnitude larger 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 fairly 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 higher the denominations R.C., 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 the constant one is needed Const, then the denominations R.C. you can choose as large as possible, but taking into account the input impedance of the next stage.
As an example, a signal from a generator - a positive square wave of 1V with a period of 2 mS - will be fed to the input of a simple integrating circuit R.C. with denominations:
R= 10 kOhm, WITH= 1 uF. Then τ
= R.C.= 10 mS.
In this case, the time constant is only five times longer than the period time, but visual 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 in shape, because the sections that do not change over 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. Value of the constant a will determine the slope of the linear function.
Let's integrate the sine wave and get a 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 portion of a function is a parabola. In its simplest form ∫xdx = x 2 /2 + Const.
The sign of the multiplier will determine the direction of the parabola.
The disadvantage of the simplest chain is that the alternating component at the output is very small relative to the input voltage.
Let us consider an Operational Amplifier (O-Amp) as an integrator according to the circuit shown in the figure.
Taking into account the infinitely large resistance of the op-amp and Kirchhoff’s rule, the equality will be valid here:
I in = I R = U in /R = - 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 will be determined based on the current of the common circuit.
At element values R.C., When τ
= 1 Sec, the output alternating voltage will be equal in value to the integral of the input. But, opposite in sign. An ideal integrator-inverter with ideal circuit elements.
RC Differentiation Circuit
Let's consider a differentiator using an Operational Amplifier.
An ideal op-amp here will ensure equal 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 current values 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 capacitor charge dU in /dt, as the rate of change of input voltage.
For a time constant R.C., equal to unity, the output voltage will be equal in value to the derivative of the input voltage, but opposite in sign. Consequently, the considered circuit differentiates and inverts the input signal.
The derivative of a constant is zero, so there will be no constant component at the output when differentiating.
As an example, let's apply a triangular signal to the differentiator input. The output will be a rectangular signal.
The derivative of the linear portion of the function will be a constant, the sign and magnitude of which is determined by the slope of the linear function.
For the simplest differentiating RC chain of two elements, 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, therefore such circuits are practically not used for calculating the derivative.
The most common use of RC differentiating and integrating circuits is to change the pulse length in logic and digital devices.
In such cases, RC denominations are calculated exponentially e-t/ RC based on the pulse length 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 it takes for the capacitor to discharge to 5% of the amplitude value.
At the output of the differentiating circuit, the amplitude voltage appears instantly after applying a pulse, since it is equal to zero at the terminals of the discharged capacitor.
This is followed by the charging process and the voltage at the resistor terminals decreases. In time 3 τ
it will decrease to 5% of the amplitude value.
Here 5% is an indicative value. In practical calculations, this threshold is determined by the input parameters of the logic elements used.
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