Differential circuit diagram. Differentiating and integrating RC chains
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 sections 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 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
With one of the arms having capacitive resistance to alternating current.
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Electrical circuits (part 1)
Lecture 27. Charging and discharging a capacitor through a resistance (RC circuit)
Lecture 29. Passage of alternating current through an RC circuit
Subtitles
We spent a lot of time discussing electrostatic fields and charge potential, or the potential energy of a stationary charge. I said before that electrons flow in electrical circuits. This is a resistor, it provides resistance and determines the speed of the current. So, we know that voltage is proportional to the current in the entire circuit.
Integrating RC circuit
If the input signal is applied to V in , and the day off is removed from V c (see figure), then such a circuit is called an integrating type circuit.
Response of an integrating type circuit to a single step action with amplitude V is determined by the following formula:
U c (t) = U 0 (1 − e − t / R C) .(\displaystyle \,\!U_(c)(t)=U_(0)\left(1-e^(-t/RC)\right).)
Thus, the time constant τ of this aperiodic process will be equal toτ = R C . (\displaystyle \tau =RC.) Integrating circuits pass the DC component of the signal, cutting off high frequencies, that is, they are low-pass filters. Moreover, the higher the time constant
τ (\displaystyle \tau)
, the lower the cutoff frequency. In the limit, only the constant component will pass through. This property is used in secondary power supplies in which it is necessary to filter out the alternating component of the mains voltage. A cable made of a pair of wires has integrating properties, since any wire is a resistor, having its own resistance, and a pair of adjacent wires also form a capacitor, albeit with a small capacitance. When signals pass through such a cable, their high-frequency component may be lost, and the greater the length of the cable, the greater the loss. (\displaystyle \tau =RC.) Differentiating RC chain (\displaystyle \tau =RC.) A differentiating RC circuit is obtained by swapping resistor R and capacitor C in the integrating circuit. In this case, the input signal goes to the capacitor, and the output signal is removed from the resistor. For a constant voltage, the capacitor represents an open circuit, that is, the constant component of the signal in a differentiating type circuit will be cut off. Such circuits are high-pass filters. And the cutoff frequency in them is determined by the same time constant
Differentiating chains have one more feature. At the output of such a circuit, one signal is converted into two successive voltage surges up and down relative to the base with an amplitude equal to the input voltage. The base is either the positive terminal of the source or the ground, depending on where the resistor is connected. When the resistor is connected to the source, the amplitude of the positive output pulse will be twice the supply voltage. This is used to multiply the voltage, and also, in the case of connecting a resistor to ground, to form a bipolar voltage from an existing unipolar one.
RC circuit- an electrical circuit consisting of a capacitor and a resistor. It can be considered as a voltage divider with one of the arms having capacitive resistance to alternating current.
Transmission coefficient 
Integrating RC circuit (Figure 2) Differential circuit Figure 1
Let's analyze the RC chain. Used as:
1. frequency filter
Passive filter
A passive electrical filter is an electrical circuit designed to isolate a specific frequency band from the signal received at its input.
High pass filter (signal attenuation)
RC circuit + op-amp (does not give attenuated signal, stable, transmittance ,strengthen the signal
Active filter - change the selectivity of the filter.
Low pass filter
Transfer coefficient 
Differentiating chain called a linear four-port network, in which the output voltage is proportional to the derivative of the input voltage. Schematic diagram of differentiating rC- the circuit is shown in Fig. 5.13, A. Output voltage u output is removed from the resistor r. According to Kirchhoff's second law
and consequently,
Basic properties and characteristics of plastic. Intrinsic and impurity conductivity. Band energy diagram. Fermi level. Generation and recombination of carriers. Lifetime and diffusion length. Diffusion and drift.
In terms of electrical resistance, semiconductors occupy an intermediate position between conductors and insulators. Semiconductor diodes and triodes have a number of advantages: low weight and size, significantly longer service life, and greater mechanical strength.
Let's consider the basic properties and characteristics of semiconductors. With regard to their electrical conductivity, semiconductors are divided into two types: with electronic conductivity and hole conductivity.
Electronically Conducting Semiconductors have so-called “free” electrons, which are weakly bound to the nuclei of atoms. If a potential difference is applied to this semiconductor, then the “free” electrons will move forward - in a certain direction, thus creating an electric current. Since the electric current in these types of semiconductors is the movement of negatively charged particles, they are called type conductors. P (from the word negative- negative).
Hole-conducting semiconductors are called semiconductors R (from the word positive- positive). The passage of electric current in these types of semiconductors can be thought of as the movement of positive charges. In semiconductors with R -conduction there are no free electrons; If a semiconductor atom loses one electron under the influence of any reason, it will be positively charged.
The absence of one electron in an atom, causing a positive charge on a semiconductor atom, is called hole (this means that a free space has formed in the atom). Theory and experience show that holes behave like elementary positive charges.
Hole conductivity consists in the fact that, under the influence of an applied potential difference, holes move, which is equivalent to the movement of positive charges. In reality, the following occurs during hole conduction. Let's assume that there are two atoms, one of which is equipped with a hole (one electron is missing in the outer orbit), and the other, located on the right, has all the electrons in place (let's call it a neutral atom). If a potential difference is applied to a semiconductor, then, under the influence of an electric field, an electron from a neutral atom, which has all the electrons in its place, will move to the left to the atom equipped with a hole. Due to this, the atom that had the hole becomes neutral, and the hole moves to the right to the atom from which the electron has left. In semiconductor devices process « filling» holes by a free electron is called recombination. As a result of recombination, both the free electron and the hole disappear, and a neutral atom is created. And so, the movement of holes occurs in the direction opposite to the movement of electrons.
In an absolutely pure (intrinsic) semiconductor, under the influence of heat or light, electrons and holes are born in pairs, therefore the number of electrons and holes in the intrinsic semiconductor is the same.
To create semiconductors with pronounced concentrations of electrons or holes, pure semiconductors are supplied with impurities, forming impurity semiconductors. There are impurities donor, giving electrons, and acceptor, forming holes (i.e., tearing electrons away from atoms). Consequently, in a semiconductor with a donor impurity, the conductivity will be predominantly electronic, or n– conductivity. In these semiconductors, the majority charge carriers are electrons and the minority charge carriers are holes. In a semiconductor with an acceptor impurity, on the contrary, the majority charge carriers are holes, and the minority charge carriers are electrons; these are semiconductors; With R-conductivity.
The main materials for the manufacture of semiconductor diodes and triodes are germanium and silicon; in relation to them, donors are antimony, phosphorus, arsenic; acceptors - indium, gallium, aluminum, boron.
Impurities, which are typically added to a crystalline semiconductor, dramatically change the physical pattern of electrical current flow.
When a semiconductor is formed with n -conductivity adds a donor impurity to a semiconductor: for example, an antimony impurity is added to a germanium semiconductor. Antimony atoms, which are donors, impart many “free” electrons to germanium, thereby becoming positively charged.
Thus, in an n-conductivity semiconductor formed by an impurity, there are the following types of electrical charges:
1 - mobile negative charges (electrons), which are the main carriers (both from the donor impurity and from their own conductivity);
2 - mobile positive charges (holes) - minority carriers arising from their own conductivity;
3 - stationary positive charges – donor impurity ions.
When a semiconductor with p-conductivity is formed, an acceptor impurity is added to the semiconductor: for example, an indium impurity is added to a germanium semiconductor. Indium atoms, which are acceptor atoms, remove electrons from germanium atoms, forming holes. The indium atoms themselves become negatively charged.
Consequently, in a semiconductor of p-conductivity there are the following types of electric charges:
1 - mobile positive charges (holes) - the main carriers arising from the acceptor impurity and from their own conductivity;
2 - mobile negative charges (electrons) - minority carriers arising from their own conductivity;
3 - stationary negative charges – acceptor impurity ions.
In Fig. 1 plates shown R-Germany (a) and n-germanium (b) with the arrangement of electric charges.
Intrinsic conductivity of semiconductors. An intrinsic semiconductor, or i-type semiconductor, is an ideally chemically pure semiconductor with a homogeneous crystal lattice. Ge Si
The crystal structure of a semiconductor on a plane can be defined as follows.
If an electron receives an energy greater than the band gap, it breaks the covalent bond and becomes free. In its place a vacancy is formed, which has a 4-equivalent
a positive charge equal in magnitude to the charge of an electron is called a hole. In an i-type semiconductor, the electron concentration ni is equal to the hole concentration pi. That is, ni=pi.
The process of formation of a pair of electron and hole charges is called charge generation.
A free electron can take the place of a hole, restoring the covalent bond and emitting excess energy. This process is called charge recombination. During the process of recombination and charge generation, the hole seems to move in the opposite direction from the direction of electron motion, therefore the hole is considered to be a mobile positive charge carrier. Holes and free electrons resulting from the generation of charge carriers are called intrinsic charge carriers, and the conductivity of a semiconductor due to its own charge carriers is called intrinsic conductivity of the conductor.
2) Impurity conductivity of conductors.
Since the conductivity of i-type semiconductors significantly depends on external conditions, in
Semiconductor devices use impurity semiconductors.
If a pentavalent impurity is introduced into a semiconductor, then 4 valence electrons restore covalent bonds with the semiconductor atoms, and the fifth electron remains free. Due to this, the concentration of free electrons will exceed the concentration of holes. The impurity due to which ni>pi is called a donor impurity.
A semiconductor with ni>pi is called an electronic type semiconductor
conductivity, or n-type semiconductor.
In an n-type semiconductor, electrons are called majority charge carriers and holes are called minority charge carriers.
When a trivalent impurity is introduced, three of its valence electrons restore a covalent bond with the atoms of the semiconductor, and the fourth covalent bond is not restored, i.e., a hole occurs.
As a result, the concentration of holes will be greater than the concentration of electrons.
An impurity at which pi>ni is called an acceptor impurity.
A semiconductor with pi>ni is called a hole-type semiconductor
conductivity, or p-type semiconductor.
In a p-type semiconductor, holes are called majority charge carriers and electrons are called minority charge carriers.
An RC circuit can change the shape of complex signals so that the output shape is completely different from the input. The amount of distortion is determined by the time constant of the RC circuit. The type of distortion is determined by the output component connected in parallel with the output. If a resistor is connected parallel to the output, then the circuit is called differentiating. used in synchronization circuits to obtain narrow pulses from rectangular, as well as for receiving switching pulses and marks. If a capacitor is connected parallel to the output, then the circuit is called integrating. used in signal conditioning circuits in radio, television, radar and computers.
The picture shows differentiating chain.
Recall that complex signals consist of a fundamental frequency and a large number of harmonics. When a complex signal enters a differentiating circuit, it affects each frequency differently. The ratio of capacitance (X s) to R is different for each harmonic. This causes each harmonic to be shifted in phase and reduced in amplitude to varying degrees. As a result, the original signal shape is distorted. The figure shows what happens to a square wave signal that passes through a differentiating circuit.
Similar to differentiating, except that a capacitor is connected parallel to the output.
The figure shows how the shape of a rectangular signal changes after passing through an integrating circuit.
Another type of circuit that changes the waveform is signal limiter. The figure shows the waveform at the limiter input: the negative part of the input signal is cut off.
A clipping circuit can be used to clip peaks of an applied signal, to produce a square wave from a sine wave, to remove positive or negative portions of a signal, or to maintain the amplitude of an input signal at a constant level. The diode is forward biased and conducts current during the positive half cycle of the input signal. During the negative half-cycle of the input signal, the diode is reverse biased and does not conduct current. The circuit is essentially a half-wave rectifier.
Using the offset voltage you can adjust the amount of the signal being cut. The parallel clipper can be offset to change the clipping level of the signal. If it is necessary to limit the signal on both the positive and negative sides, two biased diodes are used in parallel with the output. This allows you to obtain an output signal with an amplitude that does not exceed a predetermined positive and negative level. With this conversion, the output signal takes on a shape close to rectangular. Hence this circuit is called a square wave generator. The figure shows another limiter circuit that limits the signal on both the positive and negative sides using two zener diodes.
The output signal is limited on both sides by the stabilization voltages of the zener diodes. Between these limits, no zener diode conducts and the input signal passes to the output.
Sometimes it is desirable to change the DC sample level of an AC signal. The DC reference level is the level against which the AC signal is measured. A clamp can be used to clamp the high or low value of a signal at a given constant voltage. Unlike a signal limiter, a clamp does not change the waveform. Diode clamp called the constant component reducing agent.
This circuit is commonly used in radar, television, telecommunications and computers. In the circuit shown, a square wave signal is applied to the input. The purpose of the circuit is to limit the maximum value of the signal to 0 volts without changing the waveform.
A differentiating circuit is a circuit whose output voltage is proportional to the first time derivative of the input voltage:
Rice. 3.7.1. Differentiation circuit diagram
The differentiating circuit (Fig. 3.7.1) consists of a resistor R and capacitor WITH, the parameters of which are selected in such a way that the active resistance is many times less than the capacitive reactance.
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 in, then u in ≈ u C.
Value τ = R.C. called time constant of the differentiating chain.
The shorter the time constant compared to the input pulse duration, 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, which, as is known, consists of an infinite number of sinusoidal components, to the input of the differentiating circuit 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 starts charging 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 voltage. As the capacitor charges, the voltage across it begins to increase according to an exponential law:
u c = u input · (1 – e– t/τ) ;
where τ = R.C.– circuit time constant.
Voltage at the output of the differentiating circuit:
u out = u in – u c = u in – u input · (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 remains fully charged at this time, its discharge through the resistance will begin from this moment R. At the beginning of the capacitor discharge, 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 circuit output decreases exponentially.