A differential amplifier is a well-known circuit used to amplify the voltage difference of two input signals. Ideally, the output signal does not depend on the level of each of the input signals, but is determined only by their difference. When the signal levels at both inputs change simultaneously, then such a change in the input signal is called common-mode. The differential or difference input signal is also called normal or useful. A good differential amplifier has a high common mode rejection ratio(CMRR), which is the ratio of the desired output signal to the common-mode output signal, assuming that the desired and common-mode input signals have the same amplitude. CMRR is usually measured in decibels. The range of variation of the common-mode input signal specifies permissible levels voltage relative to which the input signal should change.

Differential amplifiers are used in cases where weak signals can be lost in the background noise. Examples of such signals are digital signals transmitted over long cables (a cable usually consists of two twisted wires), audio signals (in radio engineering, the concept of “balanced” impedance is usually associated with a differential impedance of 600 ohms), radio frequency signals (a two-core cable is differential), voltage electrocardiograms, signals for reading information from magnetic memory and many others. A differential amplifier at the receiving end restores the original signal if the common mode interference is not very large. Differential stages are widely used in the construction of operational amplifiers, which we discuss below. They are playing important role when developing amplifiers DC(which amplify frequencies up to DC, i.e. do not use capacitors for interstage coupling): their symmetrical circuit is inherently designed to compensate for temperature drift.

In Fig. Figure 2.67 shows the basic circuit of a differential amplifier. The output voltage is measured at one of the collectors relative to ground potential; such an amplifier is called circuit with single-pole output or difference amplifier and it is the most widespread. This amplifier can be thought of as a device that amplifies a differential signal and converts it into a single-ended signal that can be operated regular circuits(voltage repeaters, current sources, etc.). If a differential signal is needed, then it is removed between the collectors.

Rice. 2.67. Classic transistor differential amplifier.

What is the gain of this circuit? It is not difficult to calculate: let’s say a differential signal is applied to the input, and the voltage at input 1 increases by the amount uin (voltage change for a small signal relative to the input).

As long as both transistors are in active mode, the potential of point A is fixed. The gain can be determined as in the case of an amplifier with one transistor, if you notice that the input signal is applied twice to the base-emitter junction of any transistor: K diff = R k /2(r e + R e). The resistance of the resistor R e is usually small (100 Ohms or less), and sometimes this resistor is absent altogether. The differential voltage is usually amplified several hundred times.

In order to determine the common-mode signal gain, the same I/O signals must be applied to both inputs of the amplifier. If you carefully consider this case (and remember that both emitter currents flow through resistor R 1), you will get K sinf = - R k / (2R 1 + R e). We neglect resistance r e, since resistor R 1 is usually chosen to be large - its resistance is at least several thousand ohms. In fact, the resistance R e can also be neglected. CMOS is approximately equal to R 1 (r e + R e). A typical example of a differential amplifier is the circuit shown in Fig. 2.68. Let's look at how it works.

Rice. 2.68. Calculation of differential amplifier characteristics.
K diff = U out /(U 1 - U 2) = R to /2(R e + r e):
K diff = R k /(2R 1 + R e + r e);
KOSS ≈ R 1 /(R e + r e).

The resistance of the resistor R k is chosen as follows. so that the quiescent collector current can be taken equal to 100 μA. As usual, to obtain maximum dynamic range, the collector potential is set to 0.5 U kk. Transistor T 1 does not have a collector resistor, since its output signal is removed from the collector of another transistor. The resistance of resistor R 1 is chosen such that the total current is 200 μA and is equally distributed between the transistors when the input (differential) signal is zero. According to the formulas just derived, the differential signal gain is 30, and the common mode gain is 0.5. If we exclude 1.0 kOhm resistors from the circuit, then the gain of the differential signal will become equal to 150, but at the same time the input (differential) resistance will decrease from 250 to 50 kOhm (if it is necessary for the value of this resistance to be of the order of megaohms, then transistors can be used in the input stage Darlington).

Let us recall that in an asymmetrical amplifier with a grounded emitter with an output quiescent voltage of 0.5 U kk, the maximum gain is 20 U kk, where U kk is expressed in volts. In a differential amplifier the maximum differential gain(at R e = 0) half as much, i.e. numerically equal to twenty times the voltage drop across the collector resistor with a similar choice of operating point. The corresponding maximum CMRR (provided that R e = 0) is also numerically 20 times greater than the voltage drop across R 1 .

Exercise 2.13. Make sure the given ratios are correct. Design differential amplifiers to suit your own requirements.

A differential amplifier can be figuratively called a “long-tailed pair”, since if the length of the resistor is symbol is proportional to the value of its resistance, the circuit can be depicted as shown in Fig. 2.69. The “long tail” determines the suppression of the common mode signal, and the small resistances of the inter-emitter coupling (including the own resistances of the emitters) determine the amplification of the differential signal.

Biasing using a current source. The common-mode gain in a differential amplifier can be significantly reduced if resistor R 1 is replaced by a current source. In this case, the effective value of resistance R 1 will become very large, and the common-mode signal gain will be weakened almost to zero. Let's imagine that there is a common-mode signal at the input; The current source in the emitter circuit maintains the total emitter current constant, and it (due to the symmetry of the circuit) is evenly distributed between the two collector circuits. Therefore, the output signal of the circuit does not change. An example of such a scheme is shown in Fig. 2.70. For this circuit, which uses a monolithic transistor pair of type LM394 (transistors T 1 and T 2) and a current source of type 2N5963, the CMRR value is determined by the ratio of 100,000:1 (100 dB). The range of the input common-mode signal is limited to -12 and + 7 V: the lower limit is determined by the operating range of the current source in the emitter circuit, and the upper limit is determined by the quiescent collector voltage.

Rice. 2.70. Increasing the CMRR of a differential amplifier using a current source.

Do not forget that this amplifier, like all transistor amplifiers, must have DC mixing circuits. If, for example, a capacitor is used at the input for interstage coupling, then grounded base resistors must be included. Another caveat applies especially to differential amplifiers without emitter resistors: bipolar transistors can withstand no more than 6 V of reverse bias at the base-emitter junction. Then breakdown occurs; This means that if a higher differential input voltage is applied to the input, the input stage will be destroyed (provided that there are no emitter resistors). The emitter resistor limits the breakdown current and prevents destruction of the circuit, but the characteristics of the transistors can degrade in this case (coefficient h 21e, noise, etc.). In either case, the input impedance drops significantly if reverse conduction occurs.

Applications of differential circuits in DC amplifiers with single-pole output. A differential amplifier can work perfectly as a DC amplifier even with single-ended (single-ended) input signals. To do this, you need to ground one of its inputs and send a signal to the other (Fig. 2.71). Is it possible to eliminate the "unused" transistor from the circuit? No. The differential circuit provides compensation for temperature drift, and, even when one input is grounded, the transistor performs some functions: when the temperature changes, the voltage U be changes by the same amount, while no changes occur at the output and the balancing of the circuit is not disrupted. This means that the change in voltage U be is not amplified by the coefficient K diff (its amplification is determined by the coefficient K sinf, which can be reduced to almost zero). In addition, mutual compensation of voltages U be leads to the fact that at the input there is no need to take into account voltage drops of 0.6 V. The quality of such a DC amplifier deteriorates only due to the inconsistency of voltages U be or their temperature coefficients. The industry produces transistor pairs and integrated differential amplifiers with a very high degree of matching (for example, for a standard matched monolithic n-p-n pairs- for transistors of the MAT-01 type, the voltage drift U be is determined by the value of 0.15 μV/°C or 0.2 μV per month).

Rice. 2.71. The differential amplifier can operate as a precision DC amplifier with single-pole output.

In the previous circuit, you can ground any of the inputs. Depending on which input is grounded, the amplifier will or will not invert the signal. (However, due to the presence of the Miller effect, which will be discussed in Section 2.19, the circuit shown here is preferable for the high frequency range). The presented circuit is non-inverting, which means that the inverting input is grounded. The terminology associated with differential amplifiers also applies to operational amplifiers, which are the same high-gain differential amplifiers.

Using a current mirror as an active load. Sometimes it is desirable for a single stage differential amplifier, like a simple grounded emitter amplifier, to have high gain. Beautiful solution gives the use of a current mirror as an active load of an amplifier (Fig. 2.72). Transistors T 1 and T 2 form a differential pair with a current source in the emitter circuit. Transistors T 3 and T 4, forming a current mirror, act as a collector load. This ensures a high value of collector load resistance, thanks to which the voltage gain reaches 5000 and higher, provided that there is no load at the amplifier output. Such an amplifier is usually used only in circuits covered by a loop feedback, or in comparators (we will look at them in the next section). Remember that the load for such an amplifier must have a high impedance, otherwise the gain will be significantly weakened.

Rice. 2.72. Differential amplifier with current mirror as active load.

Differential amplifiers as phase splitting circuits. On the collectors of a symmetrical differential amplifier, signals appear that are identical in amplitude, but with opposite phases. If we take the output signals from two collectors, we get a phase splitting circuit. Of course, you can use a differential amplifier with differential inputs and outputs. The differential output signal can then be used to drive another differential amplifier stage, thereby increasing the CMRR value of the entire circuit significantly.

Differential amplifiers as comparators. Due to its high gain and stable performance, the differential amplifier is the main integral part comparator- a circuit that compares input signals and evaluates which one is larger. Comparators are used in the most various areas: for turning on lighting and heating, for obtaining rectangular signals from triangular ones, for comparing the signal level with a threshold value, in class D amplifiers and with pulse code modulation, for switching power supplies, etc. The main idea when constructing a comparator is that. that the transistor should turn on or off depending on the levels of the input signals. The linear gain region is not considered - the operation of the circuit is based on the fact that one of the two input transistors is in cutoff mode at any time. A typical signal capture application is discussed in the next section using a temperature control circuit that uses temperature-dependent resistors (thermistors).

Thermal fire detector is an automatic PI that responds to a certain temperature value and (or) the rate of its increase (GOST R53325-2012).

When equipping facilities automatic installations fire alarm Three types of thermal fire detectors are widely used: with sensors of maximum, differential and maximum-differential action

Classification of thermal PIs according to the nature of the reaction to a controlled sign of fire:

Maximum thermal fire detector- fire detector that generates a fire notification when the temperature exceeds environment established threshold value- detector response temperature.

Maximum differential thermal fire detector- a fire detector that combines the functions of maximum and differential thermal fire detectors.

Differential thermal fire detector- a fire detector that generates a fire notification when the rate of increase in ambient temperature exceeds the established threshold value.

Detectors with maximum action sensors are triggered at a certain, predetermined set temperature.

Detectors with differential sensors respond to a certain rate of temperature increase.

Maximum-differential detectors include maximum and differential action sensors and are triggered both at a certain, predetermined temperature and at a certain rate of its increase.

When choosing thermal fire detectors, it should be taken into account that the response temperature of maximum and maximum differential detectors must be at least 200 C higher than the maximum permissible temperature indoor air.

Thermal fire detectors are classified depending on the sensing element used.

Fusible sensor detectors are considered the most common due to their simplicity, reliability and low cost. Being a one-time action, they cannot serve as information about restoring normal conditions in controlled premises.

Currently, detectors using thermocouples as sensors are widely used. The thermocouple differential detector contains a thermopile, which provides a fire signal when there are signs of an increase in the ambient temperature above the maximum permissible. How more speed temperature rise, the sooner a fire danger signal is given.

Classification of thermal PIs according to the operating principle:

IP101 - using the dependence of the change in the value of thermal resistance on the temperature of the controlled environment;

IP-102 - using thermoEMF generated during heating;

IP-103 - using linear expansion of bodies;

IP-104 - using fusible materials;

IP-105 – using the dependence of magnetic induction on temperature;

Classification according to the configuration of the measuring zone: thermal PIs are:

A point fire detector is a fire detector that responds to fire factors in a compact area.

Multipoint fire detector (thermal) – a detector with a discrete arrangement of point sensitive elements in the measuring line.

Linear fire detector is a fire detector that responds to fire factors in an extended, linear zone.

For example:

Point thermal detector maximum 70°C IP-103-4/1 MAK-1

Device: The detector consists of a plastic protective housing and a plastic base with two mounting holes for screws, in which a temperature relay is installed directly on the screw terminals. A shunt resistor is mounted to the same terminals.

Working principle: B in good condition The detector contact system is closed. When the threshold temperature is reached, the detector contacts open, and when the temperature drops from the threshold, the contacts close again.

Multipoint thermal detector IP 102-2x2

The detector sensor consists of sensitive elements (thermocouples) evenly distributed on a long twisted wire.

Operating principle: Thermal emfs that arise when thermocouples are exposed to heat flows are summed up at the ends of the wire and converted in a special electronic unit (interface unit) into an alarm signal. If a wire with thermocouples is evenly placed over the entire ceiling area of ​​the protected room, then by scanning the heat flows in the room, fires are quickly detected. The results of fire tests showed that the response time of multipoint detectors depends little on the height of the protected premises and amounts to several tens of seconds up to a height of H = 20 m.

Linear heat detector (thermal cable)

Thermal cable device:

Linear detector(thermal cable) consists of two steel conductors, each of which is coated with thermoplastic material. The conductors are twisted together to create mechanical tension between them, and are additionally covered with an outer protective PVC sheath.

Operating principle:

The control current from the interface module constantly passes through the thermal cable. At the actuation temperature, the thermoplastic insulation material is pressed through due to the mechanical stress of the conductors, and they short-circuit. Thermal cable works as a single sensor continuous action. Linear detection has unique advantages when used in places with difficult access, places with increased pollution, dust, aggressive or explosive environments.

Scope of thermal PIs

Thermal PIs are used to protect premises whose flammable load is characterized by significant heat release during a fire. If the control zone is an extended object of complex geometric shape, linear TPIs are used.

Maximum TPI should not be used in rooms where the air temperature may be below 0ºС and in rooms intended for storing cultural property, for containing flammable materials in small quantities and/or with low calorific value.

Differential TPIs are effectively used to protect objects with low ambient temperatures. The inertia of differential detectors is lower than that of maximum detectors, which means that a fire will be detected faster. At the same time, differential TPIs should not be used to protect premises in which significant temperature changes are possible, not caused by a fire, but associated, for example, with the operation of air conditioning systems.

Maximum differential MDPI-028

Maximum differential DMD-70

Maximum differential DMD-70-S

The automatic bimetallic maximum-differential fire detector MDPI-028 is made in a waterproof design and is intended for use on ships. Structurally, the detector is built on two bimetallic elements, which deform when the ambient temperature increases and, with their loose ends, affect the contacts. Each bimetallic element is located

Automatic bimetallic maximum differential detector MDPI-028 227 ate.

Thermal maximum-differential MDPI-028, the sensitive element is two bimegallic spirals. Triggers at temperature type + 70° C (+90° C). Controlled area - from 20 to 30 m2. The ambient temperature should be between -40 and -50°C. The relative humidity of the premises should not exceed 98%. Works with ship fire alarm station TOL-10/50-S.

The MDPI-028 detector (maximum differential fire detector) in a waterproof design is intended for use in rooms with an air temperature of -40... + 50 ° C and relative humidity up to 98%. The notice gel is adapted to work in vibration conditions.

To replace morally and technically outdated fire detectors ATIM, ATP, DTL, DI-1, KI-1, RID-1, IDF-1, IDF-1M, POST-1 and control and reception equipment SKPU-1, SDPU- 1, PPKU-1M, TOL-10/100, RUOP-1, new models of modern fire detectors and control panels with significantly better performance indicators of durability, reliability and efficiency, made on a modern element base, were developed and mastered wide application. These included: radioisotope smoke fire detector RID-6M, photoelectric smoke detector DIP-1, DIP-2 and DIP-3, light ultraviolet flame fire detector IP329-2 “Amethyst”, explosion-proof thermal fire detector IP-103, multiple-action thermal magnetic contact fire detector IP105-2/1 (ITM), manual fire detector IPR detector, IP101-2 maximum differential detector, as well as PPS-3, PPK-2, RUGTI-1, PPKU-1M-01 and “Signal-42” alarm and control devices. To protect explosion and fire hazardous industries, it was developed and transferred to industrial production a new intrinsically safe receiving and control device “Signal-44”, designed for connection to an intrinsically safe fire alarm loop

Maximum-differential thermal fire detector - a thermal fire detector that combines the functions of maximum and differential thermal fire detectors.

5 Heat detector IP 129-1 Analogue maximum-differential heat detector
you. Most common heat detectors According to the principle of action, they are divided into maximum, differential and maximum-differential. The first are triggered when a certain temperature is reached, the second - at a certain rate of temperature increase, the third - from any prevailing temperature change. According to their design, heat detectors are passive, in which, under the influence of temperature, the sensitive element changes its properties (DTL, IP-104-1 - maximum action, based on the opening of spring contacts connected by lightweight solder: MDPT-028 - maximum differential on bimetallic effect, leading to deformation of the plates that open the contacts; IP-105-2/1 - on the principle of changing magnetic induction under the influence of heat; DPS-38 - differential on the use of a thermocouple thermopile).

Heat detectors according to their operating principle are divided into maximum, differential and maximum-differential. The first are triggered when a certain temperature is reached, the second - at a certain rate of temperature increase, and the third - from any significant change in temperature. Low-melting locks, bimetallic plates, tubes filled with easily expanding liquid, thermocouples, etc. are used as sensitive elements. Thermal fire detectors are installed under the ceiling in such a position that the heat flow, flowing around the sensitive element of the detector, heats it. Thermal fire detectors are not highly sensitive, so they usually do not give false alarms if the temperature in the room increases when heating is turned on or technological operations are performed.

Thermal or thermal detectors are divided into maximum, differential and maximum-differential.

Maximum differential detectors are combined, i.e. they operate simultaneously at a certain rate of temperature rise and when the temperature reaches critical temperatures indoor air.

Heat detectors according to their operating principle are divided into maximum, differential and maximum-differential.

Differential thermal detectors are triggered at a certain rate of increase in ambient temperature, which is assumed to be within 5-MO°C per minute. Maximum-differential detectors combine the properties of maximum and differential types of detectors.

Heat detectors according to their operating principle are divided into maximum, differential and maximum-differential.

Thermal automatic fire detectors are divided according to their operating principle into maximum, differential and maximum-differential. Detectors of the maximum operating principle are triggered when a certain temperature value is reached, differential - at a certain rate of increase in the temperature gradient, maximum-differential -

Thermal maximum differential detectors should not be used in following cases: the rate of change in ambient air temperature is greater than the temperature gradient of the detector (workshops, hardening, boiler rooms, etc.); there is damp dust (dust concentration is greater than permissible according to sanitary standards).

Fire detectors smoke 215 smoke optical 217 linear volumetric 221 maximum differential

Op amps are characterized by amplification, input, output, energy, drift, frequency and speed characteristics.

Gain characteristics

Gain (K U) is equal to the ratio of the output voltage increment to the differential input voltage that caused this increment in the absence of feedback (FE). It varies from 10 3 to 10 6.

The most important characteristics Op amps are amplitude (transfer) characteristics (Fig. 8.4). They are represented in the form of two curves, corresponding respectively to the inverting and non-inverting inputs. The characteristics are taken when a signal is applied to one of the inputs with a zero signal at the other. Each of the curves consists of horizontal and inclined sections.

The horizontal sections of the curves correspond to the fully open (saturated) or closed mode of the output stage transistors. When the input voltage changes in these sections, the output voltage of the amplifier remains constant and is determined by the voltages +U out max) -U out max. These voltages are close to the voltage of the power supplies.

The sloping (linear) section of the curves corresponds to the proportional dependence of the output voltage on the input. This range is called the gain region. The angle of inclination of the section is determined by the gain of the op-amp:

K U = U out / U in.

Large values ​​of the op-amp gain make it possible, when such amplifiers are covered by deep negative feedback, to obtain circuits with properties that depend only on the parameters of the negative feedback circuit.

The amplitude characteristics (see Fig. 8.4) pass through zero. The state when U out = 0 at U in = 0 is called op-amp balance. However, for real op-amps the balance condition is usually not satisfied. When Uin = 0, the output voltage of the op-amp can be greater or less than zero:

U out = + U out or U out = - U out).

Drift characteristics

The voltage (U cmo) at which U out = 0 is called input offset voltage zero (Fig. 8.5). It is determined by the voltage value that must be applied to the input of the op-amp to obtain zero at the output of the op-amp. Usually amounts to no more than a few millivolts. The voltages U cm and ∆U out (∆U out = U shift - shear stress) are related by the relation:

U cm = ∆U out / K U .

The main reason for the appearance of bias voltage is a significant spread in the parameters of the elements of the differential amplifier stage.

The dependence of the op amp parameters on temperature causes temperature drift input offset voltage. Input offset voltage drift is the ratio of the change in input offset voltage to the change in ambient temperature:

E smo = U smo / T.

Typically E cmo is 1…5 µV / °C.

Transfer characteristic of an op-amp for a common-mode signal shown in (Fig. 8.6). It shows that at sufficiently large values ​​of U sf (comparable with the voltage of the power source), the common-mode signal gain (K sf) increases sharply.

The range of input voltage used is called the common mode rejection region. Operational amplifiers are characterized by common mode rejection ratio (K oss) – differential signal gain ratio (K u d) to the common-mode signal gain factor (K u sf).

K oss = K u d / K u sf.

Common mode gain is defined as the ratio of the change in output voltage to the change in common mode that caused it.
o input signal). Common mode rejection ratio is usually expressed in decibels.

Input characteristics

Input resistance, input bias currents, difference and drift of input bias currents, as well as the maximum input differential voltage characterize the main parameters of the op-amp input circuits, which depend on the circuit of the differential input stage used.

Input bias current (I cm) – current at the amplifier inputs. Input bias currents are due to the base currents of the input bipolar transistors and the gate leakage currents for op amps with FET inputs. In other words, I cm is the currents consumed by the inputs of the op-amp. They are determined by the finite value of the input resistance of the differential stage. The input bias current (I cm), given in the reference data for the op-amp, is defined as the average bias current:

I cm = (I cm1 – I cm2) / 2.

Input shift current is the difference in displacement currents. It appears due to inaccurate matching of the current gains of the input transistors. The shear current is a variable value, ranging from several units to several hundred nanoamps.

Due to the presence of input bias voltage and input bias currents, op-amp circuits must be supplemented with elements designed to initially balance them. Balancing is carried out by applying some additional voltage to one of the inputs of the op-amp and introducing resistors into its input circuits.

Input current temperature drift – a coefficient equal to the ratio of the maximum change in the input current of the op-amp to the change in ambient temperature that caused it.

Temperature drift of input currents leads to additional errors. Temperature drifts are important for precision amplifiers because, unlike offset voltage and input currents, they are very difficult to compensate for.

Maximum differential input voltage the voltage supplied between the inputs of the op-amp in the circuit is limited to prevent damage to the transistors of the differential stage

Input impedance depends on the type of input signal. There are:

· differential input resistance (R input differential) – (resistance between the amplifier inputs);

· common-mode input resistance (Rin sf) – resistance between the combined input terminals and the common point.

The values ​​of Rin diff range from several tens of kilo-ohms to hundreds of mega-ohms. The input common-mode resistance Rin sf is several orders of magnitude greater than Rin diff.

Output characteristics

The output parameters of the op-amp are the output resistance, as well as the maximum output voltage and current.

The operational amplifier must have a small output impedance (R out) to provide high output voltages at low load resistances. Low output resistance is achieved by using an emitter follower at the op-amp output. Real Rout is units and hundreds of ohms.

Maximum output voltage (positive or negative) close to the supply voltage. Maximum output current limited by the permissible collector current of the op-amp output stage.

Energy characteristics

The energy parameters of the op-amp are assessed maximum current consumption from both power sources and, accordingly, the total power consumption .

Frequency characteristics

The amplification of harmonic signals is characterized by the frequency parameters of the op-amp, and the amplification of pulsed signals by its speed or dynamic parameters.

The frequency dependence of the op-amp gain without feedback is called amplitude-frequency response (AFC).

The frequency (f 1) at which the op-amp gain is equal to unity is called unity gain frequency .

Due to the phase shift of the output signal relative to the input created by the amplifier in the high-frequency region phase-frequency response The op-amp at the inverting input acquires an additional (over 180°) phase shift (Fig. 8.8).

To ensure stable operation of the op-amp, it is necessary to reduce the phase lag, i.e. adjust the amplitude-frequency response of the op-amp.

Speed ​​characteristics

The dynamic parameters of the op-amp are output slew rate voltage (response speed) and output voltage settling time . They are determined by the reaction of the op-amp to the impact of a voltage surge at the input (Fig. 8.9).

Output voltage slew rate is the ratio of the increment ( U out) to the time interval ( t) during which this increment occurs when a rectangular pulse is applied to the input. That is

V U out = U out / t

The higher the cutoff frequency, the faster the slew rate of the output voltage. Typical values ​​V U out – units of volts per microsecond.

Output voltage settling time (t set) – the time during which U out of the operational amplifier changes from level 0.1 to level 0.9 of the steady-state value of U out when the op-amp input is exposed to rectangular pulses. The settling time is inversely proportional to the cutoff frequency.

Operations of mathematical analysis

Amounts

The sum function is used to find sums. Function syntax:

Sum(expression, variable, lower limit of variable change, upper limit of variable change)

For example:

If you assign the value of the system variable positive infinity "inf" to the last argument, this will indicate the absence of an upper bound and an infinite sum will be calculated. Also, an infinite sum will be calculated if you assign the value of the negative infinity system variable "minf" to the argument "lower limit of variable change". The same values ​​are used in other mathematical analysis functions.

For example:

Works

To find finite and infinite products, use the product function. It has the same arguments as the sum function.

For example:

Limits

To find limits, use the limit function.

Function syntax:

limit(expression, variable, breakpoint)

If the "breakpoint" argument is set to "inf", then this will indicate the absence of a border.

For example:

To calculate one-sided limits, an additional argument is used, which has the value plus for calculating the limits on the right and minus for the left.

For example, let's study the continuity of the function arctan(1/(x - 4)). This function is undefined at the point x = 4. Let us calculate the limits on the right and left:

As we can see, the point x = 4 is a discontinuity point of the first kind for this function, since there are boundaries on the left and on the right, which are equal to -PI/2 and PI/2, respectively.

Differentials

The diff function is used to find differentials. Function syntax:

diff(expression, variable1, derivative order for variable1 [,variable2, derivative order for variable2,...])

where the expression is the function that is differentiated, the second argument is the variable with respect to which the derivative must be taken, the third (optional) is the order of the derivative (by default - the first order).

For example:

In general, only the first argument is required for the diff function. In this case, the function returns the differential of the expression. The differential of the corresponding variable is denoted by del(variable name):

As we can see from the syntax of the function, the user has the opportunity to simultaneously define several differentiation variables and set the order for each of them:

If you use a parametric function, the form of writing the function changes: after the function name the symbols ":=" are written, and the function is accessed through its name with a parameter:

The derivative can be calculated at a given point. This is done like this:

The diff function is also used to denote derivatives in differential equations, about which we're talking about below.

Integrals

To find integrals in the system, use the integrate function. To find the indefinite integral of a function, two arguments are used: the name of the function and the variable over which the integration occurs. For example:

If the answer is ambiguous, Maxima may ask an additional question:

The answer must contain the text from the question. IN in this case, if the value of the variable y is greater than "0", it will be "positive" (positive), otherwise - "negative" negative). In this case, only the first letter of the word can be entered.

To find a definite integral in a function, you must specify additional arguments: limits of the integral:

Maxima also allows for infinite integration limits. To do this, the values ​​"-inf" and "inf" are used for the third and fourth arguments of the function:

To find the approximate value of the integral in numerical form, as noted earlier, you should select the result in the output cell, call the context menu on it and select the “To Float” item from it (convert to a floating point number).

The system is also capable of calculating multiple integrals. To do this, integrate functions are nested one within the other. The following are examples of calculating double indefinite integral and double definite integral:

Solutions differential equations

In terms of its capabilities in solving differential equations, Maxima is noticeably inferior to, for example, Maple. But Maxima still allows you to solve ordinary first- and second-order differential equations, as well as their systems. To do this, depending on the purpose, two functions are used. For the general solution of ordinary differential equations, the ode2 function is used, and for finding solutions to equations or systems of equations based on initial conditions, the desolve function is used.

The ode2 function has the following syntax:

ode2(equation, dependent variable, independent variable);

The diff function is used to indicate derivatives in differential equations. But in this case, in order to display the dependence of a function on its argument, it is written as “diff(f(x), x), and the function itself is f(x).

Example. Find general solution ordinary first order differential equation y" - ax = 0.

If the value of the right side of the equation is zero, then it can be omitted altogether. Naturally, right side equations may contain an expression.

As you can see, when solving differential equations, Maxima uses the integration constant %c, which, from a mathematical point of view, is an arbitrary constant determined from additional conditions.

There is another way to solve an ordinary differential equation, which is easier for the user. To do this, run the command Equations > Solve ODE and enter the arguments of the ode2 function in the Solve ODE window.

Maxima allows you to solve second order differential equations. The ode2 function is also used for this. To denote derivatives in differential equations, the diff function is used, in which one more argument is added - the order of the equation: "diff(f(x), x, 2). For example, the solution to an ordinary second-order differential equation a·y"" + b·y" = 0 will look like:

Together with the ode2 function, you can use three functions, the use of which allows you to find a solution under certain restrictions based on the general solution of differential equations obtained by the ode2 function:

1. ic1 (result of function ode2, initial value of the independent variable in the form x = x 0, value of the function at point x 0 in the form y = y 0). Designed to solve a first order differential equation with initial conditions.
2. ic2(result of function ode2, initial value of the independent variable in the form x = x 0, value of the function at point x 0 in the form y = y 0, initial value for the first derivative of the dependent variable relative to the independent variable in the form (y,x) = dy 0). Designed to solve a second order differential equation with initial conditions
3. bc2(result of function ode2, initial value of the independent variable in the form x = x 0, value of the function at point x 0 in the form y = y 0, final value of the independent variable in the form x = x n, value of the function at point x n in the form y = y n). Designed to solve a boundary value problem for a second order differential equation.

The detailed syntax of these functions can be found in the system documentation.

Let us solve the Cauchy problem for the first order equation y" - ax = 0 with the initial condition y(n) = 1.

Let us give an example of solving a boundary value problem for a second-order differential equation y""+y=x with initial conditions y(o) = 0; y(4)=1.

It should be borne in mind that quite often the system cannot solve differential equations. For example, when trying to find a general solution to an ordinary first-order differential equation, we get:

In such cases, Maxima either displays an error message (as in in this example) or simply returns "false".

Another option for solving ordinary first- and second-order differential equations is designed to find solutions with initial conditions. It is implemented using the desolve function.

Function syntax:

desolve(differential equation, variable);

If a system of differential equations is being solved or there are several variables, then the equation and/or variables are presented in the form of a list:

desolve([list of equations], [variable1, variable2,...]);

As with the previous version, the diff function is used to denote derivatives in differential equations, which has the form “diff(f(x), x).

Initial values ​​for a variable are provided by the atvalue function. This function has the following syntax:

atvalue(function, variable = point, value at point);

In this case, it is provided that the values ​​of functions and (or) their derivatives are set to zero, therefore the syntax of the atvalue function is:

atvalue(function, variable = 0, value at point "0");

Example. Find a solution to the first order differential equation y"=sin(x) with the initial condition.

Note that even if there is no initial condition, the function will also work and produce the result:

This allows the solution to be tested for a specific initial value. Indeed, substituting the value y(0) = 4 into the resulting result, we get y(x) = 5 - cos(x).

The desolve function makes it possible to solve systems of differential equations with initial conditions.

Let us give an example of solving a system of differential equations with initial conditions y(0) = 0; z(0) = 1.

Data processing

Statistical analysis

The system makes it possible to calculate basic statistical descriptive statistics, with the help of which the most general properties empirical data. Basic descriptive statistics include mean, variance, standard deviation, median, mode, maximum and minimum values, range of variation, and quartiles. Maxima's capabilities in this regard are somewhat modest, but most of these statistics are quite easy to calculate with its help.

The most in a simple way To calculate statistical descriptive statistics is to use the "Statistics" palette.

The panel contains a number of tools grouped into four groups.

1. Statistical indicators (descriptive statistics):
   • mean (arithmetic mean);
   • median(median);
   • variance (variance);
   • deviation (standard deviation).
2. Tests.
3. Construction of five types of graphs:
   • histogram. It is used primarily in statistics to depict interval series of distributions. During its construction, parts or frequencies are plotted along the ordinate axis, and the values ​​of the attribute are plotted on the abscissa axis;
   • scatter plot (correlation diagram, correlation field, Scatter Plot) - a graph of points when the points do not connect. Used to display data for two variables, one of which is a factor and the other an outcome. With its help, a graphical representation of data pairs is carried out in the form of a set of points (“clouds”) on the coordinate plane;
   • Bar Chart - a graph in the form of vertical columns;
   • sector or pie chart (Pie Chart). Such a diagram is divided into several segments-sectors, the area of ​​each of which is proportional to their part;
   • box plot (box with a whisker, box with a whisker, Box Plot, box-and-whisker diagram). It is the one most often used to display statistical data. The information in this chart is very informative and useful. It simultaneously displays several values ​​that characterize the variation series: minimum and maximum values, average and median, first and third quartiles.
4. Tools for reading or creating a matrix. To use the palette tools, you must have initial data in the form of a matrix - a one-dimensional array. You can create it in the document with the current session and subsequently substitute its name as input in the palette tool windows in the same way as solving equations using the General Math panel. You can also directly enter the data in the input data entry windows. In this case, they are entered in the form accepted in the system, that is, in square brackets and separated by commas. It is clear that the first option is much better, since it only requires one-time data entry.

Besides the panel, all statistical tools can also be used using the corresponding functions.