Maximum differential. Maxima mathematical system

For noise immunity, complementary transmitted signals must be well balanced and have the same impedance

Differential transmission implies the presence of two complementary signals with equal amplitude and a phase shift of 180°. One of the signals is called positive (direct, non-inverse), the second is called negative (inverse). Differential transmission is widely used in electronic circuits and is essential for increasing data transfer speed. High-speed clock signals on computer motherboards and servers are transmitted over differential lines. Numerous devices such as printers, switches, routers and signal processors use LVDS (Low Voltage Differential Signaling) technology.

Compared to single-wire transmission, implementing differential transmission requires a larger number of transmitters (drivers, transmitters) and receivers (receivers), as well as a doubled number of element pins and conductors. On the other hand, using differential transmission offers several attractive advantages:

Greater temporal accuracy,
- high possible transmission speed,
- less susceptibility to electromagnetic interference,
- less noise associated with crosstalk.

When routing differential conductors, it is important that both differential traces have the same impedance, are the same length, and the distance between their edges is constant.

Using an example, let's look at a few important differential wiring concepts. Figure 1 shows the differential bus motherboard, laid between the pins of an application-specific chip (ASIC) and the connector for connecting a daughter board with memory chips. Direct signal conductor highlighted green, and inverse - red. Each conductor along its length has two vias and a serpentine section.

Rice. 1. Differential pair on motherboard PCB

The differential wiring in this figure is made taking into account several rules:

The terminals of components used to transmit or receive differential signals are located close to each other;
- on each individual layer, segments of tires of the same length are located, and the distance between the tires remains the same on different layers;
- when changing a layer, the gap between the via pads is kept to a minimum (not exceeding the distance between the buses, if feasible);
- the serpentine sections of the two buses are located in the same area so that the positive and negative signals have the same propagation delays along the entire length of the circuit.

Rounding corners and uniform lengths of differential conductors require special care.

In addition to conductors printed circuit board, the integrated circuit package contains buses that connect each pin of the package to a pin of the IC chip. The different lengths of these tires can make adjustments in some cases.

As a numerical example, consider differential buses with the following segment lengths:

for direct signal

Segment length from connector pin to first via = 3022.93 mils (76.78 mm),

Segment length between vias = 747.97 mils (19.0 mm)

Total direct signal length = 3,798.70 mils (96.49 mm);

For inverse signal

Segment length from connector pin to first via = 3025.50 mils (76.78 mm),

Segment length between vias = 817.87 mils (19.0 mm)

Segment length from second via to IC pin = 27.8 mils (0.71 mm)

Total length of direct signal circuit = 3,871.17 mils (98.33 mm).

Thus, the difference in PCB trace lengths is 72.47 mils (1.84 mm).

Some of the resulting difference can be compensated by taking into account the different lengths of the buses inside the IC package. In this case, the difference in the total lengths of the routes becomes within the specified tolerance.

Figure 2 shows that total length buses must be thought out from the point of view of reducing the difference in the lengths of differential conductors.

Rice. 2. The sum (L0 + L1) must be equal to the sum (L2 + L3) within the permissible error

Repeating again, it is desirable to keep the distance between the edges of the conductors constant throughout their entire length. A study of the differential pair shows that near the terminals of the bus connector the buses lose parallelism relative to each other. Figure 3 illustrates a wiring diagram that minimizes this drawback while maintaining parallelism on long length(the sharp angle of the inverse signal conductor formed in this case can lead to loss of its integrity with the ensuing consequences - translator's note). This design can be used in cases where differential signals must be highly coupled or when transmitting high-speed signals.

Rice. 3. Parallel wiring

When the spacing between two traces is relatively large (the coupling between the conductor and the polygon exceeds the coupling between the conductors), the pair becomes loosely coupled. Conversely, when two traces are located close enough to each other (the relationship between them is greater than the connection between the individual conductor and the polygon), then this means that the conductors of the pair are strongly connected. Strong coupling is usually not necessary to achieve the initial benefits of differential structure. However, to achieve good noise immunity, strong coupling is desirable for complementary, well-balanced signals that have symmetrical impedance relative to the reference voltage.

The concept of differential wiring in this case involves coplanar pairs (i.e., located in the same layer), having connections at the edges of the conductors. Differential signals can also be routed in another way, in which the conductors of the direct and inverse signals are located on different (adjacent!!!) layers of the board. However, this method may cause problems with impedance consistency. Figure 4 shows both of these options, as well as some critical dimensions such as width (W), edge spacing (S), conductor thickness (T), and conductor-to-polygon distance (H). These parameters, which establish the cross-sectional geometry of the differential pair, are often used (along with the material properties of the conductors and substrate dielectric) to determine impedance values ​​(for irregular, equilibrium, in-phase, and anti-phase modes) and to calculate the amount of coupling between the conductors of the pair.

Rice. 4. Geometric dimensions of the differential pair section

Abbas Riazi
DIFFERENTIAL SIGNALS ROUTING REQUIREMENTS
Printed Circuit Design & Manufacture
February-March 2004
We thank the site elart.narod.ru for providing the translation

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, order of derivative for variable1 [,variable2, order of derivative 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 of 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 represent 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 common decision 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, the right side of the equation 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.

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, provided that the desired and common mode input signals are of 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 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), sound 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), electrocardiogram voltages, signals for reading information from magnetic memory, and many others.

Rice. 2.67. Classic transistor differential amplifier.

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 play an important role in amplifier design direct current(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 a single-pole output circuit or a difference amplifier and is the most widely used. 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.

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 an amount (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 a single transistor, if you notice that the input signal is applied twice to the base-emitter junction of any transistor: . The resistance of the resistor 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 gain, the same signals must be applied to both inputs of the amplifier. If you look carefully at this case (and remember that both emitter currents flow through the resistor), you will get . We neglect resistance, since the resistor is usually chosen to be large - its resistance is at least several thousand ohms. In fact, resistance can also be neglected. CMRR is approximately equal to . A typical example The differential amplifier is the circuit shown in Fig. 2.68. Let's look at how it works.

The resistance of the resistor is chosen so that the quiescent collector current can be taken equal to . As usual, the collector potential is set to 0.5 to obtain maximum dynamic range. The transistor does not have a collector resistor, since its output signal is taken from the collector of another transistor. The resistor value is chosen such that the total current is equal and equally distributed between the transistors when the input (differential) signal equal to zero.

Rice. 2.68. Calculation of differential amplifier characteristics.

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 in the input stage you can use Darlington transistors).

Recall that in an unbalanced amplifier with a grounded emitter at an output quiescent voltage of 0.5, the maximum gain is equal to , where is expressed in volts. In a differential amplifier, the maximum differential gain (at half as much, i.e., is 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 it is also numerically 20 times the voltage drop across

Exercise 2.13. Make sure the given ratios are correct. Design a differential amplifier according to 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 common-mode signal rejection, and the small inter-emitter coupling resistances (including the emitters’ own resistances) determine the differential signal amplification.

Biasing using a current source.

The common mode gain in a differential amplifier can be significantly reduced by replacing the resistor with a current source. In this case, the effective value of the resistance will become very large, and the common-mode signal gain will be attenuated 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 (transistors and ) and a current source of type , the CMRR value is determined by the ratio dB). The input common mode signal range is limited to -12 and ; 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.

Remember that this amplifier, like all transistor amplifiers, must have DC bias 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 a reverse bias at the base-emitter junction of no more than 6 V, 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, 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: as the temperature changes, the voltages change by the same amount, while there is no change in the output and the circuit is not unbalanced. This means that the voltage change is not amplified by the coefficient Kdif (its amplification is determined by the coefficient Xinf, which can be reduced to almost zero). In addition, mutual compensation of voltages results in the fact that voltage drops of 0.6 V do not need to be taken into account at the input. The quality of such a DC amplifier deteriorates only due to voltage mismatches or their temperature coefficients. The industry produces transistor pairs and integrated differential amplifiers with very high degree coordination (for example, for a standard consistent monolithic n-p-n pairs-type transistors, the voltage drift is determined by the value or 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 form a differential pair with a current source in the emitter circuit. Transistors forming a current mirror act as a collector load. This ensures a high value of the 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 greater. Comparators are used in a wide variety of areas: to turn on lighting and heating, to obtain rectangular signals from triangular ones, to compare the signal level with a threshold value, in class D amplifiers and pulse code modulation, to switch power supplies, etc. The basic idea when building a comparator is 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).

Differential Gain and Differential Phase are measures of an amplifier's linear distortion. The differential gain is expressed in two quantities, which represent the two peak amplitudes of the subcarrier relative to the amplitude of the black level subcarrier. The differential gain is calculated from the maximum and minimum amplitude of the steps of a special test signal at the output of the demodulator using the oscillogram of the differentiated signal obtained at the output of the measuring differentiating chain with a time constant of 300 ns. Differential phase is expressed in two degrees, which represent the two peak phases of the subcarrier relative to the phase of the black level subcarrier. The differential phase value is calculated as the difference between the maximum and minimum phases of the superimposed element 1 in the level range from black to white. According to EN 50083, in any television channel, the maximum differential gain (peak to peak) must not be more than 14% and the maximum differential phase must not be more than 12%.
Other parameters include input and output impedance, power consumption and supply voltage, weight and dimensions. Let us give as an example the abbreviated specifications of two amplifiers - the main amplifier VX96 from WISI and the universal amplifier DXE 853 GA from Teleste (Table 10.2). The parameters given in the specification should not be considered standard, since they relate to specific amplifier models. Both amplifiers have a forward channel frequency range of 47 – 862 MHz and are equipped with a return channel amplifier module. The amplifiers are equipped with a local or remote power supply. Remote power is supplied through any signal port or built-in power input and can be transmitted in any direction. The VX96 amplifier is designed to create high-quality backbone sections. It is built on the basis of two hybrid chips. A low-noise Push Pull circuit is installed as the input stage, and a powerful silicon or gallium arsenide Push Pull or Power Double circuit is installed as the output stage. The DXE 853 GA amplifier can be used in trunk and home distribution sections of the cable network and has two outputs, configurable by installing taps. Both amplifiers have interstage adjustable EQ and attenuator, replaceable diplexer filters, and space for an optional EQ and attenuator (cable equivalent).

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 wanted output signal to the common-mode output signal, assuming that the wanted 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 the permissible voltage levels relative to which the input signal must vary.

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 play an important role in the design of DC amplifiers (which amplify frequencies down 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 handled by conventional circuits (voltage followers, 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) is 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 a differential amplifier to suit your own requirements.

A differential amplifier can be figuratively called a “long-tailed pair”, since if the length of the resistor in the 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 common-mode signal rejection, and the small inter-emitter coupling resistances (including the emitters’ own resistances) determine the differential signal amplification.

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 a maximum of 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 a voltage drop of 0.6 V. The quality of such a DC amplifier deteriorates only due to 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 pair of n-p-n 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. A beautiful solution is provided by using 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 used, as a rule, only in circuits covered by a feedback loop, or in comparators (we will consider 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 component comparator- a circuit that compares input signals and evaluates which one is larger. Comparators are used in a wide variety of areas: to turn on lighting and heating, to obtain rectangular signals from triangular ones, to compare the signal level with a threshold value, in class D amplifiers and pulse code modulation, to switch 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).