Bridge circuit

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Bridge circuit with voltage source

A bridge circuit  - also known as an H circuit , H bridge or full bridge - is an electrical circuit in which five two-pole connections in the form of the capital letter H are connected in the basic form. The cross connection is called the bridge branch .

principle

A bridge circuit made up of resistors can be interpreted as a parallel connection of two voltage dividers with the bridge arm between their output terminals. The advantage of the bridge circuit compared to a single voltage divider is that the voltage and current in the bridge branch can be changed not only in height, but also in polarity, depending on the setting of the resistors.

In the case of bridge circuits, a distinction is made between quarter bridges (one variable resistor), half bridges (two variable resistors) and full bridges (four variable resistors).

The measurement object  - a resistor or another impedance  - is part of the voltage divider on one side. At least one further part is made variable so that a zero adjustment of the current (or the voltage difference) can be carried out in the bridge branch, which allows a particularly precise value determination, since a zero crossing can be determined much more precisely than an extremum. The measuring instrument in the connecting branch of the bridge can usually differentiate between negative and positive values ​​and thus also provides an indication of the direction in which a change must be made. The adjustment takes place until the instrument delivers a zero display as exactly as possible. For this purpose, the sensitivity of the instrument can be switched over on some measuring bridges, so that one can switch from a coarse to a fine adjustment. The impedance of the measurement object is displayed by the setting of the adjustable bridge link (or calculated from its displayed value).

The measuring bridge can only be operated with direct voltage for measuring purely ohmic resistances . To measure impedances ( coils or capacitors ), AC voltage operation is necessary, which can also be beneficial for ohmic resistances. It is not necessary, but helpful in these cases, to display the phase position to indicate the direction of the detuning. See also at AC voltage bridge .

If you do not use a zero adjustment, but the deflection method, then the impedance can be calculated from the magnitude and phase of the diagonal voltage. In this way, the (equivalent) loss resistances of coils or capacitors can be determined without having to reproduce them in the other bridge branch.

calculation

A bridge circuit can best be described by Kirchhoff's rules . To do this, first set up the knot and mesh equations. Optionally, the relationships derived from this can also be represented in a matrix equation. A particular challenge here is the calculation of the total circuit resistance, as will be explained later.

Establishing the knot and mesh equations

When setting up the knot and mesh equations, we assume in this example that the currents flow in the direction of the voltage arrow. If this assumption is incorrect for a current, the amount of the current in question has a negative sign , but this does not change the validity of the equations. Finally, the following nodal equations result from Kirchoff's rules:

The rule of mesh gives the following equations:

Here the equations are not completely linearly independent , which is why one equation can be omitted.

In addition, the relationship applies to the individual resistances

or written out:

Here, the resistance represents the resistance of the circuit from the point of view of the voltage source.

Matrix representation of the knot and mesh equations

The matrix representation is an aid for large systems of equations and therefore especially for large circuits. In order to determine the matrix representation, the respective product of resistance and current is used for the individual voltages. From this we get:

The matrix representation is preferably used for use in computer algebra systems or in circuit simulators, since efficient solution algorithms exist with the Gaussian and Gauss-Jordan algorithms as well as Cramer's rule . In the example given, Cramer's rule can only be applied to a sub-matrix , since the determinant of the left-hand matrix would always be zero due to the upper four rows.

Calculation of the circuit resistance

The calculation of can be achieved using the relationships established using Kirchoff's rules. The following procedure is a faster variant:

  1. First it is assumed that there is an interruption. This gives the equation:
  2. Then it is set and thus short-circuited. This gives the equation:
  3. Now the resistance is determined from the point of view of , whereby the voltage source is set to infinity:
  4. This can be inserted into the following equation, which was determined in advance with the help of the formulas and simplification determined by Kirchhoff's rules:

Alternatively and with less computational effort, the star-delta transformation can also be used. In addition , rsp. , as well as , replaced by their star equivalent. This results in a simple circuit with two series resistors in parallel and the whole in series with a fifth resistor.

Matching condition

A bridge circuit is said to be balanced if and therefore no current flows from one branch of the bridge to the other. If this is the case:

The connection follows from this:

Explicit solutions

For those who visit this site because they have to solve specific tasks to bridge the gap:

The following system of equations (in matrix representation ) is not redundant, so it contains a minimum number of independent statements and can be solved according to the rules of matrix inversion.

So let the resistance values ​​and the source voltage be given.

Then the following applies to the auxiliary quantity - that is the value of the determinant of the coefficient matrix on the left -

Herewith

If the bridge is balanced , then , applies

is arbitrary. (The balanced bridge (only the one!) Can therefore be viewed in calculations as being open or short-circuited in the cross member, i.e. equal to infinity or zero.)

One of the four series resistances is determined by the other three, because

Applications

The bridge circuit serves, among other things, as the basis for the following circuits:

Energy technology or power electronics

measuring technology

Communications engineering

literature

  • Hans-Ulrich Giersch, Hans Harthus, Norbert Vogelsang: Testing electrical machines, standardization, power electronics. 5th edition, BG Teubner / GWV Fachverlage GmbH, Wiesbaden, 2003, ISBN 3-519-46821-2 .
  • Hansjürgen Bausch, Horst Steffen: Electrical engineering. Basics, 5th edition, Teubner Verlag, Wiesbaden 2004, ISBN 978-3-519-46820-2 .

Web links

Individual evidence

  1. Jörg Böttcher: Online Compendium Measurement Technology and Sensor Technology: Bridge Circuits for Resistive Sensors. Retrieved June 28, 2019 .
  2. Jörg Böttcher: Online Compendium Measurement Technology and Sensor Technology: Bridge circuits for capacitive and inductive sensors. Retrieved June 28, 2019 .