Topic: measurement of resistance with Wheatstone

BRIDGE

2. Goal of the work:

2.1. Study the method of measurements by means of a bridge circuit.

2.2. Study the method of data processing.

2.3. Finding a resistance of conductors.

 

Main concepts

Ohm's law for homogeneous subcirquit

Electric current is an ordered motion of electric charges. This ordered motion is also called flow. There are conduction currents, convection currents, displacement currents and currents in vacuum. In this section we will consider only conduction current.

Conduction current is an ordered motion of free electrons and ions. In metals current is being formed by negatively charged free electrons, in electrolytes – by ions, in semiconductors – by electrons and holes.

For a direction of current the direction of motion of positive charges (an electric intensity direction) is accepted. That's why the direction of a current in metals is opposite to the direction of electrons motion.

The main characteristics of a current are current intensity and current density.

Current intensity (or simply “ current ”) is a scalar physical quantity, which numerically is equal to an electric charge that flows through cross-sectional area of the conductor per unit of time. For direct current (DC):

; [ I ]= A, (19)

where D q – is a charge, which flows through cross-sectional area of the conductor per time intervalD t. SI unit for current is ampere (A).

Same for alternating current (AC) is:

,

where dq – is an infinitesimal charge, which flows through cross-sectional area of the conductor per infinitesimal time interval dt.

In order to create a current in a conductor we must create an electric field in it, hence, the potential difference must exist on its ends.

As it was experimentally established by Ohm, the current through a conductor is directly proportional to the potential differenceφ₁ - φ₂on its terminals, and inversely proportional to the resistance R of this conductor:

. (20)

The potential differenceφ₁ - φ₂=Dj by other words is being called voltage U, therefore equation (20) has such view I=U / R.

Resistance of a conductor - is a characteristic, which is being defined by resistivity of a material r, of which this conductor is made, but depends on a length l and on a sectional area of the conductor S.

Fig. 10– Direct current appearance mechanism

; [ R ]=W. (21)

Resistance of a conductor is being measured in Ohms.

When conductors or resistors are serially connected, their total equivalent resistance is being increased:

R _T^SER = R ₁+ R ₂,

but when they in-parallel, their total equivalent resistance is being increased and can be evaluated from formula:

,

from which

.

Let's consider, under which condition the direct current will flow through a conductor.

If two metallic bodies A and B (see Fig. 10), which charged to a potentials j₁ and j₂, we join by a conductor C, then under the influence of electric field forces electrons will start to overflow from negatively charged body B to the positively charged body A. Through the conductor C from the positively charged body A to the negatively charged body B an electric current І start to flow. Potentials of the charged bodies will be balanced and current will be decreased.

For a direct current flowing, it is necessary that on a path D charges moved against an electrostatic force (from the negatively charged body B to the positively charged body A). Such work can be done only by forces of non-electrostatic origin, for example, magnetic, chemical and other. These forces are called extraneous.

Quantity, equal to work, done by extraneous forces required to move positive unit charge inside the source against the forces of electrostatic field, is called an electromotive force (EMF) of a source.

. (22)

 

Kirchhoff’s rules

For a solution of branched circuits it is used two rules, which are algorithm for set up of equations that relate currant, voltages, and electromotive forces on elements of branched circuit.

Before usage of Kirchhoff’s rules for solution of branched circuit, for example, of bridge circuit (see Fig. 11), we should assign three topological elements of the circuit:

Junction – is a point of connection of three or greater conductors. On Fig. 11 junctions are marked by letters A, B, C and D.

Branch – is a way from one junction to a neighbour junction with its own elements. One's own current flows in each branch from initial junction to end junction through all elements of this branch. Therefore the branch's number usually same as the current's number.

For example, on Fig. 11 we see: current I ₁ flows through branch AD, current I ₂ flows through branch DС, current I ₃ flows through branch BС, current I ₄ flows through branch AB, current I ₅ flows through branch CA, current I ₆ flows through branch BD. If in the branch we have a source of EMF, then direction of current should coincide with direction of extraneous forces work (from “–“ to “+” inside the source). If in the branch a source is absent, direction of current should be set any way.

Fig. 11 – Bridge’s branched circuit

Closed loop – is a way from a junction to the same junction with its own elements. For each considered closed loop it is necessary to choose at once directions of path-tracing (or clockwise, or against) and to fix on the circuit schema. For example, on Fig. 11: the trace direction of loops ADCA and ABCA is chosen anticlockwise.

1^st Kirchhoff’s rule – is a junction’s rule:

, (23)

where – algebraic sum of a current into the junction (which is positive when the currents flow in the junction, and negative when the currents flow out of the junction);

2^nd Kirchhoff’s rule – is a closed loop’s rule:

, (24)

– algebraic sum of voltage drops on external resistors around any closed loop, and – algebraic sum of the voltage drops on internal resistance of sources (which are positive, when the direction of a current coincides with chosen direction of path-tracing);

– algebraic sum of the source electromotive force of the closed loop (which are positive, when the direction of extraneous forces work (from “–“ to “+” inside the source coincides with chosen direction of path-tracing).