3.3. 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 I₁ flows through branch AD, current I₂ flows through branch DС, current I₃ flows through branch BС, current I₄ flows through branch AB, current I₅ flows through branch CA, current I₆ 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).

4. Description of laboratory research facility and methodology of measurements

Devices and outfits (See Fig. 12): source of EMF e, microamperemeter mA (zero-indicator with null in the middle of a scale), resistor with known resistance R₁, one-decate resistor box R₂, four-decate resistor box R₃, resistor with unknown resistance R_X.

The bridge circuit of measurings of resistance is the most accurate method of measuring of resistances.

Wheatstone bridge is a circuit, shown in Fig. 12, that consists of four branches with resistors R₁, R₂, R₃, R_X. These parts are called the bridge arms. Unknown resistor R_X is connected in the arm AB, which has to be measured in this work. Resistors boxes are connected in the arms BC and CD, so R₃ and R₂ can be varied. Resistor R₁, of the arm AD is constant. So, resistors R₁, R₂, R₃ are known. Diagonal AC has a source of EMF - e; the diagonal BD has microampermeter as zero-indicator (zero is in the middle of the scale).

Fig. 12 – Installation diagram of lab 2-2 equipment

When we close the switch K, through the branches AB, AD, BC, DC the currents start to flow in directions, which shown in the Fig. 12. In the diagonal BD through microampermeter current will flow from B to D, if potential of point B is greater than potential of the point D (_B  _D), and from D to B, when _D _B. As the points B and D lie between the points A and C, their potentials are always between _A and _C (_A _C), as the point A switched to the positive pole of the source. So, by changing R₃ and R₂, we can always get the equality of potentials of points B and D (_B = _D). In this case, current in diagonal BD will be equal to zero. This state of electric circuit, when the current through microampermeter is vanishing, is called balance of the bridge.

Let's derive the relation between resistances of bridge's arms, when becomes its balance.