Counting arrow

A counting arrow or reference arrow in electrical circuit diagrams illustrates the approach for the direction of an unknown electrical current or electrical voltage . The selected reference arrow orientation determines the size value sign.

In order to be able to set up the calculation model for a circuit with unambiguous size variables, a reference arrow for the (unknown) direction of the electrical current strength (current reference arrow) and a reference arrow for the (unknown) polarity of the electrical voltage (voltage reference arrow) are entered on each component . Each reference arrow is oriented from one component pole to the other. All orientations, including those of the two reference arrows on the same component, are independent and freely selectable.

If the calculation method that takes up the selected arrowhead yields a positive value, the direction of the variable (which can be verified by measurement) is identical to the selected reference arrow direction; a negative value means that the direction of the size is opposite to the reference arrow.

Consumer counting arrow system (engine-related agreement)

Reference arrows according to the consumer counting arrow system. Each box stands for a passive or active two-pole .

The reference arrows on a two-pole form a consumer metering arrow system (VZS) if both arrowheads or both arrow ends are on the same pole and the reference arrow for the electrical power points into the two-pole (see figure opposite). The bipolar is thus formally treated as a “consumer”, regardless of its operating principle.

The consumer arrowhead is the preferred choice for passive components such as ohmic resistance , ideal capacitor and ideal coil . The current-voltage relationships for these components are , and . Consumer arrows are also permitted for active components (electric accumulator cell when discharging, solar cell , thermocouple , ...) and can also be attached in individual cases. ${\displaystyle u=R\cdot i}$ ${\displaystyle \textstyle i=C\cdot {\frac {\mathrm {d} u}{\mathrm {d} t}}}$ ${\displaystyle \textstyle u=L\cdot {\frac {\mathrm {d} i}{\mathrm {d} t}}}$

Generator counting arrow system (generator-related agreement)

Reference arrows according to the producer counting arrow system. Each box stands for an active or passive two-pole.

The reference arrows on a two-pole form a generator counting arrow system (EZS) if an arrowhead and an arrow end are on the same connection and the reference arrow for the electrical power points out of the two-pole (see figure opposite). The bipolar is thus formally treated as a “producer”, regardless of its operating principle.

The generator sweep is the preferred choice for active components (see above). Producer arrows are also permitted for passive components (see above). The current-voltage relationships are for an ohmic resistance , for an ideal capacitor and for an ideal coil . ${\displaystyle u=-R\cdot i}$ ${\displaystyle \textstyle i=-C\cdot {\frac {\mathrm {d} u}{\mathrm {d} t}}}$ ${\displaystyle \textstyle u=-L\cdot {\frac {\mathrm {d} i}{\mathrm {d} t}}}$

Electrical power

The equation for electrical power applies to both reference systems

${\displaystyle p=u\cdot i}$

in the same form. If a power value is calculated positively when the VZS is applied, the two-pole receives the electrical power ; if the value is negative, it delivers the electrical power . ${\displaystyle p}$ ${\displaystyle |p|}$

If a power value is calculated positively when using the EZS, the two-pole outputs the electrical power ; if the value is negative, it consumes the electrical power . ${\displaystyle p}$ ${\displaystyle |p|}$

Change of sizes when changing systems

In order to switch from the specified reference system to the other while the electrical state of a two-terminal remains unchanged, z. B. the current reference arrow can be turned. As a result, according to the definition of the counting arrow systems, the service reference arrow also changes its orientation. Before the move was for the performance . After the change, noted in crossed size symbols, the following applies . With and follows . The negated power value together with the turned power reference arrow describe the electrical power unchanged in terms of value and direction. ${\displaystyle p=u\cdot i}$ ${\displaystyle p'=u'\cdot i'}$ ${\displaystyle i'=-i}$ ${\displaystyle u'=u}$ ${\displaystyle p'=u\cdot (-i)=-p}$

Reversing the voltage reference arrow instead of the current reference arrow leads to the same result.

example

Network for the mutual charging of two accumulators

In the network shown, the left voltage source is set in the EZS, all other components in the VZS.

Two cases are considered:

1. At the left voltage source supplies electrical power to the right one. All values ​​including the power and the two ideal voltage sources are positive. ${\displaystyle u_{1}>u_{2}>0}$ ${\displaystyle p_{1}=u_{1}\cdot i}$ ${\displaystyle p_{2}=u_{2}\cdot i}$
2. At the value of the current intensity becomes negative, as well as the values ​​of the voltages and at the internal resistances and and the values ​​of the electrical source powers and . ${\displaystyle u_{2}>u_{1}>0}$ ${\displaystyle i}$ ${\displaystyle u_{R_{1}}=R_{1}\cdot i}$ ${\displaystyle u_{R_{2}}=R_{2}\cdot i}$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$ ${\displaystyle p_{1}=u_{1}\cdot i}$ ${\displaystyle p_{2}=u_{2}\cdot i}$

Both cases are correctly recorded with the same arrow (EZS for the left source, VZS for the right source and the internal resistances). The example illustrates that there is no coupling between the operating principle of a component (active or passive) and the counting arrow system to be used.

Derivation

Based on the rule of meshes, the following results for the example network

${\displaystyle -u_{1}+u_{R_{1}}+u_{R_{2}}+u_{2}=0}$.

With the help of Ohm's law you get

${\displaystyle -u_{1}+R_{1}\cdot i+R_{2}\cdot i+u_{2}=0}$.

The electricity can be derived from this

${\displaystyle i={\frac {u_{1}-u_{2}}{R_{1}+R_{2}}}}$

calculate by transforming the equation . The statements in the two cases considered above can now be immediately understood: The decisive factor for the sign of the current is the voltage difference in the numerator of the fraction ; the resistance sum in the denominator is always positive. The sign of the current directly affects the signs of the above-mentioned (from the power dependent) variables , , and from. ${\displaystyle i}$ ${\displaystyle u_{1}-u_{2}}$ ${\displaystyle R_{1}+R_{2}}$ ${\displaystyle i}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle u_{R_{1}}}$ ${\displaystyle u_{R_{2}}}$

Hints

1. Current and voltage reference arrows in circuit diagrams allow, together with the value sign of the variables, to infer their linear direction . They are not vectors and therefore have neither a length nor a spatial direction .
2. Reference arrows are also required for the clear definition of quantities in AC circuits.
   1. In the case of sinusoidal currents and voltages, this applies to the instantaneous values of voltage, amperage and power, as well as to the signed mean values ​​active and (displacement) reactive power as well as to the complex-valued voltage, amperage and apparent power.
   2. In the non-sinusoidal case , reference arrows only belong to the instantaneous values ​​of voltage, current strength and power as well as to the real power.
3. Even measured values ​​from devices that can display positive or negative values ​​are only fully documented when the associated reference arrows are specified. These are determined by the selected polarity of the connections and oriented from the plus to the minus pole of the device . Reference arrows can be understood as “stylized measuring devices”. The free choice of the polarity of measuring devices corresponds to the free choice of the reference arrow orientation for calculations.
4. With the usual circuit symbols for resistance, inductance, capacitance and sources, the reference arrows for current and voltage appear in the VZS with the same orientation (parallel), in the EZS they are opposite (anti-parallel).
5. Only two of the three reference arrows for voltage, current and power of a two-pole can be specified independently of one another. The third arises from the definitions of VZS or EZS.
6. The article references DIN EN 60375 in the current draft version from 2013. There is currently no valid DIN standard on the subject.

literature

• Heinrich Frohne : Introduction to Electrical Engineering - Fundamentals and Networks, pp. 119–124 . 5th revised edition. BG Teubner, 1987, ISBN 3-519-40001-4 .
• Dietrich Oeding, Bernd R. Oswald: Electrical power plants and networks, section 2.3, pp. 20-27 . 8th edition. Springer, 2016, ISBN 978-3-662-52702-3 .

Individual evidence

1. ↑ DIN EN 60375 (draft) Agreements for electrical circuits and magnetic circuits (IEC 25/464 / CD: 2013) , Section 5.2.
2. ↑ DIN EN 60375 (draft) Agreements for electrical circuits and magnetic circuits (IEC 25/464 / CD: 2013) , Section 6.2.
3. ↑ DIN EN 60375 (draft) Agreements for electrical circuits and magnetic circuits (IEC 25/464 / CD: 2013) , Section 7.2.1.
4. ↑ DIN EN 60375 (draft) Agreements for electrical circuits and magnetic circuits (IEC 25/464 / CD: 2013) , Section 7.2.2.