Electricians need to keep up-to-date with Ohm’s Law and several other mathematical theories in order to complete their day-to-day tasks. David Herres offers this refresher course.

In the Autumn 2012 edition of Electrical Connection, we discussed Ohm’s Law and formulas that electricians need to calculate power, current or voltage when the other two variables are known.

These formulas are shown graphically in the Ohm’s Law Wheel.

In the centre of the wheel are four quantities, any one of which may be unknown at the outset:
•    P (power) is quantified in Watts.
•    E (electromotive) force is quantified in Volts.
•    I (intensity) is quantified in Amps.
•    R (resistance) is quantified in Ohms.
Ohm’s Law states, as you can see in the upper right quadrant, that E=IxR. Similarly, each of the other three quantities can be expressed as a function of the known amounts.

Electricians refer to these formulas frequently. One used a lot is I=E/R.

The value of this formula is that when the voltage is known (which is just about always because you know your voltage system) it is possible to find the current in amps since the power in watts is given. Very frequently you will confront an appliance, such as a hot water heater, that is rated at 5,500W. It has two elements, each rated at that wattage. They are never on simultaneously. The amp rating, then, is 5,500 divided by the system nominal voltage. (Always use the nominal, not the measured voltage.)  This is 230. The amperage, accordingly, is 24. Since a domestic hot water heater is considered a continuous load, the 24A figure is multiplied by 1.25 to give a 30A minimum circuit size. A 30A breaker with 10AWG copper wire is used.

This is one of the most common calculations performed by an electrician, finding circuit rating in amps, breaker size and wire size based on the rating in watts of the load. Most of the time, the calculation consists of dividing watts by volts to find amps. To help remember the formula, notice that watts usually appears in the numerator while volts is in the denominator. Also beneath the bar are various factors that might be applied to the amount of voltage. These factors may be expressed as a percentage or as a decimal multiplier. Examples are power factor, efficiency (for a transformer) or the 1.73 figure associated with three-phase power circuits.

The thing to remember about the Ohm’s Law Wheel is that it is not some mysterious mechanical device geared to the cosmos, but simply a graphical representation in circular format of some fundamental numerical relationships.

Volt-amps rather than watts often appear on an appliance nameplate or in formulas. For DC and the 50Hz and 60Hz frequencies of most power systems throughout the world, watts and volt-amps are nearly the same. As the frequency increases, however, they diverge.

Another important formula depicted in the Ohm’s Law Wheel is P=I2xR. This formula, known as Joule’s Law, states that power in watts, for example the heat dissipated by any resistive load, is proportional to the amount of current in amps squared times the amount of resistance in ohms. The power may be useful work performed by a device such as a rotary electric motor (which is in part a resistive load) or it may be heat dissipated into the surrounding space. In actuality, there may or may not be useful work, such as sawing lumber, but there is always heat dissipation, usually considered a total loss.

This formula is useful not so much in performing any particular calculation as it is in helping us to understand the power flow. Whenever there is an electrical circuit that is not open, there is a power flow from the source to the load. If you visualise this transfer of power, drawing a diagram if necessary where there are multiple branching paths, it will be a great aid in troubleshooting malfunctioning equipment and locating faults. Because the amount of current is squared while the resistance is not squared, it means that I is much more important than R in determining the value of P. Also, given a constant amount of power, when R increases, I decreases. When I increases, R decreases. R and I2 are inversely proportional and they are directly proportional to P. It is helpful to understand these relationships in order to visualise the operation of all electrical circuits.

Related to this concept is the first of several theorems that focus on impedance in electrical networks. It is known as Kirchhoff’s Theorem, which consists of:
•    Kirchhoff’s Current Law, stating that the sum of electric currents that flow into any junction in an electric circuit is equal to the sum of currents that flow out; and,
•    Kirchhoff’s Voltage Law, stating that the sum of the electrical voltages around a closed circuit must be zero.
Experienced electricians are very familiar with the theorem and associated laws, even if the wording above does not have meaning for them. Every time you perform a voltage reading with your multimeter or take a current measurement with your clamp-on ammeter, you are making reference to these fundamental relationships. A voltmeter is connected in parallel to the power source of any one of the series loads. If it is a high-impedance instrument, it will draw a minute amount of current and will not affect the operation of the circuit. A direct-reading ammeter is connected in series with all of the series loads. It is a low-impedance instrument and being series connected, also will not affect the operation of the circuit.

The thing to remember is that within any parallel branch of an electrical circuit, the current is everywhere the same. If there is only a single current path, the current is the same throughout, including within the power source. Keeping this fundamental principle in mind will facilitate troubleshooting and help you make sense of measurements that you take.

Some other network theorems are Thevenin and Norton equivalence theorems. They are helpful in analysing complex circuits where Ohm’s Law is applicable but difficult to apply since there are multiple power sources and circuit branches. They are useful in managing utility-scale distribution systems.

To repair electrical equipment, it is usually essential to understand how it works. Three common components are the resistor, the capacitor and the inductor.

When they are connected in series or parallel, these formulas apply:

• Resistors in series – R_TOTAL = R₁+ R₂ + . . .
• Resistors in parallel – R_TOTAL =1/(1/R₁ + 1/R₂ + . . .)
• Capacitors in series – C_TOTAL =1/(1/C₁ + 1/C₂ + . . .)
• Capacitor in parallel – C_TOTAL = C₁+ C₂ + . . .
• Inductors in series – L_TOTAL = L₁+ L₂ + . . .
• Inductors in parallel – L_TOTAL =1/(1/L₁ + 1/L₂ + . . .)

Some simple observations will demonstrate how these formulas work.

When resistors are connected in series, it is not different from making a larger single resistor. If a single resistor is thought of as made of discrete segments, you can see that these segments are connected in series. The same analysis applies for to inductors. For capacitors, the reverse is true.

Two capacitors in parallel are equivalent to a single capacitor with larger plates, and that is why the capacitances are additive.