# Alternating current phase relationship

### AC circuits, alternating current electricity

Hello. Inductor has a property which tries to keep current constant. The inductive reactance is denoted by, Xl=2*pi*f*L. Inductive reactance is defined as. AC circuits and AC electricity, explained using animated graphs and phasor diagrams. Impedance, phase relations, resonance and RMS quantities. The relationship V = IR applies for resistors in an AC circuit, so. In AC circuits we' ll talk a lot about the phase of the current relative to the voltage.

Now we can rewrite the equation above to make it look like Ohm's law. Note the two important differences. First, there is a difference in phase: Run the animation again to make this clear.

Again, the vertical component of a phasor arrow represents the instantaneous value of its quanitity.

If they are in phase, the ratio is called resistance. Another difference between reactance and resistance is that the reactance is frequency dependent.

From the algebra above, we see that the capacitive reactance XC decreases with frequency. This is shown in the next animation: The blue shading shows q, the integral under the current curve light for positive, dark for negative. See how the lower frequency leads to a larger charge bigger shaded area before changing sign and therefore a larger VC.

Thus for a capacitor, the ratio of voltage to current decreases with frequency. We shall see later how this can be used for filtering different frequencies. Inductors and the Farady emf An inductor is usually a coil of wire. In an ideal inductor, the resistance of this wire is negligibile, as is its capacitance.

The voltage that appears across an inductor is due to its own magnetic field and Faraday's law of electromagnetic induction. So we define the self inductance of the coil thus: Now this emf is a voltage rise, so for the voltage drop vL across the inductor, we have: Again we note the analogy to Ohm's law: Remembering that the derivative is the local slope of the curve the purple linewe can see in the next animation why voltage and current are out of phase in an inductor. Again, there is a difference in phase: The animation should make this clear.

Note how this is represented on the phasor diagram. For an inductor, the ratio of voltage to current increases with frequency, as the next animation shows. Impedance of components Let's recap what we now know about voltage and curent in linear components. The impedance is the general term for the ratio of voltage to current. The table below summarises the impedance of the different components. It is easy to remember that the voltage on the capacitor is behind the current, because the charge doesn't build up until after the current has been flowing for a while.

The same information is given graphically below. It is easy to remember the frequency dependence by thinking of the DC zero frequency behaviour: At DC, a capacitor is an open circuit, as its circuit diagram shows, so its impedance goes to infinity. RC Series combinations When we connect components together, Kirchoff's laws apply at any instant. The next animation makes this clear: This may seem confusing, so it's worth repeating: This should be clear on the animation and the still graphic below: The amplitudes and the RMS voltages V do not add up in a simple arithmetical way.

Here's where phasor diagrams are going to save us a lot of work. Play the animation again click playand look at the projections on the vertical axis.

## Alternating current

Because we have sinusoidal variation in time, the vertical component magnitude times the sine of the angle it makes with the x axis gives us v t. But the y components of different vectors, and therefore phasors, add up simply: So v tthe sum of the y projections of the component phasors, is just the y projection of the sum of the component phasors.

So we can represent the three sinusoidal voltages by their phasors. While you're looking at it, check the phases. We'll discuss phase below. Now let's stop that animation and label the values, which we do in the still figure below. So we can 'freeze' it in time at any instant to do the analysis. The convention I use is that the x axis is the reference direction, and the reference is whatever is common in the circuit.

In this series circuit, the current is common. In a parallel circuit, the voltage is common, so I would make the voltage the horizontal axis. Be careful to distinguish v and V in this figure! Careful readers will note that I'm taking a shortcut in these diagrams: The reason is that the peak values VmR etc are rarely used in talking about AC: Phasor diagrams in RMS have the same shape as those drawn using amplitudes, but everything is scaled by a factor of 0.

The phasor diagram at right shows us a simple way to calculate the series voltage. The components are in series, so the current is the same in both. The voltage phasors brown for resistor, blue for capacitor in the convention we've been using add according to vector or phasor addition, to give the series voltage the red arrow.

By now you don't need to look at v tyou can go straight from the circuit diagram to the phasor diagram, like this: Now this looks like Ohm's law again: V is proportional to I. Their ratio is the series impedance, Zseries and so for this series circuit, Note the frequency dependence of the series impedance ZRC: At high frequencies, the capacitive reactance goes to zero the capacitor doesn't have time to charge up so the series impedance goes to R.

## AC Inductor Circuits

We shall show this characteristic frequency on all graphs on this page. Remember how, for two resistors in series, you could just add the resistances: That simple result comes about because the two voltages are both in phase with the current, so their phasors are parallel.

Ohm's law in AC. We can rearrange the equations above to obtain the current flowing in this circuit. So far we have concentrated on the magnitude of the voltage and current.

We now derive expressions for their relative phase, so let's look at the phasor diagram again. You may want to go back to the RC animation to check out the phases in time. At high frequencies, the impedance approaches R and the phase difference approaches zero. The voltage is mainly across the capacitor at low frequencies, and mainly across the resistor at high frequencies. Of course the two voltages must add up to give the voltage of the source, but they add up as vectors.

These lines are located in eastern Utah. High voltages have disadvantages, such as the increased insulation required, and generally increased difficulty in their safe handling.

In a power plantenergy is generated at a convenient voltage for the design of a generatorand then stepped up to a high voltage for transmission. Near the loads, the transmission voltage is stepped down to the voltages used by equipment.

Consumer voltages vary somewhat depending on the country and size of load, but generally motors and lighting are built to use up to a few hundred volts between phases. The voltage delivered to equipment such as lighting and motor loads is standardized, with an allowable range of voltage over which equipment is expected to operate.

Standard power utilization voltages and percentage tolerance vary in the different mains power systems found in the world.

High-voltage direct-current HVDC electric power transmission systems have become more viable as technology has provided efficient means of changing the voltage of DC power. Transmission with high voltage direct current was not feasible in the early days of electric power transmissionas there was then no economically viable way to step down the voltage of DC for end user applications such as lighting incandescent bulbs.

Three-phase electrical generation is very common. In practice, higher "pole orders" are commonly used. The advantage is that lower rotational speeds can be used to generate the same frequency.

If the load on a three-phase system is balanced equally among the phases, no current flows through the neutral point.

### Alternating current - Wikipedia

Even in the worst-case unbalanced linear load, the neutral current will not exceed the highest of the phase currents. Harmonics can cause neutral conductor current levels to exceed that of one or all phase conductors.

For three-phase at utilization voltages a four-wire system is often used. When stepping down three-phase, a transformer with a Delta 3-wire primary and a Star 4-wire, center-earthed secondary is often used so there is no need for a neutral on the supply side. For smaller customers just how small varies by country and age of the installation only a single phase and neutral, or two phases and neutral, are taken to the property.

For larger installations all three phases and neutral are taken to the main distribution panel. From the three-phase main panel, both single and three-phase circuits may lead off. Three-wire single-phase systems, with a single center-tapped transformer giving two live conductors, is a common distribution scheme for residential and small commercial buildings in North America.

This arrangement is sometimes incorrectly referred to as "two phase". A similar method is used for a different reason on construction sites in the UK. A third wirecalled the bond or earth wire, is often connected between non-current-carrying metal enclosures and earth ground. This conductor provides protection from electric shock due to accidental contact of circuit conductors with the metal chassis of portable appliances and tools. Bonding all non-current-carrying metal parts into one complete system ensures there is always a low electrical impedance path to ground sufficient to carry any fault current for as long as it takes for the system to clear the fault.

This low impedance path allows the maximum amount of fault current, causing the overcurrent protection device breakers, fuses to trip or burn out as quickly as possible, bringing the electrical system to a safe state. AC power supply frequencies[ edit ] Further information: A low frequency eases the design of electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type traction motors for applications such as railways.

However, low frequency also causes noticeable flicker in arc lamps and incandescent light bulbs. The use of lower frequencies also provided the advantage of lower impedance losses, which are proportional to frequency. Effects at high frequencies[ edit ] Play media A Tesla coil producing high-frequency current that is harmless to humans, but lights a fluorescent lamp when brought near it A direct current flows uniformly throughout the cross-section of a uniform wire.

An alternating current of any frequency is forced away from the wire's center, toward its outer surface. This is because the acceleration of an electric charge in an alternating current produces waves of electromagnetic radiation that cancel the propagation of electricity toward the center of materials with high conductivity.