AC Inductive Circuits

AC Applied to an Ideal Inductor:

Ideal Inductor to AC voltage - phase shift between current and voltage

As you can see, the current through the coil is 90° out of phase - and delayed. Why?

The periodically alternating current flow creates a magnetic field in the coil that also alternates periodically. This alternating magnetic field in turn results in an induction voltage, which (according to Lenz's rule) is opposite to its cause - namely the applied supply voltage.

With sinusoidal current, the slope or rate of change is maximum at the zero crossing. The self-induction voltage of the coil therefore also must have its maximum at these points.

Conversely, at maximum or minimum of the current curve, the rate of change of the current has the value 0. The value of the self-induction voltage is therefore also 0.

Real World Inductor

Experiment: A real coil is first connected to DC voltage, then to AC voltage. What you see is how at AC voltage the maximum current of 1 A can only be reached with a higher voltage. So an additional resistance has to appear at AC voltage.

You can also see that the current decreases significantly at AC voltage supply as soon as you insert an iron core. At DC voltage supply this has no influence.

Every real world inductor has some unwanted wire-resistance. So the total resístance, called Impedance is a combination of the wire-resistance and the reactance:

Real-world Inductor and its Impedance

What is the difference between a real-world inductor and an ideal inductor?

The ideal inductor has only an inductive reactance (X_L = 2 π f L). With the real-world inductor, the wire resístance is taken into account.

In the equivalent circuit diagram of a real coil ohmic resistance and reactance are in series. Nevertheless, the total resístance or impedance is determined by means of Pythagoras. The Pythagoras can be applied to right-angled triangles. This resístance triangle, like the power triangle, can be derived from the voltage triangle:

Real coil - Voltage triangle

If the current I is the common value of a series connection, each side of the voltage triangle can be multiplied or divided by the current I. This results in 2 equiangular triangles:

Real coil amd its Resistance and Power triangle

Exercises - Real World Inductor

How can you measure the impedance of a coil?

 The coil is connected to an AC voltage of known frequency, the current and voltage are measured and the impedance is calculated according to Ohm’s law.

 In order to determine the impedance, the wire resistance, i.e. the ohmic resistance of the coil, must first be determined.

  The coil is connected to an AC voltage of known frequency. You measure the current and the active power with a power meter and calculate the impedance from the measured values.

 You connect the coil to DC voltage, measure current and voltage and calculate the impedance according to Ohm’s law.

A relay labeled V~ = 12 V is accidentally connected to 12 V DC. What will happen?

 The relay will hum, but otherwise perform its function normally.

 The relay will not function because the current will be too small.

 The relay coil will heat up too much due to the excessive current and the relay will be damaged.

 The relay will operate normally because the RMS value is the same for 12V AC as it is for 12V AC.

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Exercise:

A real coil has an ohmic or wire resistance and a reactance. To determine these two resístances, the coil (see measuring circuit) is connected to a DC voltage of 30 V and in the following to AC voltage of 50 V / 50 Hz. The current is 2.5 A each.

Real world Inductor - how to determine the ohmic resistance and the Impedance

Determine
- The ohmic resístance R and the inductive reactance X_L at 50 Hz
- The inductance L
- The phase shift angle at AC voltage.