Voltage and Current Phase Relationships in an Inductive Circuit
Figure 2a: V-I Relationship of the Circuit R. Capacitors and Inductors are called Reactive Components, in which the Voltage and Current are "Out-of-Phase" with . The mnemonic "ELI the ICE man" can be helpful in keeping track of the phase between the voltage and current in an AC circuit. In a circuit with only an inductor . Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such.
This opposition to current change is called reactance, rather than resistance. Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such: The inductance L is in Henrys, and the instantaneous voltage eof course, is in volts. Figure below Pure inductive circuit: Inductor current lags inductor voltage by 90o.
- Series Resistor-Inductor Circuits
- Phase Shift
If we were to plot the current and voltage for this very simple circuit, it would look something like this: Figure below Pure inductive circuit, waveforms. Remember, the voltage dropped across an inductor is a reaction against the change in current through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current is at a peak zero change, or level slope, on the current sine waveand the instantaneous voltage is at a peak wherever the instantaneous current is at maximum change the points of steepest slope on the current wave, where it crosses the zero line.
This results in a voltage wave that is 90o out of phase with the current wave. If you are using a calculator that has the ability to perform complex arithmetic without the need for conversion between rectangular and polar forms, then this extra documentation is completely unnecessary.Phase difference in E and I in A C
Since this is a series circuit, we know that opposition to electron flow resistance or impedance adds to form the total opposition: Just as with DC, the total current in a series AC circuit is shared equally by all components.
This is still true because in a series circuit there is only a single path for electrons to flow, therefore the rate of their flow must uniform throughout. Consequently, we can transfer the figures for current into the columns for the resistor and inductor alike: And with that, our table is complete.
The exact same rules we applied in the analysis of DC circuits apply to AC circuits as well, with the caveat that all quantities must be represented and calculated in complex rather than scalar form. So long as phase shift is properly represented in our calculations, there is no fundamental difference in how we approach basic AC circuit analysis versus DC.
Now is a good time to review the relationship between these calculated figures and readings given by actual instrument measurements of voltage and current. The figures here that directly relate to real-life measurements are those in polar notation, not rectangular!
In other words, if you were to connect a voltmeter across the resistor in this circuit, it would indicate 7. To describe this in graphical terms, measurement instruments simply tell you how long the vector is for that particular quantity voltage or current.
Rectangular notation, while convenient for arithmetical addition and subtraction, is a more abstract form of notation than polar in relation to real-world measurements. As I stated before, I will indicate both polar and rectangular forms of each quantity in my AC circuit tables simply for convenience of mathematical calculation.
This effective resistance is known as the inductive reactance. This is given by: The unit of inductance is the henry. As with capacitive reactance, the voltage across the inductor is given by: Where does the energy go?
AC Inductor Circuits
One of the main differences between resistors, capacitors, and inductors in AC circuits is in what happens with the electrical energy. With resistors, power is simply dissipated as heat. In a capacitor, no energy is lost because the capacitor alternately stores charge and then gives it back again. In this case, energy is stored in the electric field between the capacitor plates. The amount of energy stored in a capacitor is given by: In other words, there is energy associated with an electric field.
In general, the energy density energy per unit volume in an electric field with no dielectric is: With a dielectric, the energy density is multiplied by the dielectric constant. There is also no energy lost in an inductor, because energy is alternately stored in the magnetic field and then given back to the circuit.
The energy stored in an inductor is: Again, there is energy associated with the magnetic field. The energy density in a magnetic field is: RLC Circuits Consider what happens when resistors, capacitors, and inductors are combined in one circuit. The overall resistance to the flow of current in an RLC circuit is known as the impedance, symbolized by Z.
The impedance is found by combining the resistance, the capacitive reactance, and the inductive reactance. Unlike a simple series circuit with resistors, however, where the resistances are directly added, in an RLC circuit the resistance and reactances are added as vectors.
This is because of the phase relationships.
Series Resistor-Inductor Circuits | Reactance and Impedance -- Inductive | Electronics Textbook
In a circuit with just a resistor, voltage and current are in phase. When all three components are combined into one circuit, there has to be some compromise.
To figure out the overall effective resistance, as well as to determine the phase between the voltage and current, the impedance is calculated like this. The impedance, Z, is the sum of these vectors, and is given by: The phase relationship between the current and voltage can be found from the vector diagram: