Brief theoretical information

In AC circuits, besides resistors, are often included inductors and capacitors, which exhibit a number of properties different from the properties of the resistors. Quantitative assessment of these properties Express the parameters of elements of AC circuits, which include, in addition to the resistance of resistor R, inductance L, capacitance C and the inductive reactance X_L = ω ∙ L and capacitive resistance X_C = 1 / (ω ∙ C), where ω is the angular frequency (ω = 2 ∙ π ∙ f, f is the AC frequency).

A characteristic property of resistors is the irreversible transformation of electric energy into heat that occurs when a leakage current on them.

In AC circuits the resistance of the resistor is called active. This resistance at the same time for the same resistor due to the influence of the surface effect has a higher value than the electrical resistance in DC circuits. However, at relatively low frequencies this difference is usually neglected.

The voltage on the resistor coincides in phase with the current. If the current through the resistor changes in accordance with the sinusoidal law I=I_m∙sin (ω ∙ T), the voltage will be similarly changed to U = U_m ∙ sin (ω ∙ t).

Inductance L, the unit of measurement of which is Henry (H), characterizes the property of the coil when flowing to accumulate energy in a magnetic field. The real circuit has an inductance, which has both conductors, and resistors, and capacitors. However, their inductance is often small, so the calculations in most cases take into account only the inductance of the coils.

When alternating current flows through an inductor having w turns, an alternating magnetic flux by the law of electromagnetic induction induces EMF e_L = − w ∙ dF / dt = − L ∙ di / dt. This EMF affects the amount of current flowing through the coil, which in practice is taken into account, introducing the concept of inductive reaction resistance X_L = ω ∙ L, the unit of measurement of Ohms and the value of which is proportional to the frequency. X_L in circuit analysis using a complex method is also called the induced resistance module.

The maximum rate of change of current of the coil corresponds to the moment of transition of the sine wave of the current through zero. With sinusoidal EMF that corresponds to its maximum value. Given the negative sign in the formula electromagnetic induction law, this fact speaks of the backwardness of the process of change of the EMF from the process of current change in the coil is perfect on the 90⁰. The process of change of voltage across the inductor, on the contrary, should be ahead of the process of changing the coil current, since the instantaneous value of the EMF of self-inductance and voltage have opposite signs.

If the AC circuit of the real coil having, in addition to inductive, and even active resistance, the current lags on the voltage by the angle ⱷ < π/2.

To simplify the analysis of electric circuits apply the complex method.

The complex inductive reactance

X = ϳ ∙ ω ∙ L = ϳ ∙ X_L = X_L ∙ e ^{ϳ ∙ π / 2},

a complex impedance the real coil

Z = R + ϳ ∙ X_L = Z ∙ e ^ (ϳ ∙ arctg (X_L/R)),

where Z is the module of full resistance of a real coil.

The capacitance of the capacitor, the unit of measurement of which is the Farad (F), characterizes its ability to store electric charges. In practice, the capacity exists between the individual conductors between the turns of coils between conductors and earth, etc., but due to the small values in the analysis believe that the capacity to have only capacitors.

In the dielectric separating the plates of a capacitor under the influence of an applied thereto voltage current flows and electric displacement i = dq / dt where q is the charge on the capacitor plates, the unit of measurement of which is Coulomb (C). This charge is proportional to the voltage on the capacitor q = C ∙ u_C. Therefore

i = C ∙ du_C / dt.

It is obvious that the current will have its maximum at the maximum rate of change of the voltage which is reached at the time of the transition curve varying sinusoidal voltage through zero. From this it follows that the process of change of the current flowing through the capacitor is ahead of the process of change of voltage at angle 90⁰. Moreover, the voltage that occurs when the accumulation of charge on the plates of the capacitor, is a factor that reduces the current through the capacitor, which often take into account using of capacitive resistance X_C=1/(ω∙C). During a comprehensive analysis of the X_C module is also called a capacitive resistance. The unit of capacitance measurement is also Ohm.

The complex capacitance resistance

X_C = 1 / (ϳ ∙ ω ∙ C) = ϳ / ( ω ∙ C) = ϳ ∙ X_C = X_C ∙ e ^{- ϳ ∙ π / 2}.

Full impedance of a dipole

Z = R + ϳ ∙ ( X_L X_C) = Z ∙ e^{ϳ ∙ arctg(X / R)},

where Z is the module of the impedance circuit sinusoidal current;

X = X_L X_C module reactance of the circuit.

Ohm's law for subcircuit in a complex form

I = U / Z ,

where U , I is the complex voltage and current.

The instantaneous power of a dipole

p = u ∙ i = U_m ∙ sin(ω ∙ t) ∙ I_m ∙ sin(ω ∙ t + ⱷ) =

(U_m ∙ I_m / 2)[cos ⱷ cos(2 ∙ ω ∙ t + ⱷ)].

The active power of a dipole

P = (1 / T) ∙ dt = U ∙ I ∙ cos ⱷ.

The reactive power of a dipole

Q = U ∙ I ∙ sin ⱷ.

Full power of a dipole

S = U_m ∙ I_m / 2 = U ∙ I.

Integrated capacity

S = U ∙ I^*∙ = P jQ = S ∙ e^{-ϳ ∙ arctg(Q / /P)} =√(P^2+Q^2 ) ∙ e^{-ϳ ∙ arctg(Q / /P)} ,

where I^* conjugate current complex.

Date: 2018-08-27; view: 28421