Normal operation mode of electrical network

Three-phase four-wire network with earthed neutral:

 

We first consider the equivalent circuit diagram and the calculated three-phase network, shown in Fig. 3. Here, for the convenience of output calculation used dependencies conduction:

Y_A = 1/ R_A +j ω C_A; Y_B=1/R_B+j ω C_B;Y _С = 1/ R _С +j ω C _С;

Y_N = 1/ R_N +j ω C_N; Y ₀= 1/ R ₀; Y_h 1/R_h= (R _h⁺ R _sh ⁺R _b)^-1,             (8.6)

 

where ω = 2πf – angular frequency of the AC; R ₀ – resistance of protective earthing.

This allows us to consider the equivalent circuit network with earthed neutral, if we take Y₀ = 1/R₀ (see Fig. 8.1), or a three-wire network with isolated neutral, if we put YN = Y0 = 0 (see Fig. 8.2).

Taking phase symmetrical voltage source, we get:

U_A=U _ph; U_B=a²U _ph; U _С =a U _ph                                       (8.7)

where U_ph – the amplitude of the phase voltages; a = e^ϳ120 = -0,5 + ϳ / 2 – vector operator, allowing to consider a phase shift of 120° of phase voltages, where | a | = 1.

Fig. 3. Schemes of a three-phase four-wire network:
a – equivalent circuit; b – design scheme

 

Applying Kirchhoff's laws, we obtain a general expression for the calculation of the current flowing through the human body, standing on the ground and touch the phase A three-phase network:

,                      (8.8)

In consideration four-wired earthed neutral network (see Fig. 1), the resistance R₀ ≤ 8 Ohm. ie, insulation resistance is much less than the conductors A, B, C, N relative to earth, or else:

Y ₀» |Y_A|, |Y_B |, | Y_C |,| Y_N | .                                             (8.9)

 

Neglecting in (8.8) are small in comparison with the values of Y₀, we get:

I_h ≈ U _ф Y_hY ₀/(Y ₀ + Y_h),                            (8.10)

 

or taking into account the relation (8.6):

                                   I_h U _ф/(R ₀ + R_h),                          (8.11)

Due to R ₀<< R_h:

I_h U _ф/ R_h,                                    (8.12)

 

or taking into account the relation (8.5):

                        I_h U _ph/(R _h⁺ R _sh +R _b) .                              (8.13)

 

Consequently, in a network with earthed neutral if human touching I_h phase is independent of the isolation resistance and capacitance of conductors A, B, C, N relative to the ground. The use of electric protective means (dielectric galoshes, insulating supports, dielectric carpets), as well as the presence of insulating the floor in the room allows you to provide the required level of security.

Under adverse circumstances (for example, in crude shoes and conductive floors) can be taken R_sh = R_b = 0. Then

                                  I_h U _ph/ R _h.                                    (8.14)

This represents a serious threat to human life.

 

Three-phase three-wire network with isolated neutral

In such networks can take Y_N = Y ₀= 0. Moreover, during normal network operation usually R_A = R_B = R_C = R, C_A = C_B = С _C = C and therefore, Y_A = Y_B = Y_C = Y. Accordingly:

                                 (8.15)

 

Turning on the conductivity of the resistance and the fact that Z = l /Y = (1/ R +j ω C)^-1 – complex conductor resistance to ground we obtain:

I_h U _ф/ (R _h + Z /3),                             (8.16)

and current amplitude:

                    (8.17)

 

Two special cases should be considered:

1. When С_А = С _B = С _C 0, which is the case in short overhead networks. With the help of formula (14) we find:

 

I_h U _ф/ (R _h + R /3),                             (8.18)

 

or based on the expression (8.5):

I_h U _ф/ (R _h+ R _sh +R _b).                                     (8.19)

 

Consequently, the current flowing through the human body depends on the phase voltage, insulation resistance to earth and resistance circuit in the human body. In damp conditions can take R _sh = R _b = 0,, and then crucial to the insulation resistance. If it meets the requirements of the «Rules for Electrical Installation», i.e. R ≥500 Ohms, then I_h cannot reach dangerous levels.

2. If R _А = R_B = R_C ∞ (it is permissible to take to cable networks) from the expression (8.17), dividing the numerator and denominator by the root to R², we obtain:

             (8.20)

In practice, the phase capacity is relatively small, so the second term of the expression under the root may not be much greater than unity. It follows that with increasing phase I_h the earth capacitance increases and may reach dangerous values.