Supply networks – Ideal and practical transformers
Part 1 – High Voltage Wires
Explain, preferably from an effect P argument, why we utilize high voltage wires to conduct current in the power grid – Meaning why high voltage as opposed to low voltage
Part 2 – Ideal Transformers
In Chapter 31-6 we’ve already seen an ideal transformer. Here we’re looking closer at the book’s derivation, see Figure 1 and compare it to Figure 31-18 in the book. We start by looking at equation 31-80 (transformation of currents) in the transformer from a slightly different angle
Figure 1 Ideal Transformer
Use Ampere-Maxwell law, under the assumption the B ⃗-field is tangential to the curve l_c, to show Bl_c=μ_c (N_1 I_1-N_2 I_2). Is the assumption reasonable?
Then show (N_1 I_1-N_2 I_2 )=Φ_c R_c where the reluctance R_c of the core is given by R_c=I_c/(μ_c A_c )
The above formula is a version of “Ohms” Law for magnetic circuits: The magnetomotive force
F_c=NI is given by F_c=Φ_c R_c (compare to the normal Ohms Law V=ri)
In an ideal transformer N_1 I_1-N_2 I_2=0 applies. What is the assumption about μ_c in an ideal transformer?
(See equation 31-80 in the book with N_1=N_p and N_2=N_s)
Now we look at the connection between current and voltage in the two coils (Equations 31-79 and 31-80): Assume the magnetic flux is ϕ(t)=Φsin(ωt)
Show Faraday’s Law gives v(t)=-N (dϕ(t))/dt and graph the phasors V, Φ in a phasor diagram
Show the phasors fulfill V=-iNωΦ, where i is the imaginary unit, and for an ideal transformer the following must apply: The two relations I_1=I_2/a_t and V_1=a_t V_2 where a_t=N_1/N_2 is turns ratio.
Part 3 – Practical Transformers
In a practical transformer (with two coils) there are 4 effects we’ve previously disregarded
Figure 2 Practical Transformer
There is resistance R_1 and R_2 in the windings
The magnetic flux “leaks” out of the core, described by a reactance X_1 and X_2
Permeability in the core is finite
There is effect loss in the core
The resistance in the windings is represented by R_1 and R_2 in Figure 2. The leakage voltage is represented by X_1 and X_2 in Figure 2. (Permeability is represented by the susceptance B_m which we disregard, likewise we disregard the conductance G_c which describes the effect loss in the core).
Show we can describe the system with an equivalent impedance Z_eq^2=R_eq^2+X_eq^2 where R_eq=R_1+(N_1^2)/(N_2^2 ) R_2 and X_eq=X_1+(N_1^2)/(N_2^2 ) X_2 as shown in Figure 3
Figure 3 Practical transformer described by an equivalent impedance