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Ideal transformer equations

By Faraday's law of induction:

V P = − N P d Φ d t {\displaystyle V_{\text{P}}=-N_{\text{P}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}} Eq. 1[a][3]
V S = − N S d Φ d t {\displaystyle V_{\text{S}}=-N_{\text{S}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}} Eq. 2

where V {\displaystyle V} is the instantaneous voltage, N {\displaystyle N} is the number of turns in a winding, dΦ/dt is the derivative of the magnetic flux Φ through one turn of the winding over time (t), and subscripts P and S denote primary and secondary.

Dividing eq. 1 by eq. 2:

Turns ratio = V P V S = N P N S = a {\displaystyle ={\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}=a} Eq. 3

where for a step-up transformer a < 1 and for a step-down transformer a > 1.[4]

By the law of conservation of energy, apparent, real and reactive power are each conserved in the input and output:

S = I P V P = I S V S {\displaystyle S=I_{\text{P}}V_{\text{P}}=I_{\text{S}}V_{\text{S}}} Eq. 4

where S {\displaystyle S} is apparent power and I {\displaystyle I} is current.

Combining Eq. 3 and Eq. 4 with this endnote[b][5] gives the ideal transformer identity:

V P V S = I S I P = N P N S = L P L S = a {\displaystyle {\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {I_{\text{S}}}{I_{\text{P}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}={\sqrt {\frac {L_{\text{P}}}{L_{\text{S}}}}}=a} Eq. 5

where L P {\displaystyle L_{\text{P}}} is the primary winding self-inductance and L S {\displaystyle L_{\text{S}}} is the secondary winding self-inductance.

By Ohm's law and the ideal transformer identity:

Z L = V S I S {\displaystyle Z_{\text{L}}={\frac {V_{\text{S}}}{I_{\text{S}}}}} Eq. 6
Z L ′ = V P I P = a V S I S / a = a 2 V S I S = a 2 Z L {\displaystyle Z'_{\text{L}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {aV_{\text{S}}}{I_{\text{S}}/a}}=a^{2}{\frac {V_{\text{S}}}{I_{\text{S}}}}=a^{2}{Z_{\text{L}}}} Eq. 7
where Z L {\displaystyle Z_{\text{L}}} is the load impedance of the secondary circuit and Z L ′ {\displaystyle Z'_{\text{L}}} is the apparent load or driving point impedance of the primary circuit, the superscript ′ {\displaystyle '} denoting impedance referred to the primary.

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