Single Phase Transformer Formulas

October 2, 2024 by Staticnotes4u

Parameter	Symbol / Formula	Where
Primary EMF Equation of Single Phase Transformer (E_p)	$E_p = 4.44 \cdot f \cdot N_p \cdot \Phi$	f= frequency N_p =Number of primary turns $\Phi$ = maximum flux
Secondary EMF Equation (E_s)	$E_s = 4.44 \cdot f \cdot N_s \cdot \Phi$	f= frequency N_s =Number of secondary turns $\Phi$ = maximum flux
Transformation ratio (K)	${\frac{N_s}{N_p} = \frac{V_s}{V_p}=\frac{I_p}{I_s}=K}$	N_s =Number of secondary turns N_p =Number of primary turns V_p =Primary voltage V_s =Secondary voltage
Turn Ratio	$\frac{N_p}{N_s} = \frac{V_p}{V_s}$	N_s =Number of secondary turns N_p =Number of primary turns V_p =Primary voltage V_s =Secondary voltage
No load Primary Current (I₀)	$I_0=\sqrt{I_m^2+I_w^2}$	I_m= Magnetising Component I_w=Active or Working or Iron loss component
Active or Working or Iron loss component (I_w)	$I_w=I_0\cos{\phi_0}$	${\phi_0}$ = Angle between V_p and I₀
Magnetising Component (I_m)	$I_m=I_0\sin{\phi_0}$	${\phi_0}$ = Angle between V_p and I₀
Impedance Ratio	$\frac{Z_1}{Z_2}=K^2$ $\frac{R_2}{R_1}=K^2$ $\frac{X_2}{X_1}=K^2$	Z₂= Secondary Impedance Z₁ = Primary Impedance
Shifting in a Transformer (Referred to Primary)	$R_{01} = R_1 + R_2' = R_1 + \frac{R_2}{K^2}$ $X_{01} = X_1 + X_2' = X_1 + \frac{X_2}{K^2}$ $Z_{01} = \sqrt{R_{01}^2 + X_{01}^2}$ $V_2'=\frac{V_2}{K}$ $I_2'={I_2}{K}$	R₀₁= Equivalent resistance referred to primary X₀₁= Equivalent reactance referred to primary Z₀₁=Equivalent impedance referred to primary R₂‘=Secondary resistance referred to primary X₂‘=Secondary reactance referred to primary V₂‘= Secondary voltage referred to primary I₂‘= Secondary current referred to primary
Shifting in a Transformer (Referred to Secondary)	$R_{02} = R_2+ R_1' = R_2 + {R_1}{K^2}$ $X_{02} = X_2 + X_1' = X_2 + {X_1}{K^2}$ $Z_{02} = \sqrt{R_{02}^2 + X_{02}^2}$ $V_1'={V_1}{K}$ $I_1'=\frac{I_1}{K}$	R₀₂= Equivalent resistance referred to secondary X₀₂= Equivalent reactance referred to secondary Z₀₂=Equivalent impedance referred to secondary R₁‘=Primary resistance referred to secondary X₁‘=Primary reactance referred to secondary V₁‘= Primary voltage referred to secondary I₁‘= Primary current referred to secondary
Percentage Voltage Regulation	1) $\%Reg_{down} = \frac{V_{nl} - V_{fl}}{V_{nl}} \times 100\%$ 2) $\%Reg_{up} = \frac{V_{nl} - V_{fl}}{V_{fl}} \times 100\%$ ***** (1) No. equation is generally used	V_nl= No load secondary voltage V_fl= Full load secondary voltage **** Voltage regulation-down (Reg_down): This happens when the secondary transformer terminal’s voltage output decreases due to a load attached to it. Voltage regulation-up (Reg_up):** This occurs when the secondary terminal of the transformer experiences an increase in voltage upon removal of the load.
Voltage Regulation for lagging and leading	For lagging p.f V_nl – V_fl = I₂R₀₂cosφ₂ + I₂X₀₂cosφ₂ For lagging p.f V_nl – V_fl= I₂R₀₂cosφ₂ -I₂X₀₂cosφ₂	R₀₂= Equivalent resistance referred to secondary X₀₂= Equivalent reactance referred to secondary
Power (Ideal Transformer)	$P_p = P_s$ or $V_p \cdot I_p = V_s \cdot I_s$	( P_p ) and ( P_s ) are the primary and secondary powers, ( V_p, V_s ) are the voltages, ( I_p, I_s ) are the currents.
Copper Loss (P_Cu)	$P_{Cu} = I_p^2 R_p + I_s^2 R_s$	I_p ,I_s are the currents, R_p, R_s are the resistances of primary and secondary windings
Core Loss or Iron loss	$P_i = P_{h} + P_{e}$	Pi = Total core loss P_h = Hysteresis loss P_e =eddy current loss
Total losses in Transformer	Total losses = P_i +P_Cu = Constant Loss + Variable Loss	P_i = Total core loss or Constant loss P_Cu= Copper loss or Variable loss
Hysteresis loss (in watts) (P_h)	$P_h = \eta B_{\text{max}}^{x} f V$	η = Steinmetz hysteresis coefficient (depends on the material) B_max = Maximum flux density in the core f= Frequency of the magnetic field (in hertz) v= Volume of the core (in cubic meters)
Eddy current loss (in watts) (Pe)	$P_e=k_e\ B_{\max}^2f^2t^2v$	k_e= Constant that depends on the material’s resistivity and geometry B_max = Maximum magnetic flux density (in teslas) f = Frequency of the magnetic field t = Thickness of the core laminations (in meters) v = Volume of the core (in cubic meters)
Efficiency (η)	$\eta = \frac{P_{out}}{P_{in}} \times 100\%$ $\eta=\frac{P_{out}}{P_{out}+\ Losses}\times 100\%$	P_out = Output power, P_in = Input power.

Leave a Comment Cancel reply