# Transformer **An electrical energy transfer device consisting of [[Inductance#Mutual inductance|magnetically coupled coils]].** The coil directly connected to the *input* voltage source is the *primary winding*, $N_{\text{P}}$, and the coil connected to the *load* is the *secondary winding*, $N_\text{S}$. > [!Efficiency] > The *efficiency* of a transformer is the percentage of input power delivered to the output. > $\eta=\frac{P_{\text{out}}}{P_{\text{in}}}\times 100$ ## Ideal transformer In an ideal transformer, - primary and secondary windings have *no resistance*. - the coils are [[Energy in a Coupled Circuit#Coupling coefficient|perfectly coupled]]; $k=1$. - the permeability of the core is *infinitely large*. - core losses are negligible. As an ideal transformer is *lossless*, the primary power is *equal* to the secondary power. Thus, [[Transfer Function|transfer functions]] for voltage and current are related to the ratio of the number of turns of the windings, $n$. $n=\frac{N_\text{S}}{N_\text{P}}=\frac{v_\text{S}}{v_\text{P}}=\frac{i_\text{P}}{i_\text{S}}$ ![[IdealTransformer.svg|400]] ## Step-up, step-down, and isolation transformers In a *step-up* transformer, the secondary voltage is *larger* than the primary voltage, and $n=\frac{N_\text{S}}{N_\text{P}}>1$ In a *step-down* transformer, the secondary voltage is *smaller* than the primary voltage, and $n=\frac{N_\text{S}}{N_\text{P}}<1$ In an *isolation* transformer, the two voltages are *equal* and the ratio of the number of turns is equal to one. ## Reflected load A transformer can be eliminated from the analysis of the circuit by *reflecting* the circuits to a side. This does not work if there are *external connections* between the the primary and secondary circuits. The input [[impedance]] as seen through a transformer towards the load from the primary side is the *reflected impedance*. $\mathbf{Z_\text{in}}=\frac{\mathbf{Z}_{L}}{n^2}$ ![[ReflectedImpedance.svg|400]] To reflect the secondary circuit to the primary side, - Secondary impedances are *divided* by $n^{2}$ - Secondary voltages are *divided* by $n$ - Secondary currents are *multiplied* by $n$ To reflect the primary circuit to the secondary side, - Primary impedances are *multiplied* by $n^{2}$ - Primary voltages are *multiplied* by $n$ - Primary currents are *divided* by $n$ ## Winding direction The direction of the windings in a transformer determines the polarity of the secondary voltage with respect to the primary voltage. ![[WindingDirection.svg]] ## Dot convention [[Dot convention]] is especially important in transformers. If *both* $\mathbf{I_{1}}$ and $\mathbf{I}_{2}$ enter or leave the dotted terminals, then a *minus* is required in front of the turns ratio. Otherwise, it is positive. If $\mathbf{V}_1$ and $\mathbf{V}_{2}$ are *both* positive or negative at the dotted terminals, then a turns ratio is left *positive*. Otherwise, a *minus* is placed in front. ![[TransformerDotConvention1.svg]] *** ![[TransformerDotConvention2.svg]]