Figure1: Losses in a Transformer.

As shown in Fig. 1, the total loss in a transformer can be divided into two types namely the copper loss and the iron loss. The iron loss is further classified into two types namely the hysteresis loss and eddy current loss.

Copper Loss (P_cu) in a Transformer

The total power loss taking place in the windings resistance of a transformers is known as the copper loss.

Copper loss = Primary copper loss + secondary copper loss

The copper loss is denoted by P_cu.

\[{{P}_{cu}}=I_{1}^{2}{{R}_{1}}+I_{2}^{2}{{R}_{2}}\]

Where R₁ and R₂ are resistances of primary and secondary respectively.

Where,

\[I_{1}^{2}{{R}_{1}}\text{ = Primary copper loss}\]

and

\[I_{2}^{2}{{R}_{2}}\text{ = Secondary copper loss}\]

The copper loss should be kept as low as possible to increase the efficiency of the transformer. To reduce the copper loss, it is essential to reduce the resistances R₁ and R₂ of the primary and secondary windings. Copper losses are also called as variable losses as they are dependent on the square of load current. The relation between copper loss at full load and that at half load is as follows:

\[{{P}_{cu(HL)}}={{\left( \frac{1}{2} \right)}^{2}}{{P}_{cu(FL)}}=\frac{{{P}_{cu(FL)}}}{4}\]

Where,

\[{{\text{P}}_{cu(FL)}}=\text{ Copper loss at full load and}\]

\[{{\text{P}}_{cu(HL)}}\text{ = Copper loss at half load}\]

Iron Loss (P_i) in a Transformer

Iron loss P_i is the power loss taking place in the iron core of the transformer. It is equal to the sum of two components called hysteresis loss and eddy current loss.

P_i = Hysteresis Loss + Eddy current loss

Hysteresis Losses in a Transformer

The hysteresis loss taking place in a magnetic material. The area enclosed by the hysteresis loop of a material represents the hysteresis loss. Hence special magnetic materials should be used in order to reduce the hysteresis loss. Materials such as silicon steel has hysteresis loops with very small area. Hence such materials are preferred for the construction of core. Commercially such steel is called as Lohys, means low hysteresis materials. Mathematically the hysteresis loss is given by:

\[\text{Hysteresis loss = }{{K}_{H}}.B_{m}^{1.67}fV\text{ watt}\]

Where:

K_H – Hysteresis constant,

B_m – Maximum flux density

f – Frequency and

V – Volume of the core

Thus the hysteresis loss is frequency dependent As we increase the frequency of operation, the hysteresis loss increase proportionally.

Eddy current losses in a Transformer

Due to the time varying flux, there is some induced emf in the transformer core. This induced emf causes some currents to flow though the core body. These currents are known as the eddy currents. The core is made of steel and has some finite resistance. Hence due to the flow of eddy currents, heat will be produced. The power loss due to the eddy currents is given by :

Eddy current loss = (Eddy current)² × r

Where,

r – Resistance of the core.

The eddy current losses are minimized by using the laminated core. The core is manufactured as a stack of laminations rather than a solid iron core. These laminations are insulated from each other by means of a varnish coating on all the laminations. Hence each lamination acts as a separate core with a small cross-sectional area, providing a large resistance to the flow of eddy currents. Mathematically the eddy current loss is given by,

\[\text{Eddy current loss = }{{K}_{E}}B_{m}^{2}{{f}^{2}}{{T}^{2}}\]

K_E – Eddy current constant,

T – core

Thus the eddy current loss also is frequency dependent. It is directly proportional to the square of operating frequency. It can be reduced by using the laminated core for transformer.

Hence the iron loss P_i of the total loss is dependent on the frequency but the copper loss P_cu is constant inspective of frequency. The iron loss is denoted by P_i. It is the sum of hysteresis and eddy current loss. Iron loss is a constant loss and does not depend on the level of load.

Total Loss in a Transformer

The total power loss taking place in a transformer can be obtained by adding the copper loss and iron losses together.

Total loss = Copper loss + Iron loss

The post Types of Losses in a Transformer – Copper Loss, Iron Loss, Hysteresis Losses & Eddy current losses appeared first on Study Book Page.

Circuit Diagram & Derivation of Series Resonance

Let, V = V_m sin ωt be the voltage applied across it and i(t) be the current flowing through it. In case of series RLC circuit, the net impedance is given by,

\[Z=R+j\left( {{X}_{L}}-{{X}_{C}} \right)\]

Where,

X_L = Inductive reactance

X_C = Capacitive reactance.

From equation (1) it is clear that there exist three effects in the circuit i.e., the resistive, inductive and capacitive effect.

However, the effect of resistance is inherent hence a series RLC circuit either behaves as inductive or capacitive depending on the relative value of X_L and X_C If X_L > X_C the circuit behaves as inductive circuit and if X_C > X_L the circuit behaves as capacitive circuit. However, if these two values are equal i.e., X_L = X_C the net reactance will be zero and the circuit is said to be under resonance.

Expression for Resonant Frequency of Series Resonance

The frequency at which resonance occurs is called as resonant frequency. The condition for resonance is,

X_L = X_C

But, we know that,

\[{{X}_{L}}=2\pi {{f}_{r}}L\]

\[{{X}_{C}}=\frac{1}{2\pi {{f}_{r}}C}\]

Where, f_r is the resonant frequency in Hz.

Substituting the above values in equation (2), we get,

\[2\pi {{f}_{r}}L=\frac{1}{2\pi {{f}_{r}}L}\]

\[f_{r}^{2}=\frac{1}{4{{\pi }^{2}}LC}\]

\[{{f}^{2}}=\frac{1}{4\pi \sqrt{LC}}\]

\[2\pi f=\frac{1}{\sqrt{LC}}\]

\[{{\omega }_{r}}=\frac{1}{\sqrt{LC}}\]

Where, ω_r is the resonant frequency in rad/sec

Resonance Curves for a series resonant circuit with variable frequency and constant R, L and C

Figure (1) shows the circuit diagram of series RLC circuit, which has a complex impedance, Z = R +jX.

Where,

\[X=j\left( \omega L-\frac{1}{\omega C} \right)\]

Figure (2) shows the variation of resistance, inductive reactance (X_L) and capacitive reactance (X_C) with frequency.

For Resistance

R is independent of frequency; hence it is represented by a straight horizontal line.

Moreover, at resonance X_C = X_L i.e., |Z| = R.

Hence, at resonance |Z| = R and thus at resonance the impedance (Z) is minimum.

For Inductance

\[{{X}_{L}}=\omega L=2\pi fL\]

It is a function of frequency i.e., inductive reactance is directly proportional to frequency (f), hence, its graph is a straight line through the origin as shown in figure (2).

For Capacitance

\[{{X}_{C}}=\frac{1}{\omega C}=\frac{1}{2\pi fC}\]

Capacitive reactance is inversely proportional to frequency (f). Its graph is a rectangular hyperbola as drawn in fourth quadrant, since, X_c is regarded as negative.

Total impedance,

\[Z=\sqrt{{{R}^{2}}+{{\left( {{X}_{L}}-{{X}_{c}} \right)}^{2}}}=\sqrt{{{R}^{2}}+{{X}^{2}}}\]

At low frequencies Z is large, the net impedance is capacitive and the power factor is leading (X_C > X_L) At high frequencies Z is again large but it is inductive (X_L > X_C) and the power factor is lagging.

Current (I)

It has a low value of current on both sides of resonant frequency as Z is large but has the maximum value of current, at resonance as Z is small, which is shown in figure (3). Hence, at resonance maximum power is dissipated by the circuit. The shape of resonance curve for various values of ‘R’ is also shown in figure (3). It is observed that, if R is large then current is less and vice-versa.

Phase Angle

At f = 0, X_L = 0 by, 0 and X_C = ∞ then the phase angle is given by.

\[\theta ={{\tan }^{-1}}\left( \frac{{{X}_{L}}-{{X}_{C}}}{R} \right)\]

\[={{\tan }^{-1}}\left( \frac{0-\infty }{R} \right)={{\tan }^{-1}}\left( -\infty  \right)=-90{}^\circ \]

As the frequency is increased from zero, X_L increases, X_C decreases and hence phase angle moves from -90º and goes towards zero.

At f = f_r, X_L = X_C and the phase angle is given by,

\[\theta ={{\tan }^{-1}}\left( \frac{0}{R} \right)=0{}^\circ \]

As the frequency is further increased beyond f_r, the phase angle increases from 0º towards +90º.

At f = ∞, X_L = ∞, X_C = 0, the phase angle is now given as,

\[\theta ={{\tan }^{-1}}\left( \frac{\infty -0}{R} \right)=90{}^\circ \]

Hence the above points can be summarized as,

1. At f = 0, X_C = ∞, θ = -90º thus the current leads the voltage by 90º.
2. At f < f_r, -90º < θ < 0, the current leads the voltage by an angle θ.
3. At f = f_r, X_L = X_C, θ = 0º and the current is in phase with voltage.
4. At f > f_r , 0º < θ < 90º and the current lags the voltage by an angle θ.
5. At f = ∞, X_L = ∞, θ = 90º and the current lags the voltage by 90º.

Figure (4) shows the variation of θ for different values of frequency.

The post What is Series Resonance? Circuit Diagram, Derivation, Formula & Resonant Frequency appeared first on Study Book Page.