7.9 Summary

• An alternating voltage \(v=v_m \sin \omega t\) applied to a resistor \(R\) drives a current \(i=i_m \sin \omega t\) in the resistor, \(i_m=\frac{v_m}{R}\). The current is in phase with the applled voltage.
• For an alternating current \(i=i_m \sin \omega t\) passing through a resistor \(R\), the average power loss \(P\) (averaged over a cycle) due to joule heating is \((1 / 2) i_m^2 R\). To express it in the same form as the dc power \(\left(P=I^2 R\right)\), a special value of current is used. It is called root mean square (rms) current and is donoted by \(I\) :
  \(
  I=\frac{i_m}{\sqrt{2}}=0.707 i_m
  \)
  Similarly, the rms voltage is defined by
  \(
  V=\frac{v_m}{\sqrt{2}}=0.707 v_m
  \)
  We have \(P=I V=I^2 R\)
• An ac voltage \(v=v_m \sin \omega t\) applied to a pure inductor \(L\), drives a current in the inductor \(i=i_m \sin (\omega t-\pi / 2)\), where \(i_m=v_m / X_L . X_L=\omega L\) is called inductive reactance. The current in the inductor lags the voltage by \(\pi / 2\). The average power supplied to an inductor over one complete cycle is zero.
• An ac voltage \(v=v_m \sin \omega t\) applied to a capacitor drives a current in the capacitor: \(i=i_m \sin (\omega t+\pi / 2)\). Here,
  \(i_m=\frac{v_m}{X_C}, X_C=\frac{1}{\omega C}\) is called capacitive reactance.
  The current through the capacitor is \(\pi / 2\) ahead of the applied voltage. As in the case of inductor, the average power supplied to a capacitor over one complete cycle is zero.
• For a series \(R L C\) circuit driven by voltage \(v=v_m \sin \omega t\), the current is given by \(i=i_m \sin (\omega t+\phi)\)
  where \(\quad i_m=\frac{v_m}{\sqrt{R^2+\left(X_C-X_L\right)^2}}\)
  and \(\phi=\tan ^{-1} \frac{X_C-X_L}{R}\)
  \(Z=\sqrt{R^2+\left(X_C-X_L\right)^2}\) is called the impedance of the ctrcuit.
  The average power loss over a complete cycle is given by
  \(
  P=V I \cos \phi
  \)
  The term \(\cos \phi\) is called the power factor.
• In a purely inductive or capacitive ctrcuit, \(\cos \phi=0\) and no power is dissipated even though a current is flowing in the circuit. In such cases, current is referred to as a wattless current.
• The phase relationship between current and voltage in an ac circuit can be shown conventently by representing voltage and current by rotating vectors called phasors. A phasor is a vector which rotates about the origin with angular speed \(\omega\). The magnitude of a phasor represents the amplitude or peak value of the quantity (voltage or current) represented by the phasor.
  The analysis of an ac circuit is facilitated by the use of a phasor diagram.
• A transformer consists of an fron core on which are bound a primary coil of \(N_p\) turns and a secondary coll of \(N_s\) turns. If the primary coil is connected to an ac source, the primary and secondary voltages are related by
  \(
  V_{\mathrm{s}}=\left(\frac{N_{\mathrm{s}}}{N_p}\right) V_p
  \)
  and the currents are related by
  \(
  I_{\mathrm{s}}=\left(\frac{N_p}{N_{\mathrm{s}}}\right) I_p
  \)
  If the secondary coll has a greater number of turns than the primary, the voltage is stepped-up ( \(V_s>V_p\) ). This type of arrangement is called a stepup transformer. If the secondary coll has turns less than the primary, we have a step-down transformer.

POINTS TO PONDER

• When a value is given for ac voltage or current, it is ordinarily the rms value. The voltage across the terminals of an outlet in your room is normally 240 V . This refers to the \(r m s\) value of the voltage. The amplitude of this voltage is
  \(
  v_m=\sqrt{2} V=\sqrt{2}(240)=340 \mathrm{~V}
  \)
• The power rating of an element used in ac circuits refers to its average power rating.
• The power consumed in an ac circuit is never negative.
• Both alternating current and direct current are measured in amperes. But how is the ampere defined for an alternating current? It cannot be derived from the mutual attraction of two parallel wires carrying ac currents, as the dc ampere is derived. An ac current changes direction with the source frequency and the attractive force would average to zero. Thus, the ac ampere must be defined in terms of some property that is independent of the direction of the current. Joule heating is such a property, and there is one ampere of rms value of alternating current in a circuit if the current produces the same average heating effect as one ampere of dc current would produce under the same conditions.
• In an ac circuit, while adding voltages across different elements, one should take care of their phases properly. For example, if \(V_R\) and \(V_C\) are voltages across \(R\) and \(C\), respectively in an \(R C\) circuit, then the total voltage across \(R C\) combination is \(V_{R C}=\sqrt{V_R^2+V_C^2}\) and not \(V_R+V_C\) since \(V_C\) is \(\pi / 2\) out of phase of \(V_R\)
• Though in a phasor diagram, voltage and current are represented by vectors, these quantities are not really vectors themselves. They are scalar quantities. It so happens that the amplitudes and phases of harmonically varying scalars combine mathematically in the same way as do the projections of rotating vectors of corresponding magnitudes and directions. The ‘rotating vectors’ that represent harmonically varying scalar quantities are introduced only to provide us with a simple way of adding these quantities using a rule that we already know as the law of vector addition.
• There are no power losses associated with pure capacitances and pure inductances in an ac circuit. The only element that dissipates energy in an ac circuit is the resistive element.
• In a \(R L C\) circuit, resonance phenomenon occur when \(X_L=X_C\) or \(\omega_0=\frac{1}{\sqrt{L C}}\). For resonance to occur, the presence of both \(L\) and \(C\) elements in the circuit is a must. With only one of these ( \(L\) or \(C\) ) elements, there is no possibility of voltage cancellation and hence, no resonance is possible.
• The power factor in a RLC circuit is a measure of how close the circuit is to expending the maximum power.
• In generators and motors, the roles of input and output are reversed. In a motor, electric energy is the input and mechanical energy is the output. In a generator, mechanical energy is the input and electric energy is the output. Both devices simply transform energy from one form to another.
• A transformer (step-up) changes a low-voltage into a high-voltage. This does not violate the law of conservation of energy. The current is reduced by the same proportion.