Network Theory ENERGY SOURCES An energy source is a device for generating electrical energy. Of course, such generation of electrical energy is possible by transformation from some other form of energy. e.g., in battery — chemical energy to electrical energy, generator — mechanical energy to electrical energy, solar cell — Solar energy to electrical energy. According to their terminal voltage-current characteristics, electrical energy sources are categorized into (i) ideal voltage sources and (ii) ideal current sources + Ideal Voltage Source An ideal voltage source is an energy source (two-terminal element) in which the terminal voltage (i) is completely independent of the current (i) through its terminals. (An ideal ac voltage source is one in which the amplitude or the fime variation of the generated voltage does not change at the terminals with the amplitude of the current drawn from it) v% * % oi, y-i characteristic ideal voltage source ‘Thus, the terminal voltage of the source remains constant for all values of terminal current, The slope of the plot is m = dv /di 0/dic= 0. This R is the called the internal resistance of source and itis zero, ic., R= 0 for an ideal voltage source, ‘Thus, the representation of energy source as an ideal voltage source is shown in Fig. (a) & (b). eel a || | Vs i @ o Ideal de voltgage source Ideal ac voltage source Whenever vs = 0, the voltage source is equivalent to a short circuit. Class Note by Santanu Das 1Network Theory If load Ru changes, it changes since ic=vs/R. [vs = constant] RL= (Ws/ Ri) x RL= vs Thus, if load Ri varies, ii changes. However, », does not alter, remains constant at. * Practical Voltage Source Almost all practical voltage sources fall short of the ideal nature. A practical voltage source is an energy source in which the voltage across the terminals falls as the current through it increases (Fig.a). Aaah YM i L___. (a) v-i characteristic (solid line), (b) Representation Accordingly a practical voltage source may be approximated as an ideal voltage source vs with a series resistance (Fig.b). The series resistance (R) accounts for the decrease in terminal voltage with the increase of terminal current, This resistance is called the internal resistance of the voltage source. The terminal voltage v depends on the terminal current by v= vs—itR ‘This equation represents a straight line with a negative slope m = dvv/ dit =~ R. This R is the called the internal resistance of source. If R_ increases, ic decreases, vx increases [R- constant]. m= i RL=[vs/(R+ Ri] x Re The less the value of R, the practical voltage source closely approaches ideal characteristics (dashed lines). R = 0 for an ideal voltage source. Class Note by Santanu DasNetwork Theory ‘© Ideal Current Source An ideal current source is an energy source in which the terminal current i is completely independent of the voltage 1 across its terminals. (An ideal ac current source generates current but neither the amplitude nor the time variation Of the terminal current changes with the terminal voltage) ‘Thus, the terminal current of this source remains constant for all values of terminal voltage. ie isl 8 vi characteristic ideal cutent source Ideal ac current source (a) (b) ‘The slope of the plot ism = di / dy, = 1/ R= 0/ dy, = 0. This R is the called the internal resistance of source and it is infinite, i.e., R = 0 for an ideal current source. ‘Thus, the representation of ideal current source is shown in the Fig.(b) When is = 0, the current source is equivalent to an open circuit. If Ri changes, vx changes since w= ii R= is RL [is = constant] WER RLS =i RL ‘Thus, if R. varies, 1 changes. However, ii does not alter, remains constant at is, © Practical Current Source Almost all practical current sources fall short of the ideal nature. In a practical current source, the terminal current falls with the increase of terminal voltage (Fig.a). ic f° @) as i v-i characteristic 7 practical current source Practical current source (a) (b) Ve Accordingly, a practical current source may be represented by an ideal current source (with a shunt resistance) in parallel with a resistance (Fig.b). The shunt resistor (R) accounts for the drop (loss / fall / reduction) in terminal current with the increase of terminal voltage. This resistance is called the internal resistance of the current source. Class Note by Santanu Das 3Network Theory The terminal current iis given by ii =is—W/R This equation represents a straight line with a negative slope (m’= dic/ dy =— I/R, or m = dv. / dic =—R). This R is the called the internal resistance of source. i) oT i) aS WA [ IF Ry increases, ic (= i, RAR + Ri) decreases, ig increases, w= i RL = i R increases. [R fixed] The higher the value of R, Practical current source closely approaches Ideal characteristics (dashed lines). R = c for an ideal current source. Class Note by Santanu Das 4Network Theory + Transformation of Energy Sources In both Practical voltage & current sources, the terminal voltage falls with the increase of the ferminal current (or, the terminal current falls with the increase in the terminal voltage). Thus, any practical energy source may be represented by either a voltage source or a current source. It is possible, therefore, fo transform (mathematically) a practical voltage source into «a practical current source and vice versa. This transformation is very useful to simy mixed sources. fy the analysis of circuits especially with A non-ideal voltage source and a non-ideal current source are said to be equivalent if their voltampere (v-i) relations are the same for all terminal conditions (for all loads) Fig.(a) and (b) show respectively a voltage source and a current source with associated internal impedances Z,, and Z; respectively. , bb i Ze : C Vs 4 @ is Zi Z, <— Source —s«-Loat-> = Source — ye Load» (a) Voltage source (©) current source ‘Transformation of energy sources If the two energy sources are equivalent, then the load current J, and /; must be equal (for all load conditions). », iZ, o $= Q Z,+Z, Z,+2Z, For equivalence of the two sources the following conditions should also be satisfied : (a) The open-circuit voltages at their terminals are equal, i.e. (b) The short-circuit currents at their terminals are equal, ic ‘Comparing Eq. (1) with (2) & (3) Zi =Z Therefore, the conditions for the equivalence of practical voltage and current sources are Zi =p is = vs /Zy vs =Ziis Class Note by Santanu Das 5Network Theory Thus, a voltage source vs in series with the impedance Z is transformed into a current source is in parallel with the impedance Z where is = vs /Z. This equivalence is shown in Fig. below. = iE 2 Transformation of a voltage source to a current source Zi =Zy vs = Zils The current flows from the equivalent current source to the terminals originally connected to the positive side of the voltage source Similarly, a current source is with a parallel impedance Z is transformed into a voltage source vs with a series a impedance Z where vs =Z is, This equivalence is shown in Fig. below. Gg ts jz G Zis Transformation of a current source to a voltage source The terminal of the equivalent voltage source will be positive to which the current source drives the current, Class Note by Santanu Das 6Network Theory + Dependent (Controlled) Sources The two types of ideal sources we have discussed are independent sources for which voltage and current are fixed and are not affected by other parts of the circuit In case of dependent sources, the source voltage or current is not fixed, but is dependent on the voltage or current existing at some other location in the circuit. The representation of dependent voltage and current sources is shown in Fig.(a) and (b). The symbol is generally a diamond shape ‘These types of sources mainly occur in the analysis of equivalent circuits. ° ue in @ o Dependent voltage source Dependent current source Source transformation can be used for independent as well as dependent sources. In case of source transformation for dependent (controlled) sources, care should be taken to keep the Control Variables intact. Class Note by Santanu Das 7