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1.5 
IdEal IndEPEndEnt two-tErmInal ElEctrIcal SourcES
Electrical sources are devices that are capable of applying a non-electrostatic force on a charge that 
moves through the source region. They can deliver energy to the charged particle or absorb energy 
from it.
1.5.1 
Ideal Independent voltage Source
A two-terminal voltage source will have non-electrostatic field at every point inside the source region. 
The charge distribution on the terminal surfaces of the source will create an electrostatic field at 
all points inside the source. The two fields cancel each other at all points at all instants under all 
conditions if the material inside the source is of infinite conductivity. The terminal voltage (which is an 
electrostatic potential difference) will always be equal to the internal electromotive force in that case.
The conducting material inside the source will have finite conductivity in practice. Charge carriers 
moving inside such material require net non-zero force to work against collisions with lattice atoms. 
This will call for a difference between the internal non-electrostatic field and the electrostatic field. 
Then, the terminal voltage will be different from the internal electromotive force. It will be less than 
the internal electromotive force if the source is delivering positive current out of its positive terminal 
and it will be more than internal electromotive force if it is absorbing positive current at its positive 
terminal. The difference between terminal voltage and internal electromotive force is termed as the 
voltage developed across the internal resistance of the source.
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1.28


CircuitVariablesandCircuitElements
A practical voltage source with time-varying internal electromotive force will require a time-
varying current flow component to support the time-varying surface charge distribution on its 
terminals. That is, a practical voltage source has a parasitic capacitance right across its terminals.
A practical voltage source will have induced electric 
field inside due to its own time-varying current as well as 
due to time-varying currents elsewhere in the circuit and in 
neighboring circuits. This will affect the voltage appearing 
at its terminals. That is, a practical voltage source has 
internal parasitic inductance too. Thus a detailed circuit 
model for a practical voltage source will be as shown in Fig. 
1.5-1. L
i
is a lumped parameter approximation for the 
inductive effect distributed within the source. R
i
is a lumped 
resistance parameter that approximates the distributed 
resistive effect within the source. C
i
is a lumped capacitance 
parameter that approximates the distributed capacitive effect within the source and at its terminals. E(t) 
is the internal electromotive force of the source. The 
+
and – signs do not signify the polarity of charges 
at the terminals. Rather, the 
+
sign indicates the point at which the potential difference is specified and 
– sign indicates the reference point for specifying the potential difference. Thus V(t) is the voltage of 
the terminal marked with 
+
with respect to the point marked with – sign at the time instant t in Volts.
An ideal voltage source is one in which all 
the three elements R
i
, L
i
and C
i
are assumed 
to be negligible. Thus the terminal voltage of 
an ideal voltage source is always equal to its 
internal electromotive force quite independent 
of magnitude or waveshape of current delivered 
or absorbed by it. Such an ideal voltage source 
is called an ideal independent voltage source 
if the electromotive force is a function of time 
only and does not depend on any other electrical 
or non-electrical variable. An ideal independent 
voltage source is specified by the following terminal equations.
v(t) 
=
E(t) , a specified function of time
i(t) 
=
Arbitrary, decided by the rest of the circuit in which this source is connected.
The symbol of a constant ideal independent voltage source (that is, a DC source) is shown in (a) 
of Fig. 1.5-2 and that of a time-varying ideal independent voltage source is shown in (b) of Fig. 1.5-2.