Problem + Hard 12-1    Source:    Problem 49 of the list of Electric Circuits I, from the School of Engineering at UFRGS, 2011, Prof. doctor Valner Brusamarello.

In the circuit of the Figure 12-01.1, the voltmeter V is known to indicate 12 volts. Calculate:

a) The voltage e.

b) The independent font value E.

Solution of the Problem + Hard 12-1

Item a

As the voltage measured by the voltmeter is 12 volts, then the current i is equal to:

Note two consequences: if the voltmeter measures 12 volts then it will not circulate current through the 6 ohms resistor that is in series with the voltage source of 12 volts. This implies that these components can be eliminated from the circuit without damage to the calculations; the current source equal to 3 i acquires the value of a 2.4 A current source. Let's redraw the circuit, simplifying what is possible. See the Figure 12-01.2 how it turned out.

It is known that e₂ = 12 volts, as it is the same value measured by the voltmeter. Let's remember that an ideal voltmeter has an internal resistance infinite, so no electric current flows through it. Remembering that currents that leave the node are considered positive and currents that arrive to the node are considered negative, you can write the equation for the node e. Then:

As we know that e₂ = 12 volts, then we can calculate the value of e, just substitute the value of e₂ in the equation above. Soon:

Item b

Knowing e, one can easily calculate the value of i₂, since:

And knowing the current i₂, we can calculate i₁, because from the circuit we have:

See the Figure 12-01.3 for the redrawn circuit with current values that circulate through it.

Knowing i₁, one can calculate e₁, or:

Now, we can calculate the current flowing through the voltage source of 105 volts and by the resistor of 16 ohms.

And finally, calculating the current flowing through the voltage source E and through the resistor 8 ohms, it is possible to calculate the value of E.

Since we know i₈, we can calculate E, because: