Sources

Sources are Nodes that provide a voltage or Current for the circuit

An ideal voltage source maintains a constant voltage no mater the change in circuit parameters

In a circuit, the voltage $𝑉 𝑇$ of the circuit is always equal to the voltage of the source $𝑉 𝑆$

𝑉 𝑆 = 𝑉 𝑇

𝑑 𝑉 𝑆 𝑑 𝑡 = 0

In reality a voltage source can not generate power nor provide infinite energy

𝑉 𝑅 𝑖 > 0

𝑉 𝑇 = 𝑉 𝑆 - 𝑉 𝑅 𝑖 \cdot 𝐼 𝑅

For a short circuit, the circuit voltage is zero

𝑉 𝑇 = 0; 𝐼 𝑆 𝐶 = 𝑉 𝑆 𝑉 𝑅 𝑖

For an open circuit, the current is zero

𝐼 𝑅 = 0; 𝑉 𝑆 = 𝑉 𝑇

The power supplied to the load by a real voltage source is as follows

𝑃 𝑅 = 𝑉 𝑅 \cdot 𝐼 𝑅 = 𝑉 2 𝑅 𝑅 = 𝐼 2 𝑅 \cdot 𝑅

𝑉 𝑅 = 𝑉 𝑆 \cdot 𝑅 𝑅 + 𝑅 𝑖

𝐼 𝑅 = 𝐼 𝑆 \cdot 𝑅 𝑖 𝑅 + 𝑅 𝑖

Maximum Power

The maximum power can be transmitted if the load resistance $𝑅$ is equal to the internal resistance $𝑅 𝑖$ of the source

An ideal current source maintain a constant current throughout the circuit

𝐼 𝑅 = 𝐼 𝑆

𝑑 𝐼 𝑆 𝑑 𝑡 = 0

A real current source can not generate power nor provide infinite energy

𝐼 𝑇 = 𝐼 𝑆 - 𝐼 𝑅 𝑖 = 𝐼 𝑆 - 𝑉 𝑇 𝑅 𝑖 𝐼; 𝑅 𝑖 𝐿 > 0

For a short circuit, the voltage is zero

𝑉 𝑇 = 0; 𝐼 𝐾 = 𝐼 𝑆

For an open circuit, the current is zero

𝐼 𝑅 = 0; 𝑉 𝑇 = 𝐼 𝑆 \cdot 𝑅 𝑖