circuit design

3. For the circuits shown below, identify all nodes and branches. Are any nodes at the same potential? Which resistors, if any, are in series? Which resistors, if any, are in parallel? (a) Nodes are where three or more wires come together. Thus points b, c, and e are nodes. Branches are the paths between nodes. Therefore the branches are bafe, be,bc,ce, and cde. Any two nodes connected by just a straight wire (no resistors or batteries in between) are at the same potential. Hence nodes b and c are at the same potential. Resistors in the branch carry the same current and are said to be in series. Here the 6­Ω and 22­Ω resistors are in series even though there is a battery between them. Resistors which share the same two nodes are said to be in parallel. The 20­Ω and 12­Ω resistors are in parallel because they have nodes c and e in common. The 10­Ω resistor is also in parallel with the 20­Ω and 12­Ω resistors because nodes b and c are equivalent.