Section 3.7 Switching Circuits

What You Will Learn Switching circuits

Electrical Circuits Electrical circuits can be expressed as logical statements. T (true) represents a closed switch (or current flow). F (false) represents an open switch (or no current flow). In a series circuit the current can take only one path. In a parallel circuit there are two or more paths the current can take.

Series Circuit Case 1: Both switches are closed; that is, p is T and q is T. The light is on, T. Case 2: Switch p is closed and switch q is open; that is, p is T and q is F. The light is off, F.

Series Circuit Case 3: Switch p is open and switch q is closed; that is, p is F and q is T. The light is off, F. Case 4: Both switches are open; that is, p is F and q is F. The light is off, F.

Series Circuit Switches in series will always be represented with a conjunction ⋀. In summary,

Parallel Circuit Case 1: Both switches are closed; that is, p is T and q is T. The light is on, T. Case 2: Switch p is closed and switch q is open; that is, p is T and q is F. The light is on, T.

Parallel Circuit Case 3: Switch p is open and switch q is closed; that is, p is F and q is T. The light is on, T. Case 4: Both switches are open; that is, p is F and q is F. The light is off, F.

Parallel Circuit Switches in parallel will always be represented with a disjunction ⋁. In summary,

Example 2: Representing a Switching Circuit with Symbolic Statements a. Write a symbolic statement that represents the circuit.

Example 2: Representing a Switching Circuit with Symbolic Statements Solution p and q are in parallel: p ⋁ q q and r are in series: q ⋀ r together we get: (p ⋀ q) ⋁ (q ⋀ r)

Example 2: Representing a Switching Circuit with Symbolic Statements b. Construct a truth table to determine when the light will be on.

Example 2: Representing a Switching Circuit with Symbolic Statements Solution

Example 3: Representing a Symbolic Statement as a Switching Circuit Draw a switching circuit that represents [(p ⋀ ~q) ⋁ (r ⋁ q)] ⋀ s. Solution

Equivalent Circuits Equivalent circuits are two circuits that have equivalent corresponding symbolic statements.

Equivalent Circuits Sometimes two circuits that look very different will actually have the exact same conditions under which the light will be on. The truth tables have identical answer columns.

Example 4: Are the Circuits Equivalent? Determine whether the two circuits are equivalent.

Example 4: Are the Circuits Equivalent? Solution p ⋁ (q ⋀ r) (p ⋁ q) ⋀ (p ⋁ r)

Example 4: Are the Circuits Equivalent? Solution The answer columns are identical.

Example 4: Are the Circuits Equivalent? Solution Therefore, p ⋁ (q ⋀ r) is equivalent to (p ⋁ q) ⋀ (p ⋁ r) and the two circuits are equivalent.