# Parallel Circuits – All Topics | PPT

## Parallel Circuits

Topics Covered in this PPT

The Applied Voltage *V**A* Is the Same Across Parallel Branches

Each Branch *I* Equals *V**A** / R*

Kirchhoff’s Current Law (KCL)

Resistance in Parallel

Conductances in Parallel

Total Power in Parallel Circuits

Analyzing Parallel Circuits with Random Unknowns

Troubleshooting: Opens and Shorts in Parallel Circuits

### The Applied Voltage *V**A* Is the Same Across Parallel Branches

**Characteristics of a Parallel Circuit**

- Voltage is the same across each branch in a parallel circuit.
- The total current is equal to the sum of the individual branch currents.
- The equivalent resistance (
*R**EQ*) is less than the smallest branch resistance. The term equivalent resistance refers to a single resistance that would draw the same amount of current as all of the parallel connected branches. - Total power is equal to the sum of the power dissipated by each branch resistance.
- A parallel circuit is formed when two or more components are connected across the same two points.
- A common application of parallel circuits is the typical house wiring of many receptacles to the 120-V 60 Hz ac power line.

Fig. 5-1: Example of a parallel circuit with two resistors. (*a*) Wiring diagram.

(*b*) Schematic diagram.

### Each Branch *I* Equals *V**A** / R*

- The current in a parallel circuit equals the voltage applied across the circuit divided by the resistance between the two points where the voltage is applied.
- Each path for current in a parallel circuit is called a
**branch**. Each branch current equals*V/R*where*V*is the same across all branches.

Fig. 5-3: Parallel circuit. (*a*) the current in each parallel branch equals the applied voltage *V**A* divided by each branch resistance *R*.

### Kirchhoff’s Current Law (KCL)

- Components connected in parallel are usually wired across one another, with the entire parallel combination connected to the voltage source.

Fig. 5-5a: The current in the mainline equals the sum of the branch currents. Note that from G to A at the top of this diagram is the negative side of the main line, and from B to F at the bottom is the positive side. (*a*) Wiring diagram. Arrows inside the lines indicate current in the main line for *R**1*; arrows outside indicate current for *R**2**.*

- This circuit structure gives the same result as wiring each parallel branch directly to the voltage source.
- The main advantage of using this structure is that it requires less wire.
- The pair of leads connecting all the branches to the voltage source terminals is the
**main line**. - All the current in the circuit must come from one side of the voltage source and return to the opposite side for a complete path.
- The amount of current in the main line is equal to the total of the branch currents.
- The total current
*I**T*in the main line is equal to the sum of the branch currents. - This is known as Kirchhoff’s current law (KCL).
- It applies to any number of parallel branches, whether the resistance in those branches are equal or not.
*I**T**= I**1**+ I**2**+ I**3**+ I**4*

**Resistance in Parallel**

- The combined equivalent resistance of a parallel circuit may be found by dividing the common voltage across all resistances by the total current of all the branches.
*R**EQ = Va/It*- A combination of parallel branches is called a
**bank**. - A combination of parallel resistances
*R**EQ*for the bank is always less than the smallest individual branch resistance because*I**T*must be more than any one branch current. - The equivalent resistance of a parallel circuit must be less than the smallest branch resistance.
- Adding more branches to a parallel circuit reduces the equivalent resistance because more current is drawn from the same voltage source.

Fig. 5-7: How adding parallel branches of resistors increases *I**T* but decreases* R**EQ*. (*a*) One resistor. (*b*) Two branches. (*c*) Three branches. (*d*) The equivalent circuit of the three branches in (*c*).

- Total Current and Reciprocal Resistance Formulas
- In a parallel circuit, the total current equals the sum of the individual branch currents:
*I**T**= I**1**+ I**2**+ I**3*+…+etc.- Total current is also equal to total voltage divided by equivalent resistance:
*I**T**=**V**T/**R**EQ*

- Total Current and Reciprocal Resistance Formulas
- The equivalent resistance of a parallel circuit equals the reciprocal of the sum of the reciprocals:
- Equivalent resistance also equals the applied voltage divided by the total current:

**Determining the Equivalent Resistance**

Fig. 5-8: Two methods of combining parallel resistances to find *R**EQ*. (*a*) Using the reciprocal resistance formula to calculate *R**EQ* as 4 Ω. (*b*) Using the total line current method with an assumed line voltage of 20 V gives the same 4 Ω for *R**EQ*.

**Special Case: Equal Value Resistors**

- If
*R*is equal in all branches, divide one resistor’s value by the number of resistors.

Fig. 5-9: For the special case of all branches having the same resistance, just divide *R* by the number of branches to find *R**EQ*. Here, *R**EQ* = 60 kΩ / 3 = 20 kΩ.

**Special Case: Two Unequal Resistors**

- When there are only two branches of a parallel circuit and their resistances are unequal, use the formula:

Fig. 5-10: For the special case of only two branch resistances, of any values, *R**EQ* equals their product divided by the sum. Here, *R**EQ* = 2400 / 100 = 24Ω.

- To find an unknown branch resistance, rewrite the formula as follows to solve for the unknown value.
*R**X =**R**×**R**EQ /**R**−**R**EQ*- These formulas may be used to simplify complex circuits.

**Conductances in Parallel**

- Conductance (
*G*) is equal to 1 /*R*. - Total (equivalent) conductance of a parallel circuit is given by:
*G**T**= G**1**+ G**2**+ G**3**+ … +*etc.

**Determining Conductance**

- Each value of
*G*is the reciprocal of*R*. Each branch current is directly proportional to its conductance. - Note that the unit for
*G*is the**Siemens**(S).

### Total Power in Parallel Circuits

- Total power is equal to the sum of the power dissipated by the individual resistances of the parallel branches:
*P**T**= P**1**+ P**2**+ P**3**+ … +*etc.

- Total power is equal to voltage times total current:
*P**T**= V**T**I**T*

**Determining Power**

Check: *P**T* = *V**T* *×** I**T*

### Analyzing Parallel Circuits with Random Unknowns

- When the voltage across one branch is known, use this voltage for all branches. There can be only one voltage across branch points with the same potential difference.
- If the values for
*I**T*and one branch current (*I**1*) are known, the value of*I**2*can be found by subtracting*I**1*from*I**T*.

### Troubleshooting: Opens and Shorts in Parallel Circuits

**Opens in Parallel Circuits**

- An open circuit in one branch results in no current through that branch.
- However, an open circuit in one branch has no effect on the other branches. This is because the other branches are still connected to the voltage source.
- An open in the mainline prevents current from reaching any branch, so all branches are affected.

**Opens in Parallel Circuits.**

- In part
**b**bulbs 2 and 3 still light. However, the total current is smaller. In part**a**no bulbs light. - Fig. 5-16: Effect of an open in a parallel circuit. (
*a*) Open path in the main line—no current and no light for all the bulbs. (*b*) Open path in any branch—bulb for that branch does not light, but the other two bulbs operate normally.

**Shorts in a Parallel Circuit**

- A short circuit has zero resistance, resulting in excessive current in the shorted branch.
- A shorted branch shorts the entire circuit.
- Current does not flow in the branches that are not shorted. They are bypassed by the short circuit path that has zero resistance.

**A Short in a Parallel Circuit**

- The other branches are shorted out. The total current is very high.
- Fig. 5-17: Effect of a short circuit across parallel branches. (
*a*) Normal circuit. (*b*) Short circuit across points H and G shorts out all the branches.