**Chapter 6 – Parallel dc Circuits** Introductory Circuit Analysis Robert L. Boylestad

**6.1 - Introduction** • There are two network configurations – series and parallel. • In Chapter 5 we covered a series network. In this chapter we will cover the parallel circuit and all the methods and laws associated with it.

**6.2 – Parallel Resistors** • Two elements, branches, or circuits are in parallel if they have two points in common as in the figure below Insert Fig 6.2

**Parallel Resistors** • For resistors in parallel, the total resistance is determined from • Note that the equation is for the reciprocal of RT rather than for RT. • Once the right side of the equation has been determined, it is necessary to divide the result into 1 to determine the total resistance

**Parallel Resistors** • For parallel elements, the total conductance is the sum of the individual conductance values. • As the number of resistors in parallel increases, the input current level will increase for the same applied voltage. • This is the opposite effect of increasing the number of resistors in a series circuit.

**Parallel Resistors** • The total resistance of any number of parallel resistors can be determined using • The total resistance of parallel resistors is always less than the value of the smallest resistor.

**Parallel Resistors** • For equal resistors in parallel: Where N = the number of parallel resistors.

**Parallel Resistors** • A special case: The total resistance of two resistors is the product of the two divided by their sum. • The equation was developed to reduce the effects of the inverse relationship when determining RT

**Parallel Resistors** • Parallel resistors can be interchanged without changing the total resistance or input current. • For parallel resistors, the total resistance will always decrease as additional parallel elements are added.

**6.3 – Parallel Circuits** • Voltage is always the same across parallel elements. V1 = V2 = E The voltage across resistor 1 equals the voltage across resistor 2, and both equal the voltage supplies by the source.

**Parallel Circuits** • For single-source parallel networks, the source current (Is) is equal to the sum of the individual branch currents. • For a parallel circuit, source current equals the sum of the branch currents. For a series circuit, the applied voltage equals the sum of the voltage drops.

**Parallel Circuits** • For parallel circuits, the greatest current will exist in the branch with the lowest resistance.

**6.4 – Power Distribution in a Parallel Circuit** • For any resistive circuit, the power applied by the battery will equal that dissipated by the resistive elements. • The power relationship for parallel resistive circuits is identical to that for series resistive circuits.

**6.5 - Kirchhoff’s Current Law** • Kirchhoff’s voltage law provides an important relationship among voltage levels around any closed loop of a network. • Kirchhoff’s current law (KCL) states that the algebraic sum of the currents entering and leaving an area, system, or junction is zero. • The sum of the current entering an area, system or junction must equal the sum of the current leaving the area, system, or junction.

**Kirchhoff’s Current Law** • Most common application of the law will be at the junction of two or more paths of current. • Determining whether a current is entering or leaving a junction is sometimes the most difficult task. • If the current arrow points toward the junction, the current is entering the junction. • If the current arrow points away from the junction, the current is leaving the junction.

**6.6 – Current Divider Rule** • The current divider rule (CDR) is used to find the current through a resistor in a parallel circuit. • General points: • For two parallel elements of equal value, the current will divide equally. • For parallel elements with different values, the smaller the resistance, the greater the share of input current. • For parallel elements of different values, the current will split with a ratio equal to the inverse of their resistor values.

**Current Divider Rule**

**6.7 - Voltage Sources in Parallel** • Voltage sources are placed in parallel only if they have the same voltage rating. • The purpose for placing two or more batteries in parallel is to increase the current rating. • The formula to determine the total current is: • at the same terminal voltage.

**Voltage Sources in Parallel** • Two batteries of different terminal voltages placed in parallel • When two batteries of different terminal voltages are placed in parallel, the larger battery tries to drop rapidly to the lower supply • The result is the larger battery quickly discharges to the lower voltage battery, causing the damage to both batteries

**6.8 - Open and Short Circuits** • An open circuit can have a potential difference (voltage) across its terminal, but the current is always zero amperes.

**Open and Short Circuits** • A short circuit can carry a current of a level determined by the external circuit, but the potential difference (voltage) across its terminals is always zero volts. Insert Fig 6.44

**6.9 – Voltmeter Loading Effects** • Voltmeters are always placed across an element to measure the potential difference. • The resistance of parallel resistors will always be less than the resistance of the smallest resistor. • A DMM has internal resistance which may alter the resistance of the network under test. • The loading of a network by the insertion of a meter is not to be taken lightly, especially if accuracy is a primary consideration.

**Voltmeter Loading Effects** • A good practice is to always check the meter resistance against the resistive elements of the network before making a measurement. • Most DMMs have internal resistance levels in excess of 10 MW on all voltage scales. • The internal resistance of a VOM depends on the scale chosen. • Internal resistance is determined by multiplying the maximum voltage of the scale setting by the ohm/volt ( / V) ratingof the meter, normally found at the bottom of the face of the meter.

**6.11 – Troubleshooting Techniques** • Troubleshooting is a process by which acquired knowledge and experience are employed to localize a problem and offer or implement a solution. • Experience and a clear understanding of the basic laws of electrical circuits is vital. • First step should always be knowing what to expect

**6.13 – Applications** • Car system • The electrical system on a car is essentially a parallel system. • Parallel computer bus connections • The bus connectors are connected in parallel with common connections to the power supply, address and data buses, control signals, and ground.

**Applications** • House wiring • Except in some very special circumstances the basic wiring of a house is done in a parallel configuration. • Each parallel branch, however, can have a combination of parallel and series elements. • Each branch receives a full 120 V or 208 V, with the current determined by the applied load.