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

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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 R T rather than for R T.  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.

1/R T = 1/1 + ¼ + 1/5 = 1 + 0.25 + 0.2 = 1.45  R T = 1/1.45 = 0.69 

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 R T R T = PRODUCT/SUM

R T = (3 x 6)/(3 + 6) = 18/9 = 2 

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.

Using a protoboard to set up the circuit

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

Measuring the voltages of a parallel dc network.

Parallel Circuits  For single-source parallel networks, the source current (I s ) 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.

Measuring the source current of a parallel network.

Measuring the current through resistor R 1.

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.

Kirchhoff’s current law.

(a) Demonstrating Kirchhoff’s current law; (b) the water analogy for the junction in (a).

I 3 = 5A and I 4 = 4A

I 1 = 1A; I 3 = I 1 = 1A; I 4 = I 2 = 4A; I 5 = I 3 + I 4 = 5A

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

Using the current divider rule to calculate current I 1 1/R T = 1/1 k + 1/10 k + 1/22 k  R T = 873  I 1 = (R T /R 1 )I T = (873/1000)(12 mA) = 10.5 mA

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

Examining the impact of placing two lead-acid batteries of different terminal voltages in parallel. I = (12 – 6)/(0.03 + 0.02) = 120A

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

I = (6V)/(12  ) = 0.5A and V = (0.5A)(10  ) = 5V

I = (6V)/(2  ) = 3A and V = 0

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 M  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) rating of the meter, normally found at the bottom of the face of the meter.

V ab = 20V Vab = (11M  )/(12M  )(20V) = 18.33V

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.

Expanded view of an automobile’s electrical system.

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.

Single phase of house wiring: (a) physical details; (b) schematic representation.

Continuous ground connection in a duplex outlet.