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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 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. GT G1 G2 G3 ... GN 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 1 RT 1 1 1 1 ... R1 R2 R3 RN 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 (I ) is equal to the sum of the individual branch currents. s Is I1 I2 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. E E Is I1 I 2 R1 R2 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. PE PR1 PR2 PR3 ... PRN 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. I in Iout 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 RT Ix IT Rx 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: E1 E2 I Rint1 Rint 2 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 (W / V) rating of 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.