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

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

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6.2 - Parallel Elements Two elements, branches, or networks are in parallel if they have two points in common as in the figure below Insert Fig 6.2

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6.3 - Total Conductance and Resistance For parallel elements, the total conductance is the sum of the individual conductances. G T = + G 1 + G 2 + G 3 +… + G n 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

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Total Conductance and Resistance Since G = 1/R the total resistance for a network can be determined by the equation below Note that the equation is for 1 divided by the total resistance rather than the total resistance Once the right side of the equation has been determined, it is necessary to divide the result into 1 to determine the total resistance

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Total Conductance and Resistance The total resistance of a parallel resistor is always less than the value of the smallest resistor Additionally, the wider the spread in numerical value between two parallel resistors, the closer the total resistance will be to the smaller resistor The equation becomes significantly easier to apply for equal resistors in parallel Total resistance of N parallel resistors of equal value is the resistance of one resistor divided by the number (N) of parallel elements

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Total Conductance and Resistance 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

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Total Conductance and Resistance Parallel elements can be interchanged without changing the total resistance or input current For parallel resistors, the total resistance will always decrease as additional elements are added in parallel

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6.4 - Parallel Circuits Total resistance is determined by R T = R 1 R 2 / (R 1 + R 2 ) and the source current by I s = E / R T. The subscript s will be used to denote a property of the source The voltage across parallel elements is the same V 1 = V 2 = E Voltage across resistor 1 is equal to the voltage across resistor 2

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Parallel Circuits For single-source parallel networks, the source current (I s ) is equal to the sum of the individual branch currents I s = I 1 + I 2

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6.5 - Kirchhoffs Current Law Kirchhoffs voltage law provides an important relationship among voltage levels around any closed loop of a network Now consider Kirchhoffs current law (KCL) Kirchhoffs current law 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 entering = I leaving

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Kirchhoffs Current Law Most common application of the law will be at the junction of two or more paths of current flow Determining whether a current is entering or leaving a junction is sometimes the most difficult task One approach to understanding the flow is to picture yourself as standing on the junction point and treating the path currents as arrows If the arrow appears to be heading toward you, the current is entering the junction If you see the tail of the arrow as it travels down its path away from you, the current is leaving the junction

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6.6 - Current Divider Rule The current divider rule (CDR) will determine how the current entering a set of parallel branches will split between the elements 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

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Current Divider Rule Current seeks the path of least resistance The current entering any number of parallel resistors divides into these resistors as the inverse ratio of their ohmic value

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Voltage sources are placed in parallel only if they have the same voltage rating The purpose for placing two or more batteries in parallel would be to increase the current rating The formula to determine the total current is: I s = I 1 + I 2 +… I N at the same terminal voltage Voltage Sources in Parallel

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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

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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 Two isolated terminals not connected by any element:

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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

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6.9 - Voltmeters: Loading Effect Voltmeters are always placed across an element to measure the potential difference The resistance of two parallel resistors will always be less than the resistance of the smallest resistor A DMM has internal resistance which will alter, somewhat, the network being measured The loading of a network by the insertion a meter is not to be taken lightly, especially if accuracy is a primary consideration

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Voltmeters: Loading Effect A good practice is to always check the meter resistance level against the resistive elements of the network before making a measurement Most DMMs have internal resistance levels in excess of 10 M W on all voltage scales Internal resistance of VOMs is sensitive to 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

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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

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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

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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

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