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image sci elec002 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

Parallel circuit CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

Simple parallel circuits

Let’s start with a parallel circuit consisting of three resistors and a single battery:

 

00092 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

paralel circuit CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across R1 is equal to the voltage across R2 which is equal to the voltage across R3 which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values:

10070 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

Just as in the case of series circuits, the same caveat for Ohm’s Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm’s Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor:

 

10071 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

10072 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

At this point we still don’t know what the total current or total resistance for this parallel circuit is, so we can’t apply Ohm’s Law to the rightmost (“Total”) column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor (“branch”) currents:

00093 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

 

prcirc4 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at point 7 to go up through R1, some more splits off at point 6 to go up through R2, and the remainder goes up through R3. Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The same thing is encountered where the currents through R1, R2, and R3 join to flow back to the positive terminal of the battery (+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through R1, R2, and R3.

This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can fill in the IT spot on our table with the sum of IR1, IR2, and IR3:

10073 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

Finally, applying Ohm’s Law to the rightmost (“Total”) column, we can calculate the total circuit resistance:

 

10074 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

 

parallel circuit4 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

Please note something very important here. The total circuit resistance is only 625 Ω: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of “rules” for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this:

10075 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit.

Just as with the series circuit, we can use computer analysis to double-check our calculations. First, of course, we have to describe our example circuit to the computer in terms it can understand. I’ll start by re-drawing the circuit:

00092 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

Once again we find that the original numbering scheme used to identify points in the circuit will have to be altered for the benefit of SPICE. In SPICE, all electrically common points must share identical node numbers. This is how SPICE knows what’s connected to what, and how. In a simple parallel circuit, all points are electrically common in one of two sets of points. For our example circuit, the wire connecting the tops of all the components will have one node number and the wire connecting the bottoms of the components will have the other. Staying true to the convention of including zero as a node number, I choose the numbers 0 and 1:

00094 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

 

circuito en paralelo 1 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

An example like this makes the rationale of node numbers in SPICE fairly clear to understand. By having all components share common sets of numbers, the computer “knows” they’re all connected in parallel with each other.

In order to display branch currents in SPICE, we need to insert zero-voltage sources in line (in series) with each resistor, and then reference our current measurements to those sources. For whatever reason, the creators of the SPICE program made it so that current could only be calculated through a voltage source. This is a somewhat annoying demand of the SPICE simulation program. With each of these “dummy” voltage sources added, some new node numbers must be created to connect them to their respective branch resistors:

00095 CALCULATION OF CURRENT IN PARALLEL CIRCUIT

 

  • REVIEW:

 

  • Components in a parallel circuit share the same voltage: ETotal = E1 = E2 = . . . En
  • Total resistance in a parallel circuit is less than any of the individual resistances: RTotal = 1 / (1/R1 + 1/R2 + . . . 1/Rn)
  • Total current in a parallel circuit is equal to the sum of the individual branch currents: ITotal = I1 + I2 + . . . In.

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