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DC Circuit Analysis of Parallel Circuits
Definitions and Terms | Introduction to Parallel Circuits | Kirchhoff's Current Law | Equivalent Parallel Resistance | Parallel Circuit Examples | Summary | Study Tools | Practice Quiz
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Definitions and Terms |
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In order to analyze and troubleshoot electric circuits, we need to distinguish between the various types of connections. Last week, we examined series circuits which connect loads (resistors) and a source to form a single loop. In a series circuit, any two components have only one point in common. This week, we examine parallel circuits, which connect two or more components to the same point (called a node). In a parallel circuit, current flows into each branch. The electrical devices in your automobile are connected in parallel and when they are turned on, they each have approximately twelve volts across them. The lights in your house are also connected in parallel, and they each have 120 volts across them.
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Introduction to Parallel Circuits |
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Two fundamental circuits form the basis of all electrical circuits: series circuits and parallel circuits. While the series circuit maintains only one common point between two circuits, a parallel circuit maintains two common points, nodes, between the circuits.
The diagram below shows an example of a parallel circuit. Note that the resistors R1 and R2 are connected together at two points (labeled 0 and 1). This arrangement forces the current in the circuit to split into two separate paths, one path through R1 and the other path through R2.
Before talking about Figure "Simple Parallel Circuit" let us discuss the basics of a parallel circuit. An example of a parallel circuit is that two resistors are connected together at each end forming a shunt connection where the current splits into two paths. One path through one resistor and the other path through the other resistor.
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"Simple Parallel Circuit"The Figure titled "Simple Parallel circuit" shows a DC voltage source of 10V connected to two resistors in parallel, R1 and R2. The voltage source is shown connected at the left of the circuit. R1 and R2 are shown on the right of the circuit and are connected at two points labeled 1 and 0. Point 1 is at the top of the resistors and point o is at the bottom. Point 0 is connected to ground and to the negative side of the DC power supply. The value of R1 and R2 are each 1.0 kOhm. The current flowing out of the battery is denoted by an arrow labeled IT shown alongside the top trace of the circuit, and the current flowing through R1 is denoted by an arrow alongside R1 and labeled I1, and the current flowing through R2 is denoted by an arrow alongside R2 and labeled I2. The current flowing back towards the negative terminal of the battery is again IT.
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The defining characteristics of a parallel circuit are:
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Kirchhoff's Current Law |
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Kirchhoff's Current Law states that the sum of the currents entering a node is equal to zero. Stating Kirchhoff's Current Law (KCL) in another way; the currents entering a node equals the currents leaving a node. By convention, currents entering a node are considered positive; currents leaving a node are considered negative.
The following figure demonstrates Kirchhoff's Current Lawwhich states that the current entering a node is equal the currents leaving the node.
As an example, consider the node below:
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"Currents at a node"The figure titled "Currents at a Node" shows a horizontal line and a vertical line intersecting in the middle and forming a node. Four arrows are drawn alongside the vertical and horizontal lines in each of the four quadrants formed by the two intersecting lines. Three arrows are pointing towards the node labeled I(1), I(2), and I(4) representing currents flowing into the node and one arrow is pointing away from the node labeled I(3) representing current flowing out of the node.
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In this example, since the currents entering the node must equal the currents leaving the node, I(1) + I(2) + I(3) = I(4).
As an example of the application of Kirchhoff's Current Law, consider the parallel circuit shown above, with a voltage source of 10V and two 1 kohm resistors in parallel. What are the branch currents (I1 and I2), the total current (IT), and the effective resistance (RT) for this circuit?
Using MultiSim to verify these calculations:
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"Simple Parallel Circuit Simulation"This circuit schematic titled "Simple Parallel Circuit Simulation" shows the "Simple Parallel Circuit" Figure described above with the DC supply voltage of 10V connected to two resistors in Parallel R1, and R2 as before. This figure shows how to measure voltage and current using the software simulation application "Multisim". A digital multi-meter (DMM) is also part of the circuit connected in series at the top horizontal part of the circuit. The DMM has Multisim label of "XMM1". Typically when double clicking on XMM within Multisim, a small window of 1" X 1 ½" will pop up within the program. That window will enable the person to click change parameters so it can measure voltage, current, resistance, dB, AC and DC. This particular XMM1 window is set up to measure both (1)DC and (2) current. In this case, the DMM is functioning as an ammeter to measure DC current. Its display shows the measured value of current to be 20.002 mA. A second digital multi-meter (DMM) is connected across the resistor R1 and R2 on the right side of the circuit schematic to measure voltage across the parallel branch of R1 and R2. This DMM has Multisim label of "XMM2". The 1" X 1 ½" pop up window labeled "XMM2" is set up to measure voltage. The display shows the measured value of voltage to be 10 V. A third digital multi-meter (DMM) is connected in series with the resistor R1 on the left of the parallel connection of R1 and R2. This third DMM has a Multisim label of "XMM3". The 1" X 1 ½" pop up window labeled "XMM3" is set up to measure current. Its display shows the measured value of current to be 10 mA.
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The simulation shows that the voltage drop across each branch is equal, and that the current is evenly divided between the two branches.
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Equivalent Parallel Resistance |
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In the example above, the total equivalent resistance was one-half of the value of each of the individual resistors. This is expected, since by combining Kirchhoff's Current Law and Ohm's Law for this circuit:
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Image description10V/RT = 10V/R1 + 10V/R2
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With further analysis, as shown in your textbook, this simplification can be shown to apply to parallel circuits with any number of branches. In those circuits, the equivalent resistance will be equal to the reciprocal of the sum of the reciprocals of the resistor values in each branch. In other words, for a parallel circuit with N branches [NOTE: In the equation below, 1/RT should be RT):
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The result of this formula is that the equivalent resistance of a parallel circuit is always LESS THAN the smallest branch resistance.
A special case of this general formula is for a simple two-branch circuit. In this case, the equivalent parallel resistance can be simplified to:
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Parallel Circuit Examples |
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To analyze a parallel circuit:
Let's look at a couple of examples and solve for total current and equivalent resistance.
The first example is a two branch circuit with different resistor values in each branch:
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"Two Branch Parallel Circuit"This circuit schematic titled "Two Branch Parallel Circuits" consists of a 10V DC supply voltage on the left of the circuit connected to two resistors in parallel, R1 and R2. The resistors R1 and R2 are shown on the right of the circuit. The value of R1 is 1.0 kOhm, and the value of R2 is 10 kOhm. The current flowing out of the battery is denoted by an arrow labeled I1 shown alongside the top trace of the circuit pointing to the right, and the current flowing through R1 is denoted by an arrow alongside R1 pointing downwards and labeled I2, and the current flowing through R2 is denoted by an arrow also pointing downwards alongside R2 and labeled I3. The current flowing back towards the negative terminal of the supply is denoted by an arrow alongside the bottom trace pointing to the left and labeled I4.
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Following the steps outlined above:
As predicted, the total resistance is less than the smallest branch resistance.
For the next example, consider a circuit with three branches:
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"Three branch Parallel Circuits"This circuit schematic titled "Three Branch Parallel Circuits" consists of a 10V DC supply voltage on the left of the circuit connected to three resistors in parallel, R1, R2, and R3 that are shown on the right of the circuit. The value of R1 is 1.0 kOhm, the value of R2 is 10 kOhm, and the value of R3 is 5.1 kOhm. The current flowing out of the battery is denoted by an arrow labeled I1 shown alongside the top trace of the circuit pointing to the right, and the current flowing through R1 is denoted by an arrow alongside R1 pointing downwards and labeled I2. The current flowing through R2 is denoted by an arrow also pointing downwards alongside R2 and labeled I3. The current flowing through R3 is denoted by an arrow also pointing downwards alongside R3 and labeled I4. The current flowing back towards the negative terminal of the supply is denoted by an arrow alongside the bottom trace pointing to the left and labeled I5.
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Using the same steps:
Once again, the total resistance is less than the smallest branch resistance and the addition of a third branch lowered the total resistance, even though the branch added has a resistance value greater than the smallest branch resistance.
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Summary |
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Some of the key terms for parallel circuits are defined below:
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Terms |
Definitions |
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Parallel Connection |
A circuit with two or more paths for current to flow, where all the components are connected between the same node and have the same voltage. |
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Node |
A place where two or more wires are connected. This is a branching point in a circuit. The current has more than one path at a node. |
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Resistance |
As more resistors are placed in a parallel circuit configuration, the resistance decreases, since the current increases. 1/Rtotal = 1/R1 + 1/R2 + 1/Rn or Rtotal = (R1*R2)/(R1 + R2) or total resistance = product over sum. |
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Conductance |
Conductance is the ease with which current flows through a component or circuit. It is the opposite of resistance. G = 1/R Symbol: G. Measured in: Siemens, that is represented by S. |
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Study Tools |
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Click on the link above to test your knowledge on this week's material. Note: These activities will open in a new pop-up window, so you may need to disable any pop-up blockers. Click here to see transcripts
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Practice Quiz |
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