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Analysis and Design of DC Resistive Parallel Circuits

Introduction to Parallel Circuits | Kirchhoff's Law | Calculating the Total Current for a Parallel Circuit | Applying Kirchhoff's Law to a Parallel Circuit | Applying the Law of Conservation of Energy to a Parallel Circuit | Summary | Practice Quiz

In order to design, diagnose, and improve electric circuits, you must be able to distinguish between the various types of connections. Last week, you examined series circuits, which connect loads (only using resistors) and a source to form a single loop. In a series circuit, any two components have only one point in common. This week, you will examine parallel circuits, in which components are connected between the same set of electrically common points.

Introduction to Parallel Circuits

Two fundamental circuits form the basis of all electrical circuits: series circuits and parallel circuits. As you learned in Week 1, the defining characteristic of a series circuit is that the components connect to form a single path for the flow of electrons. While the series circuit maintains only one common point between two circuits, a parallel circuit maintains two common points (i.e., nodes) between the circuits.

The defining characteristics of a parallel circuit are that

• all components are connected between the same set of electrically common points; and
• the elements or branches of the circuit have two common points.

Figure 2.1: Parallel Circuit Example

The simple parallel circuit example in Figure 2.1 shows the resistors are connected in parallel. According to Kirchhoff's voltage law, the voltage drop across the resistors is equal to the value of the direct current (DC) voltage source.

Kirchhoff's Law

Building upon Kirchhoff's voltage law, Kirchhoff's current law states that the algebraic sum of the currents entering and leaving a node is equal to 0. Think of water flowing into a pipe. The amount of water that enters the pipe is equal to the amount of water leaving the pipe. Therefore, in Figure 2.1

I_T = I₁ + I₂.

In the simple parallel circuit in Figure 2.1, while the voltage across the resistors has the same value as the DC voltage source, each resistor's current depends on the value of the resistor, in accordance with Ohm's law. Therefore, if the resistors have different values, then the current through each resistor is different.

Figure 2.2 highlights the split in current values. In Figure 2.2, the voltage across each resistor is 10V (shown by the voltmeter U2), which is the DC voltage applied to the circuit. In this circuit, the current is distributed equally between the two branches because the resistors have equal values. Each branch consists of a resistor with a value of 1 ohm. Each resistor's current is 10mA (shown by the ammeters U3 and U5) and the total current I_T is 20mA (shown by the ammeter U1).

Click on image to enlarge

Figure 2.2

Therefore, examining each resistor's current will allow you to calculate the total current of a circuit.

	Kirchhoff's Current Law  Please take a moment to watch this tutorial on Kirchoff's current law. Include the practice exercises.

Calculating the Total Current for a Parallel Circuit

In order to calculate the total current in a parallel circuit, first determine the total resistance of the circuit.

In a series circuit, you would total all circuit resistances (R) to derive the total resistance. In a parallel circuit, you can simplify this process by working with conductance, which is the reciprocal, or inverse, of resistance. While resistance measures the friction a component presents to the flow of electrons, conductance measures how easy it is for electrons to flow through a component. The greater the resistance is, the lower the conductance and vice versa.

Conductance, which is the reciprocal of resistance, is represented as G = 1/R.

Hence, in Figure 2.2 (above), given that measurement, the total conductance is the sum of the conductances of the individual branches. Therefore

G_T = G₁ + G₂, which is 2mS;

and

R_T = 0.5 K ohms.

Note: In this example, the total resistance R_T is smaller than the individual resistances R₁ and R₂. In general, the total resistance of a parallel circuit is smaller than the smallest resistance. In a series circuit, the total resistance is greater than the value of the largest resistance.

For a simple two-branch parallel circuit (as in figure 2.2),

The total current I_T can be calculated as V/R_T = 10V/0.5K Ohms = 20mA.

Results for two branch parallel circuit are as follows.

 

	Calculating Total Resistance of a Parallel Circuit Please take a moment to watch this tutorial that will step through the procedure for calculating the total resistance of a parallel circuit.

Applying Kirchhoff's Law to a Parallel Circuit

Kirchhoff's current law states that the algebraic sum of all currents in a node is equal to 0. In order to apply this law to a parallel circuit, it is helpful to start with the known values to calculate the branch currents.

Given the voltage V applied to a circuit with two resistors of known value R1 and R2, and the total current IT you calculated, you can calculate the branch currents as the following.

The branch current can be represented in terms of (a) the total current, (b) total resistance, and (c) branch resistance.

Calculating the current through each branch highlights the differences between series and parallel circuits; while the series circuit acts as a voltage divider, the parallel circuit acts as a current divider.

Because parallel circuit totals use the notation of conductance, consider the following:

In order to apply Kirchhoff's current law to a parallel circuit, you must assign an algebraic sign corresponding to a reference direction to each current at the node. If you are unsure of the direction of the current, assume a direction. Assume a positive sign for a current entering a node and a negative sign for a current leaving a node.

Applying the Law of Conservation of Energy to a Parallel Circuit

So far, you have determined the current through each branch of a parallel circuit using Ohm's law, and you derived the total current using Kirchhoff's current law. You can now proceed to analyze the parallel circuit by calculating the power dissipated in the resistors by following the law of conservation of energy.

The law of conservation of energy states that the power dissipated must equal the total power applied by the source(s). This applies to parallel circuits as it did to series circuits. For parallel circuits, the law of conservation of energy shows that the power delivered by the voltage source is equal to the total power dissipated by the resistors. This can be proven through our example circuit.

• Power delivered by the voltage source P = V * I_T = 10V * 20mA = 200mW.
• The power dissipated by each of the resistors is given by P₁ = V² / R₁ = 10V * 10V / 1k ohms = 100mW.

Result: Because P₁ = P₂, P = P₁ + P₂

Ideally, meters used for measuring current do not have any internal resistance, whereas meters used for measuring voltage have an infinite resistance. Given that we cannot assume the ideal, it is safe to assume internal resistance using a voltmeter, which typically has a resistance in series with the display meter. A voltmeter with an internal resistance of Rs will act as a parallel circuit when placed across a resistor. The voltmeter displays the proper voltage according to the value of the internal resistance in comparison to the resistor across which it is connected.

In Figure 2.3, the internal resistance of the voltmeter is represented as Rs and is connected across resistor R₁.

Figure 2.3: Circuit with the internal resistance Rs of the voltmeter appearing in parallel with R₁

The voltmeter in Figure 2.3 shows that if the internal resistance Rs is less than the resistance across which it is connected, it will draw more current. Thus, the smaller resistance will adversely load the circuit and produce an erroneous reading. However, if the internal resistance is significantly larger than the resistor across which the voltmeter is connected, the effective resistance will not affect the voltage because the current drawn from the source will not significantly change due to the extra internal resistance.

Similar to the voltmeter example, we can study the effect of ammeter loading on the circuit. The ammeter is connected in series with a component to determine the current flowing through it.

Figure 2.4 shows Rs as the internal resistance of the ammeter. In Figure 2.4, if the internal resistance Rs is very small, it will only have a minimal affect on the total resistance of the circuit. However, if the resistance R₁ is reduced to a small value, the internal resistance of the ammeter plays a factor.

Figure 2.4: Circuit with internal resistance Rs of the ammeter appearing in series with resistor R₁

Summary

Now that you applied Ohm's law, Kirchhoff's laws, and the law of conservation of energy to a parallel circuit and a series circuit, you will proceed to pull it all together next week. Next week, you will consider both a series and a parallel combination of resistors in a circuit and analyze these circuits using Kirchhoff's voltage and current laws.

	Series Circuits and Parallel Circuits

Practice Quiz