143Analyzing Series Circuits143
4.4 Computing Voltage, Current, Resistance, and Power
We are now ready to apply the power equations, the basic theory of series circuits, Kirch-hoff ’s Voltage Law, and Ohm’s Law to the solution of series circuit problems. Sometimes problems require a technician to simply substitute numbers into an equation and compute the result. In other cases, the technician must manipulate the available data in a creative way to obtain solutions to the circuit. In all cases, though, Ohm’s Law, Kirchhoff ’s Law, and the power equations will be valid.
Total Resistance
The electrons flowing in a series circuit must pass through every component. Each resistor opposes the current flow. That is, every resistor in a series circuit increases the total resis-tance to current flow. We can express this idea mathematically as
EXAMPLE SOLUTION
Determine the total resistance of the circuit shown in Figure 4-28.
EXAMPLE SOLUTION We apply Equation 4-2.
(4-2)
Figure 4-27. How much voltage is dropped across R1?
The total resistance in a series circuit can be computed by summing the values of individual resistors.
Figure 4-28. Calculate the value of total resistance.
EXAMPLE SOLUTION
Calculate the value of total resistance for the circuit shown in Figure 4-29.
EXAMPLE SOLUTION
Practice Problems Practice Problems
1. Calculate the total resistance in Figure 4-30.
2. What is the total opposition to cur-rent flow for the cir-cuit in Figure 4-31?
3. How much total resist-ance is in the circuit shown in Figure 4-32?
Answers to Practice Problems
Current
You will recall that current in a series circuit is the same at all points in the circuit. Thus, for example, the current through R1 (I1) will be the same as the value of total current (IT).
We can express this as an equation (Equation 4-3).
VA
Figure 4-29. How much total resistance is in this circuit?
RT = R1+R2+R3
Figure 4-32. What is the value of total resistance?
1. 224 kΩ 2. 25.5 kΩ 3. 78 kΩ
The current is the same in all parts of a series circuit.
For this reason, current is an important quantity to calculate when analyz-ing a series network.
KEY POINTS
Figure 4-31. Calculate the total opposition to current flow.
Figure 4-30. What is the total resistance in this circuit?
This equation tells us that if we know the value of current at any point in a series circuit, then we immediately know the value of current at every other point in the circuit.
EXAMPLE SOLUTION
Calculate the value of current through R1 and the value of total current in Figure 4-33.
EXAMPLE SOLUTION
The value of current through R3 in Figure 4-33 is given as 200 mA. Since the circuit is a series circuit, we immediately know that the current must be 200 mA at all other points. In equation form, we have
Total Power
Electrical power consumption often takes the form of heat dissipation. So it is with the power dissipated in a resistor. All power that is dissipated in a circuit must be supplied by the power source. The total power consumed is simply a summation of the individual powers dissipated by the various circuit components. We can state this in equation form as
Total circuit power can also be computed with any one of the basic power formulas presented in Chapter 2 provided total circuit values are used in the calculations.
EXAMPLE SOLUTION
What is the total power dissipated by the circuit shown in Figure 4-34?
EXAMPLE SOLUTION
Method 1: One way to compute the total power in a circuit is to find the sum of the powers dissi-pated by the individual resistors.
Method 2: Another way to compute total power in a circuit is to simply apply one of the basic power equations with total circuit values used as factors.
(4-3)
Figure 4-33. How much current flows through R1?
IT = I1 = I2 = I3 = 200 mA
Total power dissipation can be found by sum-ming the power dissipa-tions of each individual resistor.
Figure 4-34. Compute total power in this circuit.
The subscripts A and T are equivalent when referring to all (applied) circuit values.
Practice Problems Practice Problems
1. How much power is dissipated by the circuit shown in Figure 4-35?
2. What is the total power consump-tion for the circuit shown in Figure 4-36?
Answers to Practice Problems
Fully Specified Circuits
If the value of every component in a circuit is known, then the circuit is called a fully specified circuit. In these cases, the circuit parameters (e.g., voltage, current, and power) can be computed by substituting the appropriate numbers into a relevant equation. Ohm’s Law, Kirchhoff ’s Voltage Law, and the power equations can be used to compute values for individual components (e.g., the voltage drop across R6 or the power dissipated by R2).
The same equations can be used to compute total circuit values (e.g., total current or total power).
Figure 4-35. Calculate the power in this circuit. dissipated by this circuit?
In a fully specified circuit, all component values are known. Calculations of component voltages, cur-rents, and power dissipa-tions are normally accomplished by direct substitution of known parameters into an appro-priate circuit equation.
KEY POINTS
The first step in computing a circuit value is to select an appropriate equation. Selection of an equation is accomplished as follows:
1. Choose an equation where the value to be calculated is the only unknown factor.
2. Use known values associated with a given component to calculate other values for that same component.
3. Use total circuit values to calculate other total circuit values.
Failure to consistently apply these rules is the most common source of errors in circuit analysis.
Let’s work some examples.
EXAMPLE SOLUTION
Calculate the value of voltage across R2 in Figure 4-37.
EXAMPLE SOLUTION
Since we are required to compute a voltage value for R2, we need to select an equation where V2 is the only unknown factor. We know the value of current through R2, since the total current is given as 500 mA, and since we know the current is the same through all components in a series circuit (Equation 4-3). The value of R2 is also given on the schematic diagram. We now select the Ohm’s Law equation V = IR because cur-rent and resistance are known. To avoid using the wrong quantity type, it is impor-tant to add subscripts to all quantities. Since we are calculating values for R2, we can only use known quantities associated with R2.
EXAMPLE SOLUTION
How much total power is dissipated by the circuit in Figure 4-38?
EXAMPLE SOLUTION
Since we are computing total power, we will uti-lize other total values in the selected equation.
VA
Figure 4-37. Calculate V2 in this circuit.
V2 = I2R2
Figure 4-38. Compute the total power in this circuit.
PT = ITVT
200×10–3A×10 V
= 2 W
=
EXAMPLE SOLUTION
What is the value of current through R3 in Figure 4-39?
EXAMPLE SOLUTION
Since this is a series circuit, we know that the current is the same in all parts of the circuit. If we can find the value of current at any point, then we will also know the value of I3. Let us compute the value of I1, since we know two other factors (volt-age and resistance) associated with R1.
Since I3 is the same value as I1 (Equation 4-3), we know that I3 = 2 mA.
Practice Problems Practice Problems
1. How much current flows through R2 in Figure 4-40?
2. What is the total power in Figure 4-40?
3. What is the voltage drop across R3 in Figure 4-41?
Answers to Practice Problems
VA
Figure 4-40. Calculate I2 in this circuit.
VA
Figure 4-41. Calculate V3 in this circuit.
Partially Specified Circuits
There are many times when a technician does not know the value of every circuit compo-nent. A circuit with one or more unspecified component values is called a partially specified circuit. In general, this type of circuit is more challenging to analyze, since direct application of an equation is not always possible. To solve such a problem, a technician must creatively apply a series of equations. Each equation provides additional information. Finally, enough information will be known to allow computation of the specific quantity in question. It takes substantial practice to be able to identify the optimum sequence of calculations. When you are uncertain of the next step, and until you gain the experience needed to streamline your calculations, you are advised to calculate any value that can be calculated. Each calculated value provides you with additional data that can be used in subsequent calculations.
EXAMPLE SOLUTION
Determine the voltage drop across R2 in Figure 4-42.
EXAMPLE SOLUTION
We cannot compute V2 directly because we have no equa-tion with enough known values to solve. Since the value of R2 is given, we could compute V2 if we knew the value of I2 (i.e., V2 = I2R2). When analyzing series circuits, it is usually a good idea to compute current, since this value is common to every component.
This current will be the same at all points in the circuit (e.g., IT, I2, I3, and so on).
We can now calculate V2 by applying Ohm’s Law.
EXAMPLE SOLUTION
Calculate all missing voltages, currents, powers, and resis-tances for the circuit shown in Figure 4-43. Table 4-1 pro-vides a partially filled solution matrix.
Partially specified circuits have one or more unknown component values. The solution of this type of circuit requires creative applica-tion of basic circuit the-ory in conjunction with Ohm’s Law, Kirchhoff’s Voltage Law, and the power formulas.
Figure 4-43. Compute all missing circuit values.
V2 = I2R2
2 mA×2.7 kΩ
= 5.4 V
=
EXAMPLE SOLUTION
It is important to realize two facts: First, there are many ways (sequences) to solve the problem.
Second, there is not necessarily a single best sequence. With these thoughts in mind, let’s solve this problem by computing any circuit value for which we have sufficient data.
In a series circuit, it always a good strategy to find the value of current, since it is common to all components. The value of I3 is given as 1.2 mA, so we immediately know the value of all cur-rents (Equation 4-3).
Next, we might choose to calculate the value of R1 with Ohm’s Law.
We can apply one of the power formulas to compute P1.
Since we know the resistance of R2 and the current through it, we can calculate V2 and P2 as follows:
and,