20.6 RESISTIVE NETWORK
20.6.1 Resistances in Series
Three resistors are connected in series. A p.d. of V volt is applied across the two ends, as shown in Figure 20.36. They have resistances R1, R2 and R3. A constant current I flows through the circuit.
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| Three resistors $R_1$, $R_2$ and $R_3$ are connected in series. A potential difference V is applied across the two ends, and a constant current I flows through the circuit. |
We wish to determine the total resistance RT that exists across the two ends. To do this we make use of the following facts:
(a) If the p.d. across R1, R2 and R3 are V1, V2 and V3 respectively, then we have
$\boxed{V = V_1 + V_2 + V_3}$
20.6.1 Resistances in Series (Continued)
(b) The current flowing through every resistor is the same, i.e. I.
(c) The current flowing through RT must be I and the p.d. across it must be V.
We have
$V_1 = IR_1$, $V_2 = IR_2$, $V_3 = IR_3$ and $V = IR_T$
$IR_T = IR_1 + IR_2 + IR_3$
Hence,
$\boxed{R_T = R_1 + R_2 + R_3+...+R_n}$
20.6.2 Resistances in Parallel
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| Figure 20.37 Three resistors $R_1$, $R_2$ and $R_3$ are connected in parallel between nodes X and Y. A potential difference V is applied across the nodes, and a total current I flows into node X and splits into branch currents. |
Three resistors are connected parallel to each other. A p.d. of V volt is applied across the ends X and Y, as shown in Figure 20.37. They have resistances R1, R2 and R3. A current I flows towards the node X.
We wish to determine the total resistance RT that exists across X and Y. To do this we make use of the following facts:
(a) Kirchhoff’s current law is applicable to node X. We have
$\boxed{I = I_1 + I_2 + I_3}$
(b) The p.d. across every resistor is the same, being V volt.
(c) The current flowing through RT must be I and the p.d. across it must be V.
We have
$I_1 = \frac{V}{R_1}$, $I_2 = \frac{V}{R_2}$, $I_3 = \frac{V}{R_3}$, $I = \frac{V}{R_T}$
$\frac{V}{R_T}= \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$
$\boxed{\frac{1}{R_T}= \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}+...+\frac{1}{R_n}}$
EXAMPLE 20.8
Refer to the circuit shown in Figure 20.38. Determine the total resistance across AB.
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| Figure 20.38 A resistor network consisting of five resistors connected between points A and B. The circuit is used to determine the total resistance across AB. |
Answer
R1 and R2:
Total resistance $r_1 = R_1 + R_2$
$= 5 + 5 = 10 \ \Omega$
R3 || r1:
$\frac{1}{r_2} = \frac{1}{R_3} + \frac{1}{r_1}$
$= \frac{1}{10} + \frac{1}{10}$
$r_2=5.0 \ \Omega$
R4 and r2:
$r_3 = R_4 + r_2 $
$= 15 + 5 = 20 \ \Omega$
R5 || r3:
$\frac{1}{R_{AB}} = \frac{1}{R_5} + \frac{1}{r_3}$
$= \frac{1}{20} + \frac{1}{20}=\frac{1}{10}$
$R_{AB}=10 \ \Omega$
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Application of Resistances in Series and Parallel
Resistances in series and parallel are widely used in electrical and electronic circuits. Understanding how they work helps in designing efficient and safe systems.
1. Application of Resistances in Series
In a series circuit, resistors are connected end-to-end, and the same current flows through each resistor. The total resistance increases as more resistors are added.
- Voltage Division: Used to divide voltage into smaller parts (V = V1 + V2 + V3).
- Heating Devices: Electric heaters and irons use series resistors to control heat.
- Fuses and Safety: Series connections help protect circuits by limiting current.
- Battery Chains: Batteries connected in series increase total voltage.
2. Application of Resistances in Parallel
In a parallel circuit, resistors are connected across the same two points. The voltage across each resistor is the same, but the current is divided.
- Home Electrical Wiring: Appliances are connected in parallel so each gets the same voltage.
- Current Distribution: Current splits into branches (I = I1 + I2 + I3).
- Reliable Circuits: If one component fails, others continue to work.
- Electronic Devices: Used in circuits requiring stable voltage.
3. Combined Applications (Series + Parallel)
Most real circuits use a combination of series and parallel resistances to achieve desired voltage and current values.
- Complex Circuit Design: Used in TVs, radios, and computers.
- Power Distribution Systems: Combination ensures efficiency and safety.
- Voltage and Current Control: Engineers adjust resistance to control circuit behavior.
Conclusion
Resistances in series and parallel play a crucial role in modern electrical systems. Series circuits are useful for controlling current and dividing voltage, while parallel circuits ensure equal voltage distribution and reliability.
Conclusion: Resistances in Series and Parallel
Resistances in series and parallel are fundamental concepts in electrical circuits that determine how current and voltage behave. In a series circuit, resistors are connected in a single path, causing the same current to flow through each component while the total voltage is divided among them. The total resistance is simply the sum of all individual resistances.
In contrast, resistors in parallel are connected across the same two points, resulting in the same voltage across each resistor while the current splits into different branches. The total resistance in a parallel circuit is always less than the smallest individual resistance.
Understanding these two configurations is essential for analyzing and designing electrical systems. Series circuits are useful for controlling current and dividing voltage, while parallel circuits provide reliability and consistent voltage distribution in practical applications such as household wiring and electronic devices.
By mastering resistances in series and parallel, students and engineers can effectively solve circuit problems and build efficient electrical systems.
Frequently Asked Questions (FAQ) – Resistances in Series and Parallel
1. What is the difference between series and parallel resistors?
In a series circuit, resistors are connected in a single path, and the same current flows through each resistor. In a parallel circuit, resistors are connected across the same two points, and the voltage across each resistor is the same while the current splits into different branches.
2. What is the formula for total resistance in series?
The total resistance in a series circuit is the sum of all individual resistances:
$\boxed{R_T = R_1 + R_2 + R_3+...+R_n}$
3. What is the formula for total resistance in parallel?
The total resistance in a parallel circuit is given by:
4. Why is total resistance lower in parallel circuits?
In parallel circuits, multiple paths are available for current to flow. This reduces the overall resistance because the current can pass through several branches instead of just one.
5. Why is the current the same in series circuits?
In a series circuit, there is only one path for current to flow. Therefore, the same current must pass through each resistor.
6. Why is the voltage the same in parallel circuits?
All resistors in a parallel circuit are connected across the same two points, so they experience the same potential difference (voltage).
7. Where are series and parallel circuits used in real life?
Series circuits are used in devices like heaters and voltage dividers, while parallel circuits are commonly used in household wiring to ensure all appliances receive the same voltage.
8. Can series and parallel circuits be combined?
Yes, most real-world circuits use a combination of series and parallel resistors to achieve desired electrical properties.
9. How do you identify a series or parallel circuit?
A series circuit has only one path for current flow, while a parallel circuit has multiple branches connected across the same two points.
10. Which circuit is more reliable, series or parallel?
Parallel circuits are more reliable because if one component fails, the others can continue to operate.
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