20.4 A Complex Circuit
20.4.1 A Complex Resistive Circuit
Refer to Figure 20.10. It shows a circuit which is slightly more complex than the earlier one. It consists of a battery and several resistors.
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| Figure 20.10: A slightly more complex electrical circuit consisting of a battery and several resistors, illustrating how currents and voltages are distributed across different components in a network. |
In general, a complex circuit may be made up of several sources of electrical energy, many resistors, capacitors, inductors, diodes, transistors, and IC chips. Such complex circuits can be found on a computer motherboard.
20.4.2 Currents and Voltages in a Circuit
There are a few important points to note about the circuit shown above:
(a) Concerning Currents
(i) More than two components may join at a point. For example, three resistors join at point B, at point D, and also at points F and X. Each of these points is known as a node or junction.
(ii) Some currents may flow towards a node while others flow away from the same node at the same time. For example, at node X, one current flows towards X while two currents flow away from X.
(b) Concerning Voltage
(i) We can choose a battery together with some components, or just components alone (without battery), to form a closed path. This closed path is known as a loop.
Example of loops: ABXFGA, ABCDFGA, ABCDXFGA, BCDXB, BCDEFXB, XDEFX.
(ii) A p.d. is set up across every component found in a loop.
Example: In loop BCDXB, there are four resistors. A p.d. exist across each resistor. The p.d.'s are $V_2$, $V_3$, $V_4$, and $V_5$.
(c) Determining a particular p.d.
(i) suppose we wish to calculate a particular p.d., like $V_5$. One method to do that is to choose a loop that has $V_5$ in it, like loop BCDXB. Besides choosing this loop, we need also to choose a few other loops. After that we apply Kirchooff's laws to all those chosen loops.
20.5 Kirchhoff's Laws
20.5.1 Kirchhoff's Current Law
1. Definition
The Current Law states that:
The algebraic sum of all the currents at a node in a circuit is zero, i.e.,
$\boxed{\sum_{i=1}^{n}I_i = 0}$
i.e., $\boxed{I_1+I_2+I_3+...+I_n = 0}$
2. Sign Convention for Currents
Before we apply the law, we need to distinguish currents flowing towards a node from those flowing away from the node. To do that:
we assign a positive (or negative) sign to a numerical value of each current that flows towards a node. Then, a negative (or positive) sign will be assigned to the numerical value of a current that flows away from the same node.
Example 20.3
Refer to Figure 20.11. Point X is a node in a circuit. Two currents flow towards it while three currents flow away from it. The magnitudes of currents I1, I2, I4, and I5 are 5 A, 2 A, 4 A, and 8 A respectively.
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| Figure 20.11. Point X is a node in an electrical circuit where two currents flow into the node and three currents flow out of it, illustrating the conservation of current at a junction. |
Determine the magnitude of current I3. Is the direction of current flow for I3 shown in the diagram correct?
Answer
Applying Kirchhoff’s current law to the currents at node X, we get:
$I_1+ I_2 + I_3 + I_4 + I_5 =0$
Suppose currents flowing towards X have positive values. Then we have:
$(+5) + (−2) + I3 + (+4) + (−8) = 0$
$I_3 = +1 \ A$
Since the value of $I_3$ is positive, it means that the current is flowing towards node X. Hence, the direction shown in the diagram is incorrect.
Example 20.4
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| Figure 20.12 The diagram below represents a section of an electrical circuit with five junctions. |
Answer
Figure 20.13 illustrates the labeling of five junctions (A, B, C, D, and E) to facilitate the application of Kirchhoff’s current law.
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| Figure 20.13 illustrates the labeling of five junctions (A, B, C, D, and E). |
At junction A:
There are three paths connected to junction A. Since two currents of 1 A and 2 A flow into the junction, the remaining path must carry current leaving the junction (from A to C) according to Kirchhoff’s current law. Therefore, the outgoing current is equal to the sum of the incoming currents.
Suppose currents flowing towards A have positive values. Then we have:
$1 + 2 + I_Y=0$
$I_Y=-3 \ A$
Thus, the magnitude of the current $I_Y = 3 \ A$ flows away from junction A toward point C.
At junction B:
There are three paths connected to junction B. Since two currents of 2 A and 4 A flow away from the junction, the remaining path must carry current flowing into the junction (from B to C) according to Kirchhoff’s current law.
Suppose currents flowing towards B have positive values. Then we have:
$(-2) + (-4) + I_Z=0$
$I_Z=+6 \ A$
Thus, the magnitude of the current $I_Z = 6 \ A$ flows from point C toward point B.
At junction C:
There are three paths connected to junction C. Since a current of $I_Y=3 \ A$ flows into the junction and a current of $I_Z=6 \ A$ flows away from the junction. Suppose currents flowing towards C have positive values. Then we have:
$(+3) + (-6) + I_X=0$
$I_X=+3 \ A$
Thus, the magnitude of the current $I_X = 3 \ A$ flows from point C toward point D.
At junction D:
There are three paths connected to junction D. A current of $I_X = 3 \ A$flows away from the junction, while a current of 4 A flows into the junction. Suppose currents flowing towards D have positive values. Then we have:
$(-3) + (+4) + I_P=0$
$I_P=-1 \ A$
Thus, the magnitude of the current $I_P = 1 \ A$ flows away from junction D toward point E.
At junction E:
There are three paths connected to junction E. Two currents, $I_P=1 \ A$ and 2 A, flow into the junction. Suppose currents flowing towards E have positive values. Then we have:
$(+1) + (+2) + I=0$
$I=-3 \ A$
Thus, the magnitude of the current $I=3 \ A$ flows away from junction E.
Example 20.5
The diagram below represents a section of an electrical circuit with six junctions. Determine the magnitudes and directions of the currents $i_b$, $i_d$, $i_e$ and $i_f$!
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| Figure 20.14 The diagram below represents a section of an electrical circuit with six junctions. |
Figure 20.15 illustrates the labeling of six junctions (P, Q, R, X, Y and Z) to facilitate the application of Kirchhoff’s current law.
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| Figure 20.15 illustrates the labeling of six junctions (P, Q, R, X, Y and Z). |
$i_a + i_b + i_c=0$
$(-2)+ i_b+ (-3)=0$
$i_b=+5 \ A$
Thus, the magnitude of the current $i_b= 5 \ A$ flows from point X toward point Z.
At junction P:
There are three paths connected to junction P. Two currents, $i_c=3 \ A$ and $i_h=1 \ A$, flow into the junction. Therefore, the current in the remaining path, $i_e$, flows away from junction P toward point R to satisfy Kirchhoff’s current law. Therefore, the outgoing current is equal to the sum of the incoming currents.
Suppose currents flowing towards P have positive values. Then we have:
$i_c + i_e + i_h=0$
$(+3)+ (+1) + i_e=0$
$i_e=-4 \ A$
Thus, the magnitude of the current $i_e = 4 \ A$ flows away from junction P toward point R.
At junction Q:
There are three paths connected to junction Q. Two currents, $i_g=6 \ A$ and $i_h=1 \ A$, flow away from the junction. Therefore, the current in the remaining path, $i_f$, flows into junction Q to satisfy Kirchhoff’s current law. Suppose currents flowing towards Q have positive values. Then we have:
$i_f+i_g+i_h=0$
$i_f + (-6) + (-1)=0$
$i_f=+7 \ A$
Thus, the magnitude of the current $i_f = 7 \ A$ flows from point Y toward point Q.
At junction Y:
There are three paths connected to junction Y. A current of $i_a = 2 \ A$ flows into the junction, while a current of $i_f = 7 \ A$ flows away from the junction. Suppose currents flowing towards Y have positive values. Then we have:
$i_a+i_d+i_f=0$
$(+2) + i_d + (-7)=0$
$i_d=+5 \ A$
Thus, the magnitude of the current $I_X = 3 \ A$ flows from point X toward point Y.
20.5.2 Potential Rise and Potential Fall in a Loop
1. Encountering Potential Rise and Potential Fall as We Move Around a Loop
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| Figure 20.16 A simple one-loop circuit with a battery and two resistors showing the potential rise across the battery and the potential drops across the resistors. |
(b) Let us move along this circuit, starting from point J on the circuit. We move through the battery to K, then from L to M through resistor $R_1$, and from N to P through resistor $R_2$. Finally, we move from P back to J along the conductor below those three components to complete the loop.
(c) When we have completed the loop, we would have encountered three voltages, namely:
(i) E, the emf of the battery
(ii) $V_1$, the p.d. across $R_1$
(iii) $V_2$, the p.d. across $R_2$
(d) As we move through each component, we may experience a rise or a fall in the voltage which exists across the component. In the direction of motion along the loop mentioned above, we experience:
(i) a potential rise of E volts as we move from the negative terminal to the positive terminal of the battery
(ii) a potential drop of $V_1$ volts as we move from L to M
(iii) a potential drop of $V_2$ volts as we move from N to P
Such potential rise and potential fall along the loop are shown in Figure 20.12
2. Determining Potential Rise or Fall
(a) D.C. Source of Electrical Energy
Refer to Figure 20.17(a). If we move from the negative terminal to the positive terminal through a battery of emf $E$ volt, we will encounter a potential rise of $E$ volt.
%20a%20battery%20and%20(b)%20a%20resistor..png) |
| Figure 20.17 (a) Potential rise across a battery when moving from the negative terminal to the positive terminal. (b) Potential rise across a resistor when moving opposite to the direction of current flow. |
(b) Across Resistors
When a current I flows through a resistor of resistance R, a p.d. of V volt, where:
$V = IR$
will be produced across the resistor. Refer to Figure 20.17(b). If we move through the resistor in a direction which is opposite to the direction of current flow, we will encounter a potential rise of V volt.
3. Voltage Arrow
In order to help us recognise potential rise and potential fall easily in a circuit, we can draw voltage arrows, as follows:
(a) Draw an arrow beside each component found in a loop which is of interest to us.
(b) (i) For a battery, draw an arrow that points at the positive terminal. We will experience a potential rise if we move through the battery following the direction of the arrow (from negative terminal to positive terminal). Conversely, if we move in the opposite direction (from positive terminal to negative terminal), we will experience a potential fall.
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| Figure 20.18 Arrows showing the direction of potential rise for a battery and a resistor. |
(ii) For a resistor, draw an arrow that points in the direction of current entering the resistor. We will experience a potential rise if we move through the resistor following the direction of the arrow.
Figure 20.18 shows two such arrows.
Applications in Real Life Circuits
1. Application of A Complex Circuit
Complex circuits are widely used in modern electronic devices. These circuits consist of multiple components such as resistors, capacitors, transistors, and integrated circuits working together to perform specific functions.
For example, in a computer motherboard, complex circuits are used to control data processing, memory storage, and communication between hardware components. Each node in the circuit connects multiple components, allowing current to be distributed efficiently throughout the system.
Understanding complex circuits helps engineers design reliable electronic systems such as smartphones, laptops, and industrial control systems.
2. Application of Kirchhoff’s Current Law (KCL)
Kirchhoff’s Current Law is very useful in analyzing electrical circuits, especially at junctions (nodes) where multiple currents meet.
In practical applications, KCL is used to:
- Analyze current distribution in electrical networks
- Design and troubleshoot electronic circuits
- Ensure proper current flow in power supply systems
For example, in household electrical wiring, KCL helps ensure that the total current entering a junction is equal to the total current leaving it, preventing overload and ensuring safety.
Engineers also use KCL in circuit simulation software to calculate unknown currents in complex networks.
3. Application of Potential Rise and Potential Fall in a Loop
The concept of potential rise and potential fall is important in understanding how energy is transferred within an electrical circuit.
In real-world applications:
- Batteries provide a potential rise, supplying energy to the circuit
- Resistors and other components cause a potential fall, consuming electrical energy
This principle is used in designing circuits such as:
- Battery-powered devices like flashlights and remote controls
- Power distribution systems
- Electronic devices where voltage must be controlled across components
By understanding potential rise and fall, engineers can ensure that electrical energy is properly distributed and that components operate within safe voltage limits.
Conclusion
In conclusion, understanding complex circuits, Kirchhoff’s Current Law, and the concept of potential rise and potential fall is essential in the study of electrical circuits.
A complex circuit consists of many interconnected components, where currents and voltages interact at different nodes and loops. Kirchhoff’s Current Law helps us analyze how current is distributed at junctions, ensuring that the total current entering a node is equal to the total current leaving it.
Meanwhile, the concept of potential rise and potential fall explains how electrical energy is supplied and used within a loop. A source such as a battery provides energy, while components like resistors consume that energy.
By mastering these principles, students and engineers can better analyze, design, and troubleshoot electrical circuits in both simple and advanced applications found in everyday electronic devices.
Frequently Asked Questions (FAQ)
1. What is a complex circuit?
A complex circuit is an electrical circuit that contains multiple components such as resistors, batteries, capacitors, and other electronic devices connected in various configurations. It usually has multiple nodes and loops, making the analysis more advanced than simple circuits.
2. What is a node in a circuit?
A node (or junction) is a point in a circuit where two or more components are connected. At a node, currents can either enter or leave the junction.
3. What does Kirchhoff’s Current Law (KCL) state?
Kirchhoff’s Current Law states that the total current entering a node is equal to the total current leaving the node. In other words, the algebraic sum of currents at a node is zero.
4. Why is Kirchhoff’s Current Law important?
KCL is important because it helps in analyzing and solving complex circuits. It ensures that current is conserved at every junction, which is essential for designing safe and efficient electrical systems.
5. What is meant by potential rise in a circuit?
Potential rise refers to an increase in electrical potential energy as charge moves through a component, such as a battery, from the negative terminal to the positive terminal.
6. What is potential fall?
Potential fall (or voltage drop) is the decrease in electrical potential energy as charge moves through components like resistors, where energy is used or dissipated.
7. What is the difference between potential rise and potential fall?
Potential rise occurs when energy is supplied to the charges (e.g., by a battery), while potential fall occurs when energy is used or lost in components like resistors.
8. Where are these concepts used in real life?
These concepts are used in everyday electronic devices such as smartphones, computers, power supplies, and household electrical systems to ensure proper current flow and voltage distribution.
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