Electrical circuit

1. Series and parallel circuit:

• a) Kirchhoff’s second law; the conservation of energy:
• Kirchhoff’s second law states that in any closed loop, the sum of the e.m.f. is equal to the sum of the products of the current and the resistance. In equation form, this is:
• [math]\Sigma \varepsilon = \Sigma I R[/math]
• Kirchhoff’s second law is a consequence of the conservation of energy.
• 
• Figure 1 The e.m.f. provided by the cell must be the same as the total drop in potential differences across all of the resistors in the loop
• Kirchhoff’s Second Law, also known as the Conservation of Energy, states that the total energy entering a system is equal to the total energy leaving the system, plus any energy stored or dissipated within the system.
• Mathematically, this is expressed as:
• [math]\sum E_{\text{in}}=\sum E_{\text{out}}+\sum E_{\text{stored}}+\sum E_{\text{dissipated}}[/math]
• Where:
• [math]E_{\text{in}}[/math]is the total energy entering the system
• [math] E_{\text{out}}[/math]is the total energy leaving the system
• [math]E_{\text{stored}}[/math]is the energy stored within the system (e.g., in capacitors, inductors)
• [math]E_{\text{dissipated}}[/math]is the energy dissipated within the system (e.g., as heat, due to resistance)
• This law applies to any closed system, and is a fundamental principle in physics and engineering.
• In electrical circuits, Kirchhoff’s Second Law ensures that:
• The sum of the voltage rises (energy gained) around a loop is equal to the sum of the voltage drops (energy lost).
• The total power entering a circuit is equal to the total power leaving, plus any power stored or dissipated.
• b) Kirchhoff’s first and second laws applied to electrical circuits:
• When using Kirchhoff’s first and second laws, you must be careful about the directions that the currents are travelling in.
• For more complex problems involving two sources of e.m.f. you need to make sure that the current flowing in one direction is taken as positive and the current flowing in the opposite direction is taken as negative.
• For example, you may decide that the current flowing clockwise is positive, and so any current flowing anticlockwise must be taken as negative.
• Kirchhoff’s Laws are fundamental principles in electrical circuit analysis. Here’s how Kirchhoff’s First and Second Laws apply to electrical circuits:
• Kirchhoff’s First Law (KVL):
• – Also known as the Voltage Law
• – States that the sum of voltage rises (energy gained) around a closed loop is equal to the sum of voltage drops (energy lost)
•  Mathematically:
• [math]\sum V_{\text{rises}} = \sum V_{\text{drops}}[/math]
• – Applies to any closed loop in a circuit
• Kirchhoff’s Second Law (KCL):
• – Also known as the Current Law
• – States that the total current entering a node (junction) is equal to the total current leaving that node
•  Mathematically:
• [math]\sum I_{\text{in}} = \sum I_{\text{out}}[/math]
• – Applies to any node in a circuit
• Kirchhoff’s Second Law (KVL) – Conservation of Energy:
• – States that the total energy entering a circuit is equal to the total energy leaving, plus any energy stored or dissipated
• Mathematically:
• [math]\sum E_{\text{in}} = \sum E_{\text{out}} + \sum E_{\text{stored}} + \sum E_{\text{dissipated}}[/math]
• – Applies to any closed system (circuit)
• By applying these laws, you can:
• – Analyze complex electrical circuits
• – Calculate unknown voltages, currents, and resistances
• – Understand energy transfer and conservation in circuits
• – Design and optimize electrical systems

c) Total resistance of two or more resistors in series;[math]R = R_1 + R_2 + R_3 + \dots[/math]

• Consider the resistor network shown in Figure 2 and 3.
• Figure 3 represents the single resistor that could replace the three resistors in series in Figure 2.
• Using Kirchoff’s Circuit laws and Ohm’s law leads to:
• [math]\varepsilon = V_1 + V_2 + V_3[/math]
• 
  Figure 2 Circuit diagram showing resistor combinations in series.
• and
• [math]\varepsilon = V_T[/math]
• where;
• [math]V_T = V_1 + V_2 + V_3 \qquad (1)[/math]
• So,
• [math]V=IR[/math]
• Put in equation according to resistor and volt then
• [math]IR_T = IR_1 + IR_2 + IR_3[/math]
• [math]IR_T = I(R_1 + R_2 + R_3)[/math]
• So,
• [math]R_T=R_1+ R_2 + R_3[/math]
• and for a series network of n resistors:
• 
  Figure 3 Circuit diagram showing equivalent resistor
• [math]R_T = R_1 + R_2 + R_3 + \dots + R_n[/math]
• or, using sigma notation
• [math]R_T = \sum_{i=1}^n R_i[/math]
• d) Total resistance of two or more resistors in parallel;[math]\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots[/math]
• Consider the following circuit:
• 
• Figure 10 Circuit diagram showing resistors connected in parallel
• In the right-hand circuit one resistor, RT, has been used to replace all three resistors arranged in parallel in the left-hand circuit.
• Again, using Kirchoff’s Circuit laws and Again, using Kirchoff’s Circuit laws and the definition of resistance,
• [math]V=IR[/math]
• Kirchoff’s First Circuit law says:
• [math]I_T = I_1 + I_2 + I_3[/math]
• And
• [math]I = \frac{V}{R}[/math]
• So as the potential difference, , is the same across all of the resistors:
• [math]I_T = \frac{V_T}{R_1} + \frac{V_T}{R_2} + \frac{V_T}{R_3}[/math]
• [math]I_T = V_T \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right)[/math]
• Rearranging
• [math]\frac{I_T}{V_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}[/math]
• [math]\frac{I_T}{V_T} = \frac{1}{R_T}[/math]
• So,
• [math]\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}[/math]
• For a network of n resistors connected in parallel
• [math]\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}[/math]
• or using sigma notation:
• [math]\frac{1}{R_T} = \sum_{i=1}^n \frac{1}{R_i}[/math]
• To summarize, the total resistance of a series of resistors is equal to the sum of its individual resistances.
• The reciprocal of the total resistance for resistors connected in parallel is equal to the reciprocal of each individual resistance.
• The resistance of a parallel arrangement of resistors is always smaller than the resistance of any individual resistor in that combination.

e) Analysis of circuits with components, including both series and parallel:

• The analysis of circuits with components, including both series and parallel circuits.
• ⇒ Series Circuit:
• A series circuit has components connected one after the other, forming a single loop.
• Characteristics:
• – Current is the same throughout the circuit
• – Voltage is divided among components
• – If one component fails, the entire circuit is broken
• Analysis:
• – Use Kirchhoff’s Voltage Law (KVL) to find the voltage across each component
• – Use Ohm’s Law (V=IR) to find the current and resistance of each component
• ⇒ Parallel Circuit:
• A parallel circuit has components connected between the same two nodes, forming multiple paths.
• Characteristics:
• – Voltage is the same across each component
• – Current is divided among components
• – If one component fails, the other components remain functional
• Analysis:
• – Use Kirchhoff’s Current Law (KCL) to find the current through each component
• – Use Ohm’s Law (V=IR) to find the voltage and resistance of each component
• ⇒ Combined Series-Parallel Circuit:
• A circuit with both series and parallel components.
• Analysis:
• – Use KVL and KCL to identify series and parallel components
• – Analyze each series and parallel section separately
• – Combine the results to find the overall circuit behavior
• By analyzing circuits with components in both series and parallel configurations, you can:
• – Understand how current and voltage distribute throughout the circuit
• – Calculate component values and circuit behavior
• – Design and optimize electrical systems
• f) Analysis of circuits with more than one source of e.m.f.
• Analyzing circuits with multiple sources of electromotive force (e.m.f.) can be a bit more complex, but it’s still manageable using the techniques we’ve discussed.
• – Identify the sources of e.m.f. and their polarities.
• – Apply Kirchhoff’s Voltage Law (KVL) to each loop in the circuit.
• – Use the principle of superposition to analyze the effect of each source separately.
• – Combine the results to find the overall circuit behavior.
• Techniques:
• – Mesh Current Analysis: Use KVL to find the currents in each mesh (loop).
• – Nodal Voltage Analysis: Use KCL to find the voltages at each node (junction).
• – Thevenin’s Theorem: Convert each source and its series resistance to a Thevenin equivalent.
• – Norton’s Theorem: Convert each source and its parallel resistance to a Norton equivalent.
• Label each source and its polarity clearly.
• – Use different colors or symbols to distinguish between the sources.
• – Check your work by verifying that the total current and voltage at each node are consistent with the circuit behavior.
• By following these steps and techniques, you can analyze circuits with multiple sources of e.m.f. and gain a deeper understanding of how they behave.

2. Internal resistance:

a) Source of e.m.f.; internal resistance:

• Electromotive Force (EMF):
• – EMF is the voltage generated by a cell or battery when no current is flowing through it.
• – Measured in volts (V), it’s the “open-circuit voltage” of a cell or battery.
• – EMF is the maximum voltage a cell or battery can provide.
• Internal Resistance:
• – Internal resistance is the opposition to current flow within a cell or battery.
• – Measured in ohms (Ω), it depends on the cell’s or battery’s chemistry, age, and other factors.
• – Internal resistance causes a voltage drop when current flows, reducing the available voltage.
• Relationship between EMF and Internal Resistance:
• – When current flows, the internal resistance reduces the voltage available from the EMF.
• – The higher the internal resistance, the greater the voltage drop.
• – The lower the internal resistance, the closer the available voltage is to the EMF.
• Internal resistance, or r, is a constant feature of real power sources, such batteries and lab power packs.
• The power supply’s internal resistance produces a potential difference as current passes through it, which causes electrical energy to be converted to heat energy.
• This is one of the causes of the warming up of portable electronics like tablets after extended usage.
• Since internal resistance is located “inside” a power source, it cannot be tested directly.
• It can only be measured by applying its electrical characteristic.
• Figure 5 illustrates a circuit that may be used to do this.
• 
• Figure 5 A circuit used to measure the internal resistance and electromotive force of a real power supply and a real power supply
• According to Kirchoff’s Second Circuit Law, the electromotive force, ε, must equal the total of the potential differences in the circuit.
• In this circuit, there are two potential differences: the pd across the internal resistor and the one across the external variable resistor (V).
• The pd across this resistor is equal to Ir, however this cannot be determined precisely.
  This implies that:
• [math]\varepsilon = V + Ir[/math]
• The current, I, can be measured directly using an ammeter; ε and r are both constants, so the equation can be rewritten as:
• [math]V = \varepsilon – Ir[/math]
• Or
• [math]V = -rI + \varepsilon[/math]
• It is the equation for a straight line with a negative gradient that looks like this: y = mx + c.
• The electromotive force, ε, is the graph’s y-intercept, and the gradient is negative and equal to –r, the internal resistance, if an electrical characteristic is created using values of V and I from different values of R (the external load resistance).
• ⇒ Voltmeters and ammeters:
• Voltmeters and ammeters are essential instruments in electronics for measuring voltage and current, respectively.
• Voltmeters:
• – Measure the voltage between two points in a circuit.
• – Typically connected in parallel with the component or circuit being measured.
• Types:
• – Analog voltmeters (pointer-type)
• – Digital voltmeters (display numerical values)
• – Units: Volts (V)
• Ammeters:
• – Measure the current flowing through a circuit or component.
• – Typically connected in series with the component or circuit being measured.
• Types:
• – Analog ammeters (pointer-type)
• – Digital ammeters (display numerical values)
• Units: Amperes (A)
• ⇒ Techniques and procedures used to determine the internal resistance of a chemical cell or other source of e.m.f.
• To determine the internal resistance of a chemical cell or other source of e.m.f., the following techniques and procedures can be used:
• – Open-Circuit Voltage Measurement: Measure the voltage across the cell when it’s not connected to any load (open-circuit). This gives the electromotive force (e.m.f.) of the cell.
• – Short-Circuit Current Measurement: Connect a low-resistance ammeter across the cell and measure the current when the cell is short-circuited. This gives the short-circuit current.
• – Load Test: Connect a variable load (resistor) across the cell and measure the voltage and current at different load settings. Plot the voltage vs. current graph and find the internal resistance (slope of the graph).
• – Wheatstone Bridge: Use a Wheatstone bridge configuration to measure the internal resistance of the cell.
• – AC Bridge Method: Use an AC bridge circuit to measure the internal resistance of the cell.
• – Impedance Measurement: Measure the impedance of the cell using an impedance analyzer or a vector impedance meter.
• – Current-Voltage (I-V) Curve: Plot the I-V curve of the cell and find the internal resistance from the slope of the curve.
• – Electrochemical Impedance Spectroscopy (EIS): Use EIS to measure the internal resistance and other electrochemical properties of the cell.

3. Potential divisor:

• a) Potential divider circuit with components:
• A fundamental concept in electronics, used to reduce a voltage level or divide a voltage ratio.
• A potential divider is a resistor network that divides an input voltage ([math]V_{\text{in}}[/math]) into a smaller output voltage ([math]V_{\text{out}}[/math]). It consists of two resistors,[math]R_1[/math] and [math]R_2[/math] , connected in series.
• Working:
• The input voltage ([math]V_{\text{in}}[/math] ) is applied across the series combination of [math]R_1[/math] and [math]R_2[/math].
• The voltage across[math]R_2(V_{\text{out}})[/math] is proportional to the ratio of to the total resistance [math](R_1 + R_2)[/math].
• By adjusting [math]R_1[/math] and [math]R_2[/math], the output voltage ([math]V_{\text{out}}[/math] ) can be set to a specific value.
• Types:
• Fixed Potential Divider: Uses fixed resistors to divide the voltage.
• Variable Potential Divider: Uses a potentiometer (variable resistor) to adjust the output voltage.
• The potential divider formula assumes ideal resistors and neglects loading effects.
• In practice, the output voltage may vary due to resistor tolerances and loading.
• 
• Figure 6 Circuit diagram of a potential divider
• Generally, we are trying to vary the potential difference [math]V_1 \, (\text{across } R_1)[/math] by varying[math]R_2[/math] .
• Assuming that the cell has negligible internal resistance, then the total resistance of the circuit [math][/math] is given by:
• [math]R_T = R_1 + R_2[/math]
• by using the Ohm’s law
• [math]I = \frac{V}{R} = \frac{\varepsilon}{R_T} = \frac{\varepsilon}{R_1 + R_2}[/math]
• Then considering the resistor R1 we can write:
• [math]V_1 = I R_1[/math]
• Substituting:
• [math]V_1 = \frac{\varepsilon}{R_1 + R_2} \times R_1[/math]
• [math]V_1 = \frac{\varepsilon R_1}{R_1 + R_2}[/math]
• From this equation it can be seen that if ε and [math]R_1[/math] are fixed, then [math]V_1[/math]  only depends on [math]R_2[/math] . In fact, as [math]R_2[/math]  increases,[math]V_1[/math]  decreases, and vice versa.
• ⇒ Potential dividers as sensors:
• Potential dividers can be used as sensors to measure various physical parameters, such as:
• 1. Temperature: By using a thermistor or a temperature-dependent resistor in the potential divider circuit, the output voltage can be made to vary with temperature.
• 2. Light: A photoresistor or light-dependent resistor can be used to measure light intensity.
• 3. Pressure: A pressure-dependent resistor or a piezoresistor can be used to measure pressure changes.
• 4. Position: A potentiometer can be used as a position sensor, where the output voltage indicates the position of the wiper.
• 5. Force: A force-sensing resistor or a piezoresistor can be used to measure force or weight.
• 6. Humidity: A humidity-dependent resistor can be used to measure humidity levels.
• 7. pH: A pH-dependent resistor can be used to measure the acidity or basicity of a solution.
• The potential divider sensor circuit typically consists of:
• 1. A voltage source ([math]V_{\text{in}}[/math])
• 2. A sensor resistor ( [math]R_{\text{sense}}[/math]) that changes value with the physical parameter being measured
• 3. A fixed resistor ([math]R_{\text{fixed}}[/math] )
• 4. A voltage output ([math]V_{\text{out}}[/math] ) that varies with the sensor resistance
• By measuring the output voltage ([math]V_{\text{out}}[/math] ), the physical parameter can be inferred. Potential divider sensors are simple, low-cost, and widely used in various applications.
• 
• Figure 7 A thermistor potential divider circuit and the resistance–temperature graph for a ntc thermistor
• [math]R_2[/math] is replaced by a thermistor coupled to an electronic thermometer.
• Negative temperature coefficient, or NTC, thermistors make up the majority of thermistors.
• This indicates that, as the temperature rises, their resistance falls, as seen in Figure 12, which also includes a circuit schematic illustrating the connections between the parts:
• As the temperature rises, the resistance of the thermistor,[math]R_2[/math], decreases and [math]V_1[/math] increases, which has an impact on the potential difference across the fixed resistor [math]R_1[/math] in figure 11.
• As a result, rising temperatures lead to rising pd. Conversely, if the voltmeter is connected across the thermistor, a rise in temperature will result in a fall in pd.
• The voltmeter is often connected across the fixed resistor since most applications call for the pd to rise with temperature.
• Using a light-dependent resistor yields a comparable result (LDR).
• These parts serve as excellent light sensors since they alter in resistance in response to changes in light intensity.
• Since semi-conducting materials are used to make LDRs, light may flow through them and release electrons from their structural bonds, lowering the LDR’s resistance.
• LDRs can have resistances as high as megaohms while operating in the dark, but they can also have resistances as low as a few hundred ohms when operating in the light.
• When an LDR is used in place of [math]R_2[/math] in a potential divider circuit, the output voltage [math]V_1[/math] across the fixed resistor will grow in tandem with the intensity of the light.
• 
• Figure 8 An LDR: its electrical circuit symbol (a) resistance– light intensity graph (b) and its use in a potential divider circuit (c).
• ⇒ Techniques and procedures used to investigate potential divider circuits which may include a sensor such as a thermistor or an LDR.
• To investigate potential divider circuits with a sensor like a thermistor or LDR, follow these techniques and procedures:
• – Circuit Assembly: Build the potential divider circuit with the sensor (thermistor or LDR) and two resistors.
• – Voltage Division: Apply a voltage source ([math]V_{\text{in}}[/math]) and measure the output voltage ([math]V_{\text{out}}[/math]) using a multimeter.
• – Sensor Characterization: Measure the resistance of the sensor (thermistor or LDR) at different temperatures or light levels.
• – Circuit Analysis: Calculate the output voltage ([math]V_{\text{out}}[/math]) using the voltage division rule and compare it with the measured value.
• – Sensor Calibration: Create a calibration curve for the sensor by plotting its resistance vs. temperature or light level.
• – Circuit Simulation: Use software like SPICE or Multisim to simulate the circuit and compare the results with the experimental data.
• – Parameter Sweep: Vary the resistances or voltage source and measure the output voltage to understand the circuit’s behavior.
• – Data Acquisition: Use a data logger or oscilloscope to record the output voltage and sensor readings.
• – Data Analysis: Plot the data and analyze the results to understand the circuit’s behavior and sensor characteristics.
• Some specific techniques for thermistors and LDRs:
• Thermistor:
• – Measure the temperature coefficient of resistance (TCR)
• – Calculate the thermistor’s resistance at different temperatures
• LDR:
• – Measure the light sensitivity and dark resistance
• – Calculate the LDR’s resistance at different light levels