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Energy Conservation in Circuits
Energy conservation is a fundamental principle in electrical circuits. Energy cannot be created or destroyed, only transferred between different forms. In circuits, energy is transferred between sources (batteries, power supplies), storage elements (capacitors, inductors), and dissipative elements (resistors).
Power and Energy
⚡ Power Transfer
Power is the rate at which energy is transferred in a circuit.
Power can be supplied by sources or dissipated by loads.
Power Formulas
$$P = IV = I^2R = \frac{V^2}{R}$$
Energy Transfer
$$E = Pt = IVt$$
Power is measured in watts (W) and energy in joules (J). The power dissipated in a resistor is always positive, while power supplied by a source can be positive or negative depending on the direction of current flow.
Example: Power Calculation
Problem: A 12V battery supplies 2A to a circuit. Calculate the power supplied and the energy transferred in 5 minutes.
Step 1: Calculate Power
$$P = IV = (2A)(12V) = 24W$$
Step 2: Calculate Energy
$$E = Pt = (24W)(5 \times 60s) = 7200J = 7.2kJ$$
Answer
Power supplied = 24W, Energy transferred = 7.2kJ
Energy Conservation Principle
🎯 Conservation Law
In any circuit, the total energy supplied equals the total energy dissipated plus the energy stored.
Energy cannot be created or destroyed, only transferred between forms.
Conservation Equation
$$\sum P_{supplied} = \sum P_{dissipated} + \sum P_{stored}$$
Energy Forms in Circuits
- Electrical Energy: Energy stored in electric fields (capacitors)
- Magnetic Energy: Energy stored in magnetic fields (inductors)
- Thermal Energy: Energy dissipated as heat in resistors
- Chemical Energy: Energy stored in batteries
Example: Energy Conservation
Problem: A 10V battery charges a 100μF capacitor through a 1kΩ resistor. Verify energy conservation during the charging process.
Step 1: Calculate Final Stored Energy
$$U_{stored} = \frac{1}{2}CV^2 = \frac{1}{2}(100μF)(10V)^2 = 5mJ$$
Step 2: Calculate Energy Supplied
$$E_{supplied} = CV^2 = (100μF)(10V)^2 = 10mJ$$
Step 3: Calculate Energy Dissipated
$$E_{dissipated} = E_{supplied} - U_{stored} = 10mJ - 5mJ = 5mJ$$
Verification
Energy supplied (10mJ) = Energy stored (5mJ) + Energy dissipated (5mJ) ✓
Answer
Energy conservation is verified: 10mJ = 5mJ + 5mJ
Energy Storage in Circuit Elements
⚡ Energy Storage
Circuit elements can store energy in electric and magnetic fields.
Capacitors store electrical energy, inductors store magnetic energy.
Capacitor Energy Storage
$$U_C = \frac{1}{2}CV^2$$
Inductor Energy Storage
$$U_L = \frac{1}{2}LI^2$$
Energy Transfer During Transients
- Charging: Energy flows from source to capacitor
- Discharging: Energy flows from capacitor to resistor
- Oscillations: Energy oscillates between capacitor and inductor
Example: Energy Storage Analysis
Problem: A 50μF capacitor is charged to 20V, then discharged through a 100Ω resistor. Calculate the energy stored and dissipated.
Step 1: Calculate Stored Energy
$$U_{stored} = \frac{1}{2}CV^2 = \frac{1}{2}(50μF)(20V)^2 = 10mJ$$
Step 2: Energy Dissipation
During discharge, all stored energy is dissipated as heat in the resistor.
Answer
Stored energy = 10mJ, Dissipated energy = 10mJ
Power Analysis in Complex Circuits
Power in Different Elements
- Resistors: Always dissipate power (P = I²R > 0)
- Batteries: Can supply or absorb power depending on current direction
- Capacitors: Store energy, power can be positive or negative
- Inductors: Store energy, power can be positive or negative
Example: Complex Circuit Power Analysis
Problem: In a circuit with a 12V battery, 2Ω resistor, and 4Ω resistor in series, calculate the power supplied and dissipated.
Step 1: Find Current
$$I = \frac{V}{R_{total}} = \frac{12V}{6Ω} = 2A$$
Step 2: Calculate Power Supplied
$$P_{supplied} = IV = (2A)(12V) = 24W$$
Step 3: Calculate Power Dissipated
$$P_{R1} = I^2R_1 = (2A)^2(2Ω) = 8W$$
$$P_{R2} = I^2R_2 = (2A)^2(4Ω) = 16W$$
$$P_{total} = P_{R1} + P_{R2} = 8W + 16W = 24W$$
Verification
Power supplied (24W) = Power dissipated (24W) ✓
Answer
Power supplied = 24W, Power dissipated = 24W
Efficiency Considerations
Power Efficiency
$$\text{Efficiency} = \frac{P_{useful}}{P_{total}} \times 100\%$$
Energy Efficiency
$$\text{Efficiency} = \frac{E_{useful}}{E_{total}} \times 100\%$$
Factors Affecting Efficiency
- Resistance: Higher resistance increases power dissipation
- Current: Higher current increases power dissipation
- Voltage Drop: Voltage drops across components reduce efficiency
- Heat Loss: Energy lost as heat reduces efficiency
Example: Efficiency Calculation
Problem: A motor draws 5A from a 24V supply and delivers 100W of mechanical power. Calculate the efficiency.
Step 1: Calculate Input Power
$$P_{input} = IV = (5A)(24V) = 120W$$
Step 2: Calculate Efficiency
$$\text{Efficiency} = \frac{100W}{120W} \times 100\% = 83.3\%$$
Answer
The motor efficiency is 83.3%.
Practical Applications
Power Distribution
- Transmission Lines: Minimize power loss over long distances
- Power Supplies: Efficient conversion of electrical energy
- Battery Systems: Optimal energy storage and delivery
Energy Storage
- Capacitor Banks: Store electrical energy for later use
- Battery Systems: Chemical energy storage
- Flywheels: Mechanical energy storage
Energy Recovery
- Regenerative Braking: Recover energy during deceleration
- Heat Recovery: Convert waste heat to useful energy
- Energy Harvesting: Capture ambient energy sources
Key Takeaways
- Conservation: Total energy supplied equals total energy dissipated plus stored
- Power Transfer: P = IV = I²R = V²/R for different circuit elements
- Energy Storage: Capacitors store electrical energy, inductors store magnetic energy
- Efficiency: Ratio of useful power/energy to total power/energy
- Practical Importance: Essential for power system design and optimization
- Verification Tool: Energy conservation provides a check for circuit analysis