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ES
Elements of Electrical Engineering
Lecture
Jeffrey Miller, Ph.D.
Outline
• Chapter 6.1-6.
Inductors
- An electrical component that opposes any change in electrical current
- Composed of a coil of wire wound around a supporting core whose material may be magnetic or non-magnetic
- A time-varying magnetic field induces a voltage in any conductor linked by the field
- Inductance
- Relates the induced voltage to the current
Inductors
- Inductance is the circuit parameter used to describe an inductor
- The letter L is used for inductance, measured in henrys (H) - henry => Wb/A => V * s / A
Inductors
v = L di/dt where v is in volts, L in henrys, i in amps, and t in seconds
If the current is in the opposite direction of the voltage drop across the inductor, the equation will be negated
Inductors
- Looking at the equation v = L di/dt
- If the current is constant...
- The voltage across the inductor is 0
- So the inductor behaves as a short-circuit in the presence of a constant (DC) current
- Current cannot change instantaneously in an inductor because that would require an infinite voltage
Inductor Sample Problem
- The independent current source in the circuit below generates no current for t < 0 and a pulse 10te-^5 t^ A for t > 0 - At what instant of time is the current maximum? - Express the voltage across the inductor as a function of time.
Inductor Sample Problem
To find when the current is maximum, find the derivative of the current with respect to time i = 10te-5t di/dt = 10 (1 * e-5t^ + t * (-5e-5t) ) = 10 e-5t^ (1 – 5t) The current is maximum when di/dt = 0 10 e-5t^ (1 – 5t) = 0 10 e-5t^ can never be 0, so 1 - 5t = 0 t = 1/5 seconds
Power and Energy in the Inductor
Power p = vi v = L di/dt p = Li di/dt
Energy p = dw/dt = Li di/dt dw = Li di Integrate both sides w = ½ Li^2
Capacitors
- An electrical component that consists of two conductors separated by an insulator or dielectric material, which stores electrical charge
- A time-varying electric field produces a displacement current in the space occupied by the electric field
- Capacitance
- The displacement current is equal to the conduction current at the terminals of the capacitor
Capacitors
- Capacitance is the circuit parameter used to describe a capacitor
- The letter C is used for capacitance, measured in farads (F) - farad => C/V => A * s / V
Capacitors
- Applying a voltage to the terminals of a capacitor cannot move a charge through the dielectric, but it can displace a charge within the dielectric
- As the voltage varies with time, the displacement of charge also varies with time, causing displacement current
- The current is proportional to the rate at which the voltage across the capacitor varies with time i = C dv/dt
Capacitors
- The current i = C dv/dt abides by the passive sign convention, so it would be negated if the current is not in the direction of the voltage drop
- If the voltage is constant...
- The current across the capacitor is 0
- So the capacitor behaves as an open-circuit in the presence of a constant voltage
- Voltage cannot change instantaneously in a capacitor because that would produce infinite current
Voltage Across Capacitor
We know i = C dv/dt, but how can we
express the voltage as a function of the
current?
i = C dv/dt
i dt = C dv
Integrate both sides
v(t) = (1/C) * ∫ i dt + v(t 0 )
Power and Energy in the Capacitor
Power p = vi i = C dv/dt p = Cv dv/dt
Energy p = dw/dt = Cv dv/dt dw = Cv dv Integrate both sides w = ½ Cv^2
Capacitor Sample Problem
- The voltage pulse described by the following equations is impressed across the terminals of a 0.5μF capacitor 0 t ≤ 0s v(t) = 4t V 0s ≤ t ≤ 1s 4e-^ (t-^1 )V t ≥ 1s Derive the expressions for the capacitor current, power, and energy. Specify the intervals of time when energy is being stored in and delivered by the capacitor.
Homework
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