Lecture 13

Capacitive and Inductive elements

Capacitor and Inductive elements

• Capacitor: A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.

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• The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. 

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• The physical form and construction of practical capacitors vary widely.

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• Most capacitors contain at least two electrical conductors often in the form of metallic plates or surfaces separated by a dielectric medium

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• The nonconducting dielectric acts to increase the capacitor's charge capacity. Materials commonly used as dielectrics include glass, ceramic, plastic film, paper, mica, air, and oxide layers. 

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Capacitor and Inductive elements

• Capacitors are widely used as parts of electrical circuits in many common electrical devices. Unlike a resistor, an ideal capacitor does not dissipate energy, although real-life capacitors do dissipate a small amount.

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• When an electric potential difference (a voltage) is applied across the terminals of a capacitor, for example when a capacitor is connected across a battery, an electric field develops across the dielectric, causing a net positive charge to collect on one plate and net negative charge to collect on the other plate.

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• No current actually flows through the dielectric. However, there is a flow of charge through the source circuit. If the condition is maintained sufficiently long, the current through the source circuit ceases.

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• If a time-varying voltage is applied across the leads of the capacitor, the source experiences an ongoing current due to the charging and discharging cycles of the capacitor.

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Capacitor and Inductive elements

• In the hydraulic analogy, a capacitor is analogous to a rubber membrane sealed inside a pipe – this animation illustrates a membrane being repeatedly stretched and un-stretched by the flow of water, which is analogous to a capacitor being repeatedly charged and discharged by the flow of charge.

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• Today, capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass.

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• For direct-current source, the charging process of the capacitor will eventually cease.

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Capacitor and Inductive elements

• Linear time invariant capacitor element -The charge on the electrode plate of the capacitor element is proportional to the voltage at both ends.

q = C u

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where C is the capacitance of a capacitor element.

capacitance unit: Farad (F)

1F = 10^6 uF

1uF = 10^6 pF

Capacitor and Inductive elements

• The relation between voltage and current on a capacitor (both current and voltage take associative reference directions).

C

u

+

-

i

Differential form of voltage-current relationship

(VCR) on a capacitor

• At any time, the current depends on the change rate of the voltage at both ends of the capacitor.

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• If du/dt > 0 (voltage is increasing), i>0. 🡪 capacitor is charging.
• If du/dt < 0 (voltage is decreasing), i<0. 🡪 capacitor is discharging.
• If du/dt = 0 (voltage is constant), i=0. 🡪 equivalently open-circuited.

Capacitor and Inductive elements

• At any time, the current i has limited value, which means u is a continuous function.

du/dt 🡪 infinity, i🡪 infinity

Capacitor and Inductive elements

• Integral form of a capacitor’s VCR (voltage-current relation)

• At any moment, the voltage value at both ends of capacitor depends on the current values of all previous moments. Therefore, capacitive elements have the function of memorizing (recording) current, so they are called memory elements.

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• Question: if you want to calculate the voltage value of a certain capacitor after t₀, what you need to know?

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• When the current and voltage take the non-associative reference directions,

u(t₀): Initial voltage / initial state

Capacitor and Inductive elements

• The power and energy storage of a capacitor

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• Power p = u*i = u * C*du/dt (u and i take the associative reference directions)

• If du/dt > 0 (voltage is increasing), i>0 (p>0). 🡪 capacitor is charging and absorbing power.

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• If du/dt < 0 (voltage is decreasing), i<0 (p<0). 🡪 capacitor is discharging and emitting power.

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• Capacitive elements (capacitor) can absorb the energy supplied from the circuit and convert it into electric field energy for storage in a period of time, and release the energy back to the circuit in another period of time. Therefore, the capacitor element is an energy storage element and does not consume energy.

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• Different from power sources (voltage or current sources), it is a passive element.

Capacitor and Inductive elements

• The power and energy storage of a capacitor

• Change of energy stored by a capacitor element

Capacitor and Inductive elements

Example: to solve i(t), power p(t) and energy W(t)

i

Capacitor and Inductive elements

W(J)

t（s）

1

2

0

Capacitor and Inductive elements

Capacitor and Inductive elements

i

• The actual capacitor model

C

R

Capacitor and Inductive elements

• Inductive element (inductor)

• An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil.

• An electric current flowing through a conductor generates a magnetic field surrounding it. The magnetic flux linkage Ψ_B generated by a given current i(t) depends on the geometric shape of the circuit. Their ratio defines the inductance L = Ψ_B/i.

Weber-Ampere

Capacitor and Inductive elements

• Linear time-invariant inductance element

• At any time, the current passing through the inductive element is proportional to its flux chain.

• Their ratio defines the inductance L = Ψ/i.

• Unit: Henry (H), uH, mH…

Capacitor and Inductive elements

• The relation between voltage and current on an inductor (both current and voltage take associative reference directions).

• At any time, the voltage depends on the change rate of the current at both ends of the inductor.

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• If di/dt > 0 (current is increasing), u>0. 🡪 inductor is charging.
• If di/dt < 0 (current is decreasing), u<0. 🡪 inductor is discharging.
• If di/dt = 0 (current is constant), u=0. 🡪 equivalently short-circuited.

• At any time, the current u in an actual circuit has limited value, which means i is a continuous function.

Differential form of voltage-current relation

(VCR) on an inductor

Capacitor and Inductive elements

• Integral form of an inductor’s VCR (voltage-current relation)

• At any moment, the current value through an inductor depends on the voltage values of all previous moments. Therefore, inductive elements have the function of memorizing (recording) voltage, so they are called memory elements.

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• Question: if you want to calculate the current value of a certain inductor after t₀, what you need to know?

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• When the current and voltage take the non-associative reference directions,

i(t₀): Initial current / initial state

Capacitor and Inductive elements

• The power and energy storage of an inductor

(u and i take the associative reference directions)

• If di/dt > 0 (current is increasing), u>0 (p>0). 🡪 inductor is storing more magnetic-field energy and absorbing power.

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• If di/dt < 0 (current is decreasing), u<0 (p<0). 🡪 capacitor is releasing magnetic-field engergy and emitting power.

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• Inductive elements (inductor) can absorb the energy supplied from the circuit and convert it into magnetic-field energy for storage in a period of time, and release the energy back to the circuit in another period of time. Therefore, the inductive element is an energy storage element and does not consume energy.

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• Different from power sources (voltage or current sources), it is a passive element.

Capacitor and Inductive elements

• The power and energy storage of an inductor

(u and i take the associative reference directions)

• Change of energy (from t₀ to t) stored by an inductive element

Capacitor and Inductive elements

• The actual inductor model

Capacitor and Inductive elements

• The serial and parallel connection of capacitors

• Equivalent capacitance when two capacitors are connected in a serial way.

Capacitor and Inductive elements

• Voltage distribution on each capacitor.

• The serial and parallel connection of capacitors

Capacitor and Inductive elements

• The serial and parallel connection of capacitors

• Equivalent capacitance when two capacitors are connected in a parallel way.

• Current distribution on each capacitor.

Capacitor and Inductive elements

• The serial and parallel connection of inductors

• Equivalent inductance when two inductors are connected in a serial way.

Capacitor and Inductive elements

• The serial and parallel connection of inductors

• Equivalent inductance when two parallel inductors