Passive components and circuits Lecture 6

Index Inductance Inductance as circuit element DC regime behavior AC regime behavior Transient regime behavior RLC series circuit RLC parallel circuit

Web addresses: http://en.wikipedia.org/wiki/RL_circuit http://www.play-hookey.com/ac_theory/ac_rl_series.html http://en.wikibooks.org/wiki/Circuit_Theory/RLC_Circuits#Series_RLC_Circuit http://members.aol.com/_ht_a/RAdelkopf/rl.html http://www.tpub.com/neets/book2/4l.htm http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcpar.html

Inductance as circuit element The main electrical property of the inductance – to generate magnetic field when a current is flowing through it. Measurement unit - Henri [H]. Practical units: starting nH and H up to mH and H. Conductive element Magnetic flux lines Magnetic substrate Conductive element

Inductance as circuit element The electronic component characterized by the inductance is called inductor (coil). Inductance L of a particular coil depends on its geometrical dimensions (A – turn area, L – length or l – width of the winding on the support), number of turns N, relative permeability r of the core, working temperature, etc. The theoretical formula for a coil with one turn after another, disposed on one layer, linear shape, is:

The energy stored in inductance The inductance doesn’t dissipate power, but it stores electric energy:

Series connection The equivalent inductance of a series connection is equal with the sum of all inductances:

Parallel connection: The equivalent inductance of a parallel connection is given by the following relation:

DC regime behavior In DC regime, the inductances are equivalent with short circuit: Electronic circuit Electronic circuit

AC regime behavior In AC regime, inductances are equivalent with impedances ZL. Inductive reactance

AC regime behavior Inductive reactance (impedance) depends on the frequency. In AC, the imittances of the circuits with inductances depend on the signal frequency. In consequence, the circuits with inductances have the signal filtering property.

RL high-pass filter For R=1K and L=160H is obtained: Exercises:: Calculate vo=f(vi). Identify, from the previous course, the circuit with the same transfer function.

RL high-pass filter - frequency characteristics

RL low-pass filter For R=1K and L=160H is obtained: Exercises:: Calculate vo=f(vi). Identify, from the previous course, the circuit with the same transfer function.

RL low-pass filter - frequency characteristics

High frequency inductance behavior At a very high frequency, the inductive reactance becomes much higher than the resistances from the circuits. In this case, the inductance is equivalent with open-circuit.

High frequency shocks In some circuits, the inductances are used to separate AC high frequency components between two circuits (AC high frequency -> open-circuits), without affecting the DC components (DC -> short-circuits). In these situations, the inductances are called high frequency shocks.

Transient regime behavior In this case, the transient regime consists in the modification of a DC circuit state in a new DC state. During these modifications, the inductance cannot be considered open-circuit or short-circuit. The transient regime analysis presume determining the way the inductances charge and discharge. In transient regime, the circuit operations are described by differential equations.

Determining the current when a constant voltage is applied Considering the switch K on position 1. The current through the inductance is zero. At the time t=t0, the switch is moved on position 2. After enough time, t, the current through inductor will be E/R. The transient regime is taking place between these two DC states.

Determining the current when a constant voltage is applied Solution of differential equations Circuit time constant

Current variation

Voltage variation

Significance of time constant If the transient process has the same slope like in the origin (initial moment), the final values of voltages and currents will be obtain after a time equal with the circuit time constant. As can be seen in the previous figures, the charging process continues to infinite. Practically, the transient regime is considered to be finished after 3 (95% from the final values) or 5 (99% from the final values).

Example (E=1V, R=1K, L=1mH)

Inductance discharging At the initial time, consider the switch on position 2. The current through the inductor is E/R. At a reference time moment t=t0, The switch K is moved on position 1. After enough time, t, the current becomes zero. The transient regime is the time between these two DC states.

Inductance discharging Solution of differential equation

Example (E=1V, R=1K, L=1mH)

Observation At the switching between 2 and 1 positions, we can have an open circuit and the current becomes zero instantaneously. That means di/dt. This phenomenon determines an over voltage across the inductance which can be dangerous for other circuits. Over voltage protection – introducing a diode in circuit.

The RL circuits behavior when pulses are applied Consider a pulses signal source applied to a series RL circuit. In analyzing of circuit behavior, we consider both the voltage across the inductor, vL(t), and the voltage across the resistor, vR(t). Applying this signal source, the phenomenon of charging and discharging described to transient regime is repetitive.

Case A – the time constant is much lower than the pulses duration

Case B – the time constant is much greater than the pulses duration

Integrating circuit If the output voltage is the voltage across the resistor, the effect under the input signal is an attenuation of edges, similarly with the integration mathematical operation. In this situation,(when vO(t)=vR(t)), the circuit is called integration circuit. The integration effect is higher in case B , when the time constant is greater than the pulse duration. The integration function in transient regime corresponds to low-pass filtering in AC regime.

Derivative circuit If the output voltage is the voltage across the inductance, the circuit effect under the input signal is an accentuation of edges, similarly with the derivative mathematical operation. In this situation,(when vO(t)= vL(t)), the circuit is called derivative circuit. The derivative effect is higher in case A , when the time constant is lower than the pulse duration. The derivative function in transient regime corresponds to high-pass filtering in AC regime.

RLC series circuit – AC regime behavior The equivalent impedance between AB terminals is: Modulus of this impedance is:

RLC series circuit – AC regime behavior We can notice that when the frequency is 0 or , the modulus is . In DC, the capacitance is equivalent with open-circuit. At very high frequency, the inductance is equivalent with open-circuit. The imaginary part of impedance becomes zero at the frequency: This frequency is called resonance frequency. From the energetic point of view, at this frequency the energy is transferred between inductor and capacitor.

RLC series circuit – AC regime behavior The derivate of impedance modulus becomes zero at the resonance frequency. The resonance frequency is an extreme point for impedance modulus (minimum, for this case). At the resonance frequency, the impedance is pure resistive.

Modulus of ZSech for R=10, L=10H, C=100nF Exercise: Represent |ZSech| versus frequency at the logarithmic scale.

RLC parallel circuit – AC regime behavior The equivalent impedance between AB terminals is: Modulus of this impedance is:

RLC parallel circuit – AC regime behavior We can notice that when the frequency is 0 or , the modulus is 0. In DC, the inductance is equivalent with short-circuit. At very high frequency, the capacitance is equivalent with short-circuit. The imaginary part of impedance becomes zero at the frequency: This frequency is called resonance frequency. From the energetic point of view, at this frequency the energy is transferred between inductor and capacitor.

RLC parallel circuit – AC regime behavior The derivate of impedance modulus becomes zero at the resonance frequency. The resonance frequency is an extreme point for impedance modulus (maximum, for this case). At the resonance frequency, the impedance is pure resistive.

Modulus of ZPech for R=100, L=10H, C=100nF Exercise: Represent |ZPech | versus frequency at the logarithmic scale.

Quality factor - Q These two structures (series and parallel RLC circuits) are used in order to obtain Band-pass and Band-reject filters. The selectivity of these circuits respective to some frequencies is characterized by quality factor. The quality factor is defined as the ratio between resonance frequency and 3dB frequency band.

Homework For the following circuit and each situation (from table), determine the circuit function. Make an essay: “Complementarity's between inductance and capacitance behavior in electronic circuits”