LCR Circuits ( Physics )

Induction and Inductance The process of generating electrical current in a conductor by placing the conductor in a changing magnetic field is called electromagnetic induction  or just induction. Faraday also noticed that the rate at which the magnetic field changed also had an effect on the amount of current or voltage that was induced. Faraday's Law for an uncoiled conductor states that the amount of induced voltage is proportional to the rate of change of flux lines cutting the conductor. Faraday's Law for a straight wire is shown below. Where: VL = the induced voltage in voltsdø/dt = the rate of change of magnetic flux in webers/second  One henry is the amount of inductance that is required to generate one volt of induced voltage when the current is changing at the rate of one ampere per second.  The alternating current running through the coil creates a magnetic field in and around the coil that is increasing and decreasing as the current changes. The magnetic field forms concentric loops that surround the wire and join to form larger loops that surround the coil as shown in the image below.  Self-Inductance and Inductive Reactance By studying this image of a coil, it can be seen that the number of turns in the coil will have an effect on the amount of voltage that is induced into the circuit. Increasing the number of turns or the rate of change of magnetic flux increases the amount of induced voltage. Therefore,Faraday's Law must be modified for a coil of wire and becomes the following. Where: VL = induced voltage in voltsN = number of turns in the coildø/dt = rate of change of magnetic flux in            webers/second Where: VL = the induced voltage in voltsL = the value of inductance in henriesdi/dt = the rate of change of current in amperes per second Inductive Reactance The reduction of current flow in a circuit due to induction is called inductive reactance.  Self inductance is defined as the induction of a voltage in a current-carrying wire when the current in the wire itself is changing. Mutual Inductance(The Basis for Eddy Current Inspection) The values of L1 and L2 depend on the geometrical arrangement of the circuit (i.e. number of turns in the coil) and the conductivity of the material. The constant M, called the mutual inductance of the two circuits, is dependent on the geometrical arrangement of both circuits Circuits and Phase A circuit can be thought of as a closed path in which current flows through the components that make up the circuit. The current (i) obeys Ohm's Law, which is discussed on the page oncurrent flow. The simple circuit below consists of a voltage source (in this case an alternating current voltage source) and a resistor. The graph below the circuit diagram shows the value of the voltage and the current for this circuit over a period of time. This graph shows one complete cycle of an alternating current source. From the graph, it can be seen that as the voltage increases, the current does the same. The voltage and the current are said to be "in-phase" since their zero, peak, and valley points occur at the same time. They are also directly proportional to each other. In the circuit below, the resistive component has been replaced with an inductor. When inductance is introduced into a circuit, the voltage and the current will be "out-of-phase," meaning that the voltage and current do not cross zero, or reach their peaks and valleys at the same time. When a circuit has an inductive component, the current (iL) will lag the voltage by one quarter of a cycle. One cycle is often referred to as 360o, so it can be said that the current lags the voltage by 90o.  This phase shift occurs because the inductive reactance changes with changing current.  Recall that it is the changing magnetic field caused by a changing current that produces inductive reactance.  When the change in current is greatest, inductive reactance will be the greatest, and the voltage across the inductor will be the highest.  When the change in current is zero, the inductive reactance will be zero and the voltage across the inductor will be zero.  Be careful not to confuse the amount of current with the amount of change in the current.  Consider the points where the current reaches it peak amplitude and changes direction in the graph below (0o, 180o, and 360o).  As the current is changing directions, there is a split second when the change in current is zero.  Since the change in current is zero, no magnetic field is generated to produce the inductive reactance.  When the inductive reactance is zero, the voltage across the inductor is zero.  The resistive and inductive components are of primary interest in eddy current testing since the test probe is basically a coil of wire, which will have both resistance and inductive reactance. However, there is a small amount of capacitance in the circuits so a mention is appropriate. This simple circuit below consists of an alternating current voltage source and a capacitor. Capacitance in a circuit caused the current (ic) to lead the voltage by one quarter of a cycle (90o current lead). When there is both resistance and inductive reactance (and/or capacitance) in a circuit, the combined opposition to current flow is known as impedance. Impedance will be discussed more on the next page. Impedance Electrical Impedance (Z), is the total opposition that a circuit presents to alternating current. Impedance is measured in ohms and may include resistance (R), inductive reactance (XL), and capacitive reactance (XC) The phase shift is always relative to the resistance line since the resistance line is always in-phase with the voltage. If more resistance than inductive reactance is present in the circuit, the impedance line will move toward the resistance line and the phase shift will decrease. If more inductive reactance is present in the circuit, the impedance line will shift toward the inductive reactance line and the phase shift will increase. The relationship between impedance and its individual components (resistance and inductive reactance) can be represented using a vector as shown below. The amplitude of the resistance component is shown by a vector along the x-axis and the amplitude of the inductive reactance is shown by a vector along the y-axis. The amplitude of the the impedance is shown by a vector that stretches from zero to a point that represents both the resistance value in the x-direction and the inductive reactance in the y-direction. Eddy current instruments with impedance plane displays present information in this format. The impedance in a circuit with resistance and inductive reactance can be calculated using the following equation. If capacitive reactance was present in the circuit, its value would be added to the inductance term before squaring. The phase angle of the circuit can also be calculated using some trigonometry. The phase angle is equal to the ratio between the inductance and the resistance in the circuit. With the probes and circuits used in nondestructive testing, capacitance can usually be ignored so only inductive reactance needs to be accounted for in the calculation. The phase angle can be calculated using the equation below. If capacitive reactance was present in the circuit, its value would simply be subtracted from the inductive reactance term. or At the resonant frequency, the total impedance of the circuit appears to come only from resistance since XL and XC cancel out.    Every circuit containing capacitance and inductance has a resonant frequency that is inversely proportional to the square root of the product of the capacitance and inductance. LCR Circuit -By: M. Abdul Mumeed. +966507433412 14 © Md. Abdul Mumeed, Senior Secondary Teacher in PHYSICS at I. I. School, Riyadh, K.S.A.