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AC Circuits 1 1 Capacitance in AC Circuits i iC dV d C ( V sin 2ft ) 2fC V cos 2ft dt dt X peak voltage V C Vc Y peak current 2fC V The current leads the voltage by 90o . V sin 2ft During the positive half-cycle of the voltage waveform, plate X of the capacitor becomes positively charged and plate Y negatively charged. During the negative half-cycle, X receives a negative charge and Y a positive one. There is therefore an alternating flow of charge or alternating current, i, through the capacitor Thus, unlike the case for a dc voltage where the capacitor eventually charges up to the value of the dc supply voltage and no more current flows, a capacitor continually conducts an ac current. Thus we may say that a capacitor passes ac and blocks dc. The opposition to the flow of ac current in a capacitor is known as the capacitive reactance and is the equivalent of resistance in a resistor circuit. capacitive reactance V I v rms 1 irms 2fC capacitive reactance is measured in Ohms. Note that it is inversely proportional to the frequency of the source. Capacitor In a purely capacitive a.c. circuit the current IC leads the applied voltage VC by 900 In a purely capacitive a.c. circuit the opposition to current flow is known as Capacitive Reactance, XC Capacitive Reactance is measured in ohms VC XC IC 1 XC 2fC Capacitor 1 XC 2fC Thus as frequency rises the capacitive reactance decreases non linearly XC reactance frequency Hz Capacitor In a purely capacitive a.c. circuit the current IC leads the applied voltage VC by 900 90 0 VC IC (Self ) Inductance A piece of wire wound in the form of a coil possesses an electrical property known as inductance. The property arises from the observable phenomenon that if a current flowing through the coil changes for some reason then an emf e is induced in the coil which tries to oppose the current change. The magnitude of the induced emf is proportional to the rate of change of current. The constant of proportionality is known as the inductance of the coil L and is measured in a unit called the henry. Mathematically we can summarise this with the equation e L di dt 6 Inductance in AC Circuits vL L di L d L ( I sin 2ft ) 2fL I cos 2ft dt dt peak current I voltage iL vL VL (t) or IL (t) peak voltage 2fL I The voltage leads the current by 90o . current Let i L ( t ) I sin 2ft The opposition to the flow of ac current in an inductor is known as the inductive reactance and is the equivalent of resistance in a resistor circuit. inductive reactance V I v rms 2fL irms inductive reactance is measured in Ohms. Note that it is proportional to the frequency of the source. 7 Inductor In a purely inductive a.c. circuit the current IL lags the applied voltage VL by 900 o 90 IL VL Inductor In a purely inductive a.c. circuit the current IL lags the applied voltage VL by 900 L IL VL Circuit Diagram VL 90 0 IL Phasor Diagram Inductor In a purely inductive a.c. circuit the current IL lags the applied voltage VL by 900 In a purely inductive a.c. circuit the opposition to current flow is known as Inductive Reactance, XL Inductive Reactance is measured in ohms VL XL IL X L 2fL Inductor X L 2fL Thus as frequency rises the inductive reactance increases linearily reactance frequency Resistance and Inductance in series VR IS VL VS VS VL As the current IS flows through both components it should be used as the reference for the phasor diagram VR IS Resistance and Inductance in series VR VL Circuit Diagram IS VS VS VL VR IS Phasor diagram applied voltage VS the ratio current flowing IS Is the opposition to current flow in the circuit However as the current and voltage are not in phase this opposition to current flow is known as Impedance Z (Ω) As the current flowing lags the applied voltage the circuit is said to be inductively reactive Resistance and Capacitance in series VC VR VR IS IS VS VC VS As the current IS flows through both components it should be used as the reference for the phasor diagram Resistance and Capacitance in series VC VR VR IS IS VS applied voltage VS the ratio current flowing IS VC VS Is the opposition to current flow in the circuit However as the current and voltage are not in phase this opposition to current flow is known as Impedance.(Z) (Ω) As the current flowing leads the applied voltage the circuit is said to be capacitively reactive Voltage Triangle VR VR IS VC VS VS Phasor diagram for CR circuit VC Voltage triangle for CR circuit If each of the voltages are divided by IS then opposition to current flow of each element is obtained. V i.e. R Re sistance IS VC capacitive reactance IS VS impedance IS Impedance Triangle VR R IS VC Z XC VS Phasor diagram for CR circuit Impedance triangle for CR circuit Using the same steps the voltage triangle and impedance triangle for R and L in series may be obtained Impedance Triangle VR R IS VC XC Z VS Phasor diagram for CR circuit Impedance triangle for CR circuit Using Pythagoras' s theorem Z R 2 X C XC and from trigonome try tan R 2 Find the inductive reactance XL 25.46 mH 6 10 V 50Hz Find the Impedance Z Find the current flowing Find the voltage across the inductor Find the voltage across the resistor Phase angle between VS and IS 3 X L 2fL 2 50Hz 25.46 10 H 8 Z R 2 X L 2 62 82 10 IS VS 10V 1A Z 10 VL I S X L 1A 8 8V VR I S R 1A 6 6V XL 8 arctan arctan arctan 1.333 53.130 R 6 25.46 mH 6 10 V 50Hz X L 8 Z 10 I S 1A VL 8V VR 6V 53.130 The point Z could be specified to the origin by :- Z 6 j8 OR 0 1053.13 Construct a reactance/resistance/impedance vector for the above diagram R 20, X C 15, X L 10 Construct a reactance/resistance/impedance vector for the above diagram XL add X L and X C R XC X L Z Z R2 ( X C X L )2 XC R 20, X C 15, X L 10 Note that as XC > XL the circuit Is said to be capacitively reactive XL R XC X L Z XC The series circuit could be replaced by a 20Ω resistor in series with a capacitive reactance of 5Ω the point Z could be defined by Z 20 j10 - j15 or Z 20 - j5 in cartesian form or 20.62 - 14.04o in polar form