Reactance and Impedance

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1 Reactance and Impedance Capacitance in AC Circuits Professor Andrew H. Andersen 1

2 Objectives Describe capacitive ac circuits Analyze inductive ac circuits Describe the relationship between current and voltage in an RC circuit Determine impedance and phase angle in a series RC circuit Analyze a series RC circuit Determine the impedance and phase angle in a parallel RC circuit Describe the relationship between current and voltage in an RL circuit Determine impedance and phase angle in a series RL circuit Analyze a series RL circuit Determine impedance and phase angle in a parallel RL circuit 13 February 2004 Impedance 3 Capacitors in AC Circuits The instantaneous capacitor current is equal to the capacitance times the instantaneous rate of change of the voltage across the capacitor This rate of change is a maximum positive when the rising sine wave crosses zero This rate of change is a maximum negative when the falling sine wave crosses zero The rate of change is zero at the maximum and minimum of the sine wave 13 February 2004 Impedance 4 Professor Andrew H. Andersen 2

3 Capacitive Reactance, XC Capacitive reactance (XC) is the opposition to sinusoidal current, expressed in ohms The rate of change of voltage is directly related to frequency As the frequency increases, the rate of change of voltage increases, and thus current ( i ) increases An increase in i means that there is less opposition to current (XC is less) XC is inversely proportional to i and to frequency The relationship between capacitive reactance, capacitance and frequency is: 1 X C = - j 2 π f C where j = -1 XC is in ohms (Ω) f is in hertz (Hz) C is in farads (F) 13 February 2004 Impedance 5 Phase Relationship Between Current and Voltage in a Capacitor In a capacitive circuit with a sinusoidal voltage, the current leads the voltage by 90 in a purely capacitive ac circuit 13 February 2004 Impedance 6 Professor Andrew H. Andersen 3

4 Sinusoidal Response of RC Circuits When a circuit is purely resistive, the phase angle between applied voltage and total current is zero When a circuit is purely capacitive, the phase angle between applied voltage and total current is 90 When there is a combination of both resistance and capacitance in a circuit, the phase angle between the applied voltage and total current is somewhere between 0 and 90, depending on relative values of resistance and capacitance 13 February 2004 Impedance 7 Impedance in an AC Circuit Purely Resistive Purely Capacitive An RC Circuit 13 February 2004 Impedance 8 Professor Andrew H. Andersen 4

5 Analysis of Series RC Circuits The application of Ohm s law to a series RC circuits involves the use of the quantities V, I, and Z as follows: Z = R - j XC V = I Z V I = Z V Z = I 13 February 2004 Impedance 9 Impedance Phasor Diagram for a Series RC Circuit In the series RC circuit, the total impedance is the phasor sum of R and -jxc 2 2 Impedance: Z = R + X C - -1 X C Phase angle: θ = tan (NOTE: θ is negative) R 13 February 2004 Impedance 10 Professor Andrew H. Andersen 5

6 Impedance in a Series RC Circuit Rectangular Polar Z = 47 j100 Z = February 2004 Impedance 11 Variation of Z and θ with Frequency In a series RC circuit; as frequency increases: XC decreases Z decreases θ decreases R remains constant 13 February 2004 Impedance 12 Professor Andrew H. Andersen 6

7 KVL in a Series RC Circuit From KVL, the sum of the voltage drops must equal the applied voltage (VS) Since VR and VC are 90 out of phase with each other, they must be added as phasor quantities Magnitude of source voltage: 2 2 V S= V R + V C Phase angle between VR and VS: -1 - V C θ = tan VR 13 February 2004 Impedance 13 Voltage Phasor Diagram VS = 10 j15 VS = February 2004 Impedance 14 Professor Andrew H. Andersen 7

8 Relationships of I and V in a Series RC Circuit In a series circuit, the current is the same through both the resistor and the capacitor The resistor voltage is in phase with the current, and the capacitor voltage lags the current by February 2004 Impedance 15 Analysis of Parallel RC Circuits The application of Ohm s law to parallel RC circuits involves the use of the quantities Z, V, and I as: ( R ) (- j XC ) Z = R - j XC V S= V R= VC I T= I R+ IC V S Z = I T Note : All quantities are phasors 13 February 2004 Impedance 16 Professor Andrew H. Andersen 8

9 Conductance, Susceptance and Admittance Conductance, Susceptance, and Admittance all have the Sieman as the unit of measurement (formerly the mho) Remember that we are dealing with phasor quantities Conductance is the reciprocal of resistance: 1 G = R Susceptance is the reciprocal of capacitive reactance: Admittance is the reciprocal of impedance: B C = 1 X 1 Y = Z C NOTE: All quantities are phasors 13 February 2004 Impedance 17 Relationships of the I and V in a Parallel RC Circuit The applied voltage, VS, appears across both the resistive and the capacitive branches KCL States the total current IT, divides at the node into the two branch currents, IR and IC Remember that the currents are phasors IT = IR +j IC I = IT θ 13 February 2004 Impedance 18 Professor Andrew H. Andersen 9

10 I and Z in the Parallel RC Circuit ( ) (100) -j Z = = j Z = 20 - j40 = V S 5 0 I T = = Z ma I T = 50 + j100 ma = V S 5 0 I R = = I R = 50 V S 5 0 I C = = C I R = 100 T R C I T = 50 + j100 ma I T = R ma = 50mA X ma = j100ma I = I + I ma 13 February 2004 Impedance 19 Kirchhoff s Current Law In the parallel RC circuit Current through the resistor is in phase with the voltage Current through the capacitor leads the voltage, and thus the resistive current by 90 Total current is the phasor sum of the two branch currents Magnitude of IT is: Phase angle: I = I + I 2 2 T R C θ = tan -1 I I C R 13 February 2004 Impedance 20 Professor Andrew H. Andersen 10

11 Current Phasor Diagram 13 February 2004 Impedance 21 Find Z 13 February 2004 Impedance 22 Professor Andrew H. Andersen 11

12 Inductance in AC Circuits Inductive Reactance Inductive reactance is the opposition to sinusoidal current, expressed in ohms The inductor offers opposition to current, and that opposition varies directly with frequency The formula for inductive reactance, XL, is: XL = j 2 π f L The analysis of the RL circuit is the same for the RC except that the all the signs of the imaginary quantities are the opposite 13 February 2004 Impedance 24 Professor Andrew H. Andersen 12

13 Phase Relationship Between I and V in an Inductor The current lags inductor voltage by 90 The curves below are for a purely inductive circuit 13 February 2004 Impedance 25 Z in the Series RL Circuit 13 February 2004 Impedance 26 Professor Andrew H. Andersen 13

14 Series RL Phasor Diagrams Impedance Voltage Z = R + jx L 2 2 Z = R + X L V = V + jv S R L V = V + V 2 2 S R L 13 February 2004 Impedance 27 Series RL Circuit ZT = j ZT = February 2004 Impedance 28 Professor Andrew H. Andersen 14

15 Relationship Between I and V in Series RL 13 February 2004 Impedance 29 Parallel RL Circuit 13 February 2004 Impedance 30 Professor Andrew H. Andersen 15

16 Parallel RL Circuit 13 February 2004 Impedance 31 Professor Andrew H. Andersen 16

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