# Chapter 10. RC Circuits ISU EE. C.Y. Lee

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1 Chapter 10 RC Circuits

2 Objectives 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 Analyze a parallel RC circuit Analyze series-parallel RC circuits Determine power in RC circuits 2

3 Sinusoidal Response of RC Circuits The capacitor voltage lags the source voltage Capacitance causes a phase shift between voltage and current that depends on the relative values of the resistance and the capacitive reactance 3

4 Impedance and Phase Angle of Series RC Circuits The phase angle is the phase difference between the total current and the source voltage The impedance of a series RC circuit is determined by both the resistance (R) and the capacitive reactance (X C ) (Z C = 1/j C= jx C ) (Z= R+Z C = R jx C ) (Z = R) (Z = jx C ) 4 (Z = R jx C )

5 Impedance and Phase Angle of Series RC Circuits In the series RC circuit, the total impedance is the phasor sum of R and jx C Impedance magnitude: Z = R 2 + X 2 C Phase angle: θ = tan -1 (X C /R) 5

6 Impedance and Phase Angle of Series RC Circuits Example: Determine the impedance and the phase angle Z = (47) 2 + (100) 2 = 110 Ω θ = tan -1 (100/47) = tan -1 (2.13) =

7 Analysis of Series RC Circuits The application of Ohm s law to series RC circuits involves the use of the quantities Z, V, and I as: V = I = Z = IZ V Z V I 7

8 Analysis of Series RC Circuits Example: If the current is 0.2 ma, determine the source voltage and the phase angle X C = 1/2π( )( ) = 15.9 kω Z = ( ) 2 + ( ) 2 = 18.8 kω V S = IZ = (0.2mA)(18.8kΩ) = 3.76 V θ = tan -1 (15.9k/10k) =

9 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 90 I V R V S V C 9

10 KVL in a Series RC Circuit From KVL, the sum of the voltage drops must equal the applied voltage (V S ) V S I V R V C Since V R and V C are 90 out of phase with each other, they must be added as phasor quantities V R = IR V S = V 2 R + V2 C θ = tan -1 (V C /V R ) V C = I( jx C ) V S = IZ = I(R jx C ) 10

11 KVL in a Series RC Circuit Example: Determine the source voltage and the phase angle V S = (10) 2 + (15) 2 = 18 V θ = tan -1 (15/10) = tan -1 (1.5) =

12 Variation of Impedance and Phase Angle with Frequency For a series RC circuit; as frequency increases: R remains constant X C decreases Z decreases θ decreases f R Z 12

13 Variation of Impedance and Phase Angle with Frequency Example: Determine the impedance and phase angle for each of the following values of frequency: (a) 10 khz (b) 30 khz (a) X C = 1/2π( )( ) = 1.59 kω Z = ( ) 2 + ( ) 2 = 1.88 kω θ = tan -1 (1.59k/1.0k) = 57.8 (b) X C = 1/2π( )( ) = 531 kω Z = ( ) 2 + (531) 2 = 1.13 kω θ = tan -1 (531/1.0k) =

14 Impedance and Phase Angle of Parallel RC Circuits Total impedance in parallel RC circuit: Z = (RX C ) / ( R 2 +X 2 C ) Phase angle between the applied V and the total I: θ = tan -1 (R/X C ) 14 V = + Z R jx C 1 jx C + R = Z R( jx C ) RX C Z = X C + jr RX = IZ = I X C + C jr

15 Conductance, Susceptance and Admittance Conductance is the reciprocal of resistance: G = 1/R Capacitive susceptance is the reciprocal of capacitive reactance: B C = 1/X C Admittance is the reciprocal of impedance: Y = 1/Z 15

16 Ohm s Law Application of Ohm s Law to parallel RC circuits using impedance can be rewritten for admittance (Y=1/Z): V = I Y I =VY Y = I V 16

17 Relationships of the I and V in a Parallel RC Circuit The applied voltage, V S, appears across both the resistive and the capacitive branches Total current, I tot, divides at the junction into the two branch current, I R and I C I C = V/( jx C ) I tot = V/Z = V((X C +jr)/rx C ) I R = V/R V s, V R, V C 17

18 KCL in a Parallel RC Circuit From KCL, Total current (I S ) is the phasor sum of the two branch currents Since I R and I C are 90 out of phase with each other, they must be added as phasor quantities I C = V/( jx C ) I tot = V/Z = V((X C +jr)/rx C ) I tot = I 2 R + I2 C θ = tan -1 (I C /I R ) 18 I R = V/R

19 KCL in a Parallel RC Circuit Example: Determine the value of each current, and describe the phase relationship of each with the source voltage I R = 12/220 = 54.5 ma I C = 12/150 = 80 ma I tot = (54.5) 2 + (80) 2 = 96.8 ma θ = tan -1 (80/54.5) = V s

20 Series-Parallel RC Circuits An approach to analyzing circuits with combinations of both series and parallel R and C elements is to: Calculate the magnitudes of capacitive reactances (X C ) Find the impedance (Z) of the series portion and the impedance of the parallel portion and combine them to get the total impedance 20

21 21 流 (Z) (Z') (Z") 聯 Z Z' jz" ( ) R i jx m R Z + + = ( ) m i e m e i e i e X R R X R R R R R Z = ( ) m i e m e X R R X R j + + 路

22 路 22

23 Z-plot ( Z ) 2 + ( Z ) 2 Z = c c Z s 流 流 Z 流 流 率 f C ( 率 ) Z" 率 參 數 23

24 Z-plot 24

25 RC Lag Network The RC lag network is a phase shift circuit in which the output voltage lags the input voltage φ = 90 o tan 1 X C R 25

26 RC Lead Network The RC lead network is a phase shift circuit in which the output voltage leads the input voltage 26 φ = tan 1 X C R

27 Frequency Selectivity of RC Circuits A low-pass circuit is realized by taking the output across the capacitor, just as in a lag network 27

28 Frequency Selectivity of RC Circuits The frequency response of the low-pass RC circuit is shown below, where the measured values are plotted on a graph of V out versus f. 10 V 28

29 Frequency Selectivity of RC Circuits A high-pass circuit is implemented by taking the output across the resistor, as in a lead network 29

30 Frequency Selectivity of RC Circuits The frequency response of the high-pass RC circuit is shown below, where the measured values are plotted on a graph of V out versus f. 10 V 30

31 Frequency Selectivity of RC Circuits The frequency at which the capacitive reactance equals the resistance in a low-pass or high-pass RC circuit is called the cutoff frequency: 1 f C = 2πRC 31

32 Coupling an AC Signal into a DC Bias Network 32

33 Summary When a sinusoidal voltage is applied to an RC circuit, the current and all the voltage drops are also sine waves Total current in an RC circuit always leads the source voltage The resistor voltage is always in phase with the current In an ideal capacitor, the voltage always lags the current by 90 33

34 Summary In an RC circuit, the impedance is determined by both the resistance and the capacitive reactance combined The circuit phase angle is the angle between the total current and the source voltage In a lag network, the output voltage lags the input voltage in phase In a lead network, the output voltage leads the input voltage A filter passes certain frequencies and rejects others 34

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