Capacitive Circuits Topics Covered in Chapter : Sine-Wave V C Lags i C by 90 o 18-2: X C and R in Series 18-3: Impedance Z Triangle 18-4: RC Phase-Shifter Circuit 18-5: X C and R in Parallel Chapter 18 © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 18  18-6: RF and AF Coupling Capacitors  18-7: Capacitive Voltage Dividers  18-8: The General Case of Capacitive Current i C McGraw-Hill© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

18-1: Sine-Wave V C Lags i C by 90 o  For any sine wave of applied voltage, the capacitor’s charge and discharge current i c will lead v c by 90°. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 18-1: Capacitive current i c leads v c by 90°. (a) Circuit with sine wave V A across C. (b) Waveshapes of i c 90° ahead of v c. (c) Phasor diagram of i c leading the horizontal reference v c by a counterclockwise angle of 90°. (d) Phasor diagram with i c as the reference phasor to show v c lagging i c by an angle of −90°.

18-1: Sine-Wave V C Lags i C by 90 o  The value of i c is zero when V A is at its maximum value.  At its high and low peaks, the voltage has a static value before changing direction. When V is not changing and C is not charging or discharging, the current is zero.  i c is maximum when v c is zero because at this point the voltage is changing most rapidly.

18-1: Sine-Wave V C Lags i C by 90 o  i c and v c are 90° out of phase because the maximum value of one corresponds to the zero value of the other.  The 90° phase angle results because i c depends on the rate of change of v c. i c has the phase of dv/dt, not the phase of v.  The 90° phase between v c and i c is true in any sine wave ac circuit. For any X C, its current and voltage are 90° out of phase.

18-1: Sine-Wave V C Lags i C by 90 o  The frequency of v c and i c are always the same.  The leading phase angle only addresses the voltage across the capacitor. The current is still the same in all parts of a series circuit. In a parallel circuit, the voltage across the generator and capacitor are always the same, but both are 90° out of phase with i c.

18-2: X C and R in Series  When a capacitor and a resistor are connected in series, the current is limited by both X C and R.  Each series component has its own series voltage drop equal to IR for the resistance and IX C for the capacitive reactance.  For any circuit combining X C and R in series, the following points are true: 1. The current is labeled I rather than I C, because I flows through all the series components.

18-2: X C and R in Series 2. The voltage across X C, labeled V C, can be considered an IX C voltage drop, just as we use V R for an IR voltage drop. 3. The current I through X C must lead V C by 90°, because this is the phase angle between the voltage and current for a capacitor. 4. The current I through R and its IR voltage drop are in phase. There is no reactance to sine-wave alternating current in any resistance. Therefore, I and IR have a phase angle of 0°.

18-2: X C and R in Series  Phase Comparisons  For a circuit combining series resistance and reactance, the following points are true: 1. The voltage V C is 90° out of phase with I. 2. V R and I are in phase. 3. If I is used as the reference, V C is 90° out of phase with V R.  V C lags V R by 90° just as voltage V C lags the current I by 90°.

18-2: X C and R in Series  Combining V R and V C ; the Phasor Voltage Triangle  When voltage wave V R is combined with voltage wave V C the result is the voltage wave of the applied voltage V T.  Out-of-phase waveforms may be added quickly by using their phasors. Add the tail of one phasor to the arrowhead of another and use the angle to show their relative phase. V R 2 + V C 2 V T =

18-2: X C and R in Series Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 18-3: Addition of two voltages 90° out of phase. (a) Phasors for V C and V R are 90° out of phase. (b) Resultant of the two phasors is the hypotenuse of the right triangle for V T.

18-2: X C and R in Series VRVR VCVC VTVT Voltage Phasors  R XCXC ZTZT Impedance Phasor  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Phasor Voltage Triangle for Series RC Circuits

18-2: X C and R in Series Waveforms and Phasors for a Series RC Circuit Note: Since current is constant in a series circuit, the current waveforms and current phasors are shown in the reference positions.  VRVR I I  I VCVC Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18-3: Impedance Z Triangle  R and X C may be added using a triangle model as was shown with voltage.  Adding phasors X C and R results in their total opposition in ohms, called impedance, using symbol Z T. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 18-4: Addition of R and X C 90° out of phase in a series RC circuit to find the total impedance Z T.

18-3: Impedance Z Triangle  Z takes into account the 90° phase relationship between R and X C. R 2 + X C 2 Z T =  Phase Angle with Series X C and R  The angle between the applied voltage V T and the series current I is the phase angle of the circuit.  The phase angle may be calculated from the impedance triangle of a series RC circuit by the formula tan Θ Z = − XCXC R

18-3: Impedance Z Triangle The Impedance of a Series RC Circuit The impedance is the total opposition to current flow. It’s the phasor sum of resistance and reactance in a series circuit I = 2 A V T = 100 R = 30  X C = 40  = 2 A Z VTVT I = = R XCXC Z = 50 W R 2 + X C 2 =Z = Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18-3: Impedance Z Triangle The Tangent Function Θ opposite adjacent negative angle Θ opposite adjacent positive angle Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18-3: Impedance Z Triangle The Phase Angle of a Series RC Circuit I = 2 A V T = 100 R = 30  X C = 40  30  40  50   = −53°  = Tan −1  R XCXC Θ = Tan −1 V T lags I by 53° I VCVC VTVT −53°

18-3: Impedance Z Triangle Source Voltage and Current Phasors Note: The source voltage lags the current by an amount proportional to the ratio of capacitive reactance to resistance. I VTVT Θ < 0  I VTVT X C < R Θ = −45  I VTVT X C = R Θ < − 45  I VTVT X C > R Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18-4: RC Phase-Shifter Circuit  The RC phase-shift circuit is used to provide a voltage of variable phase to set the conduction time of semiconductors in power control circuits.  Output can be taken across R or C depending on desired phase shift with respect to V IN.  V R leads V T by an amount depending on the values of X C and R.  V C lags V T by an amount depending on the values of X C and R.

18-4: RC Phase-Shifter Circuit Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 18-5: An RC phase-shifter circuit. (a) Schematic diagram. (b) Phasor triangle with IR, or V R, as the horizontal reference. V R leads V T by 46.7° with R set at 50 kΩ. (c) Phasors shown with V T as the horizontal reference.

18-5: X C and R in Parallel 18-5: X C and R in Parallel  The sine-wave ac charge and discharge currents for a capacitor lead the capacitor voltage by 90°.  The sine-wave ac voltage across a resistor is always in phase with its current.  The total sine-wave ac current for a parallel RC circuit always leads the applied voltage by an angle between 0° and 90°.

18-5: X C and R in Parallel  Phasor Current Triangle  The resistive branch current I R is used as the reference phasor since V A and I R are in phase.  The capacitive branch current I C is drawn upward at an angle of +90° since I C leads V A and thus I R by 90°. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 18-7: Phasor triangle of capacitive and resistive branch currents 90° out of phase in a parallel circuit to find the resultant I T.

18-5: X C and R in Parallel  Phasor Current Triangle (Continued)  The sum of the I R and I C phasors is indicated by the phasor for I T, which connects the tail of the I R phasor to the tip of the I C phasor.  The I T phasor is the hypotenuse of the right triangle.  The phase angle between I T and I R represent the phase angle of the circuit. I R 2 + I C 2 I T =

18-5: X C and R in Parallel  Impedance of X C and R in Parallel  To calculate the total or equivalent impedance of X C and R in parallel, calculate total line current I T and divide into applied voltage V A : Z EQ = VAVA ITIT

18-5: X C and R in Parallel Impedance in a Parallel RC Circuit V A = 120 R = 30  X C = 40  I T = 5 A 4 A 3 A5 A  = 24 ΩZ EQ = ITIT VAVA Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18-5: X C and R in Parallel  Phase Angle in Parallel Circuits Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig Tan Θ I = ICIC IRIR Use the Tangent form to find Θ from the current triangle. Tan Θ I = 10/10 = 1 Θ I = Tan 1 Θ I = 45°

18-5: X C and R in Parallel  Parallel Combinations of X C and R  The series voltage drops V R and V C have individual values that are 90° out of phase.  They are added by phasors to equal the applied voltage V T.  The negative phase angle −Θ Z is between V T and the common series current I.  The parallel branch currents I R and I C have individual values that are 90° out of phase.  They are added by phasors to equal I T, the main-line current.  The positive phase angle Θ I is between the line current I T and the common parallel voltage V A.

18-5: X C and R in Parallel  Series circuit impedance (Z T ) in Ohms, Ω  Voltage lags current.  Becomes more resistive with increasing f.  Becomes more capacitive with decreasing f.  Parallel circuit impedance (Z EQ ) in Ohms, Ω  Voltage lags current.  Becomes more resistive with decreasing f.  Becomes more capacitive with increasing f.  Parallel Combinations of X C and R  Resistance (R) in Ohms, Ω  Voltage in phase with current.  Capacitive Reactance (X C ) in Ohms, Ω  Voltage lags current by 90°.

18-5: X C and R in Parallel Summary of Formulas Series RCParallel RC X C = 1 2 π f C X C = 1 2 π f C V T =V R 2 + V C 2 I T =I R 2 + I C 2 R 2 + X C 2 Z T = Z EQ = VAVA ITIT tan Θ = − XCXC R tan Θ = ICIC IRIR

18-6: RF and AF Coupling Capacitors  C C is used in the application of a coupling capacitor.  The C C ’s low reactance allows developing practically all the ac signal voltage across R.  Very little of the ac voltage is across C C.  The dividing line for C C to be a coupling capacitor at a specific frequency can be taken as X C one- tenth or less of the series R.

18-7: Capacitive Voltage Dividers  When capacitors are connected in series across a voltage source, the series capacitors serve as a voltage divider.  Each capacitor has part of the applied voltage.  The sum of all the series voltage drops equals the source voltage.  The amount of voltage across each capacitor is inversely proportional to its capacitance.

18-7: Capacitive Voltage Dividers Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 18-9: Series capacitors divide V T inversely proportional to each C. The smaller C has more V. (a) An ac divider with more X C for the smaller C.

18-7: Capacitive Voltage Dividers Fig. 18-9: Series capacitors divide V T inversely proportional to each C. The smaller C has more V. (b) A dc divider. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18-8: The General Case of Capacitive Current i C  The capacitive charge and discharge current i c is always equal to C(dv/dt).  A sine wave of voltage variations for v c produces a cosine wave of current i.  Note that v c and i c have the same waveform, but they are 90° out of phase.

18-8: The General Case of Capacitive Current i C  X C is generally used for calculations in sine-wave circuits.  Since X C is 1/(2πfC), the factors that determine the amount of charge and discharge current are included in f and C.  With a nonsinusoidal waveform for voltage v c, the concept of reactance cannot be used. (Reactance X C applies only to sine waves).