Sinusoidal Response of RC Circuits When both resistance and capacitance are in a series circuit, the phase angle between the applied voltage and total current is between 0 and 90, depending on the values of resistance and reactance.

Impedance In a series RC circuit, the total impedance is the phasor sum of R and XC. R is plotted along the positive x-axis. XC is plotted along the negative y-axis. R R Z is the diagonal XC XC Z Z It is convenient to reposition the phasors into the impedance triangle.

Impediance

Impediance

Impediance Example Sketch the impedance triangle and show the values for R = 1.2 kW and XC = 960 W. R = 1.2 kW 39o XC = 960 W Z = 1.33 kW

Examples

Examples

Analysis of Series RC Ohm’s law is applied to series RC circuits using Z, V, and I. Because I is the same everywhere in a series circuit, you can obtain the voltages across different components by multiplying the impedance of that component by the current as shown in the following example.

Examples

Examples

Analysis of Series RC Example Assume the current in the previous example is 10 mArms. Sketch the voltage phasor diagram. The impedance triangle from the previous example is shown for reference. The voltage phasor diagram can be found from Ohm’s law. Multiply each impedance phasor by 10 mA. VR = 12 V R = 1.2 kW x 10 mA = 39o 39o XC = 960 W VC = 9.6 V Z = 1.33 kW VS = 13.3 V

Phase Relationships

Variation of Phase Angle Phasor diagrams that have reactance phasors can only be drawn for a single frequency because X is a function of frequency. As frequency changes, the impedance triangle for an RC circuit changes as illustrated here because |XC| decreases with increasing f. This determines the frequency response of RC circuits.

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Examples

Examples

Examples

Applications For a given frequency, a series RC circuit can be used to produce a phase lag by a specific amount between an input voltage and an output by taking the output across the capacitor. This circuit is also a basic low-pass filter, a circuit that passes low frequencies and rejects all others. R VR Vout Vin Vout C Vin Vout Vin

Examples

Applications Reversing the components in the previous circuit produces a circuit that is a basic lead network. This circuit is also a basic high-pass filter, a circuit that passes high frequencies and rejects all others. This filter passes high frequencies down to a frequency called the cutoff frequency. C Vout Vin Vin R Vout Vout VC Vin

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Parallel Capacitor Impedance Explain why the capacitance of parallel capacitors is the sum of capacitance of each capacitor.

Sinusoidal Response of Parallel RC For parallel circuits, it is useful to introduce two new quantities (susceptance and admittance) and to review conductance. Conductance is the reciprocal of resistance. Capacitive susceptance is the reciprocal of capacitive reactance. Admittance is the reciprocal of impedance.

Admittance

Admittance

Sinusoidal Response of Parallel RC In a parallel RC circuit, the admittance phasor is the sum of the conductance and capacitive susceptance phasors. The magnitude can be expressed as From the diagram, the phase angle is BC Y BC VS G G

Sinusoidal Response of Parallel RC Some important points to notice are: G is plotted along the positive x-axis. BC is plotted along the positive y-axis. Y is the diagonal BC Y BC VS G G

Sinusoidal Response of Parallel RC Example Draw the admittance phasor diagram for the circuit. The magnitude of the conductance and susceptance are: BC = 0.628 mS VS R 1.0 kW C 0.01 mF f = 10 kHz Y = 1.18 mS G = 1.0 mS

Analysis of Parallel RC Ohm’s law is applied to parallel RC circuits using Y, V, and I. Because V is the same across all components in a parallel circuit, you can obtain the current in a given component by simply multiplying the admittance of the component by the voltage as illustrated in the following example.

Analysis of Parallel RC Example If the voltage in the previous example is 10 V, sketch the current phasor diagram. The admittance diagram from the previous example is shown for reference. The current phasor diagram can be found from Ohm’s law. Multiply each admittance phasor by 10 V. BC = 0.628 mS x 10 V = IC = 6.28 mA Y = 1.18 mS IS = 11.8 mA G = 1.0 mS IR = 10 mA

Phase Angle of Parallel RC Notice that the formula for capacitive susceptance is the reciprocal of capacitive reactance. Thus BC and IC are directly proportional to f: IR IC IS q As frequency increases, BC and IC must also increase, so the angle between IR and IS must increase.

Equivalent Series and Parallel RC For every parallel RC circuit there is an equivalent series RC circuit at a given frequency. The equivalent resistance and capacitive reactance are shown on the impedance triangle: Req = Z cos q XC(eq) = Z sin q Z

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Examples

Phase Relationships

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Examples

Series-Parallel RC Circuits Series-parallel RC circuits are combinations of both series and parallel elements. These circuits can be solved by methods from series and parallel circuits. R1 C1 R2 C2 Z1 Z2 For example, the components in the green box are in series: The components in the yellow box are in parallel: The total impedance can be found by converting the parallel components to an equivalent series combination, then adding the result to R1 and XC1 to get the total reactance.

The Power Triangle

The Power Triangle Recall that in a series RC circuit, you could multiply the impedance phasors by the current to obtain the voltage phasors. The earlier example is shown for review: VR = 12 V R = 1.2 kW x 10 mA = 39o 39o XC = 960 W VC = 9.6 V Z = 1.33 kW VS = 13.3 V

The Power Triangle Example Multiplying the voltage phasors by Irms gives the power triangle (equivalent to multiplying the impedance phasors by I2). Apparent power is the product of the magnitude of the current and magnitude of the voltage and is plotted along the hypotenuse of the power triangle. Example The rms current in the earlier example was 10 mA. Show the power triangle. VR = 12 V VC = 9.6 V VS = 13.3 V 39o Ptrue = 120 mW x 10 mA = 39o Pr = 96 mVAR Pa = 133 mVA

Examples

Examples

Power Factor The power factor is the relationship between the apparent power in volt-amperes and true power in watts. Volt-amperes multiplied by the power factor equals true power. Power factor is defined mathematically as PF = cos  The power factor can vary from 0 for a purely reactive circuit to 1 for a purely resistive circuit.

Apparent Apparent power consists of two components; a true power component, that does the work, and a reactive power component, that is simply power shuttled back and forth between source and load. Ptrue (W) Some components such as transformers, motors, and generators are rated in VA rather than watts. Pr (VAR) Pa (VA)

Quiz Impedance Phase angle Capacitive suceptance (BC) Admittance (Y) The total opposition to sinusoidal current expressed in ohms. The angle between the source voltage and the total current in a reactive circuit. The ability of a capacitor to permit current; the reciprocal of capacitive reactance. The unit is the siemens (S). A measure of the ability of a reactive circuit to permit current; the reciprocal of impedance. The unit is the siemens (S).

Selected Key Terms Power factor Frequency response Cutoff frequency The relationship between volt-amperes and true power or watts. Volt-amperes multiplied by the power factor equals true power. In electric circuits, the variation of the output voltage (or current) over a specified range of frequencies. The frequency at which the output voltage of a filter is 70.7% of the maximum output voltage.

Quiz 1. If you know what the impedance phasor diagram looks like in a series RC circuit, you can find the voltage phasor diagram by a. multiplying each phasor by the current b. multiplying each phasor by the source voltage c. dividing each phasor by the source voltage d. dividing each phasor by the current

Quiz 2. A series RC circuit is driven with a sine wave. If the output voltage is taken across the resistor, the output will be in phase with the input. lead the input voltage. lag the input voltage. none of the above

Quiz 3. A series RC circuit is driven with a sine wave. If you measure 7.07 V across the capacitor and 7.07 V across the resistor, the voltage across both components is a. 0 V b. 5 V c. 10 V d. 14.1 V

Quiz 4. If you increase the frequency in a series RC circuit, a. the total impedance will increase b. the reactance will not change c. the phase angle will decrease d. none of the above

Quiz 5. Admittance is the reciprocal of a. reactance b. resistance c. conductance d. impedance

Quiz 6. Given the admittance phasor diagram of a parallel RC circuit, you could obtain the current phasor diagram by a. multiplying each phasor by the voltage b. multiplying each phasor by the total current c. dividing each phasor by the voltage d. dividing each phasor by the total current

Quiz 7. If you increase the frequency in a parallel RC circuit, a. the total admittance will decrease b. the total current will not change c. the phase angle between IR and IS will decrease d. none of the above

Quiz 8. The magnitude of the admittance in a parallel RC circuit will be larger if a. the resistance is larger b. the capacitance is larger c. both a and b d. none of the above

Quiz 9. The maximum power factor occurs when the phase angle is a. 0o b. 30o c. 45o d. 90o

Quiz 10. When power is calculated from voltage and current for an ac circuit, the voltage and current should be expressed as a. average values b. rms values c. peak values d. peak-to-peak values

Quiz Answers: 1. a 2. b 3. c 4. c 5. d 6. a 7. d 8. d 9. a 10. b