RL-RC Circuits & Applications SVES Circuits Theory

Introduction In this chapter, phasor algebra will be used to develop a quick, direct method for solving both the series and the parallel ac circuits. Describe the relationship between current and voltage in an RC & RL circuits Determine impedance and phase angle in RC and RL circuits 2

Impedance and the Phasor Diagram Resistive Elements Use R=0° in the following polar format to ensure the proper phase relationship between the voltage and the current resistance: The boldface Roman quantity ZR, having both magnitude and an associate angle, is referred to as the impedance of a resistive element. ZR is not a phasor since it does not vary with time. Even though the format R0° is very similar to the phasor notation for sinusoidal current and voltage, R and its associated angle of 0° are fixed, non-varying quantities. 3

Resistive ac circuit.

Resistive ac circuit Voltage is 100 volts Peak

Waveforms for Last Example Resistive

Phasor diagram of Example Resistive 20.0 100

Analysis of Resistive Circuits The application of Ohm’s law to series circuits involves the use of the quantities Z, V, and I as: V = IZ I = V/Z Z = V/I R = Z

Impedance and the Phasor Diagram Capacitive Reactance (XC) Use C = – 90° in the following polar format for capacitive reactance to ensure the proper phase relationship between the voltage and current of an capacitor: The boldface roman quantity Zc, having both magnitude and an associated angle, is referred to as the impedance of a capacitive element. 6

Impedance and the Phasor Diagram ZC is measured in ohms and is a measure of how much the capacitive element will “control or impede” the level of current through the network. This format like the one for the resistive element, will prove to be a useful “tool” in the analysis of ac networks. Be aware that ZC is not a phasor quantity for the same reason indicated for a resistive element. 7

Analysis of Capacitive ac Circuit The current leads the voltage by 90 in a purely capacitive ac circuit

Capacitive ac circuit.

Capacitive ac circuit, Voltage is 15 volts peak

Waveforms for Example current leads the voltage by 90 degrees

Phasor diagrams for Example 7.50 15.00

Impedance and the Phasor Diagram Inductive Reactance (XL) Use L = 90° in the following polar format for inductive reactance to ensure the proper phase relationship between the voltage and the current of an inductor: The boldface roman quantity ZL, having both magnitude and an associated angle, in referred to as the impedance of an inductive element. 4

Impedance and the Phasor Diagram ZL is measured in ohms and is a measure of how much the inductive element will “control or impede” the level of current through the network. This format like the one for the resistive element, will prove to be a useful “tool” in the analysis of ac networks. Be aware that ZL is not a phasor quantity for the same reason indicated for a resistive element. 5

Inductive ac circuit.

Inductive ac circuit Voltage is 24 volts Peak

Inductor Waveforms for Example voltage leads the current by 90 degrees

Phasor diagrams for Example. 24.0 V 8.0

Three cases of impedance R – C series circuit

Illustration of sinusoidal response with general phase relationships of VR, VC, and I relative to the source voltage. VR and I are in the phase; VR leads VS; VC lags VS; and VR and VC are 90º out of phase.

Impedance of a series RC circuit.

Development of the impedance triangle for a series RC circuit.

Impedance of a series RC circuit.

Impedance of a series RC circuit.

Phase relation of the voltages and current in a series RC circuit.

Voltage and current phasor diagram for the waveforms

Voltage diagram for the voltage in a R-C circuit

Voltage diagram for the voltage in a R-C circuit

An illustration of how Z and XC change with frequency.

As the frequency increases, XC decreases, Z decreases, and  decreases As the frequency increases, XC decreases, Z decreases, and  decreases. Each value of frequency can be visualized as forming a different impedance triangle.

Illustration of sinusoidal response with general phase relationships of VR, VL, and I relative to the source voltage. VR and I are in phase; VR lags VS; and VL leads VS. VR and VL are 90º out of phase with each other.

Impedance of a series RL circuit.

Development of the Impedance triangle for a series RL circuit.

Impedance of a series RL circuit.

Impedance of a series RL circuit.

Phase relation of current and voltages in a series RL circuit.

Voltage phasor diagram for the waveforms .

Voltage and current phasor diagram for the waveforms

Voltages of a series RL circuit.

Voltages of a series RL circuit.

Reviewing the frequency response of the basic elements.

Frequency Selectivity of RC Circuits Frequency-selective circuits permit signals of certain frequencies to pass from the input to the output, while blocking all others A low-pass circuit is realized by taking the output across the capacitor, just as in a lag network A high-pass circuit is implemented by taking the output across the resistor, as in a lead network

The RC lag network (Vout = VC)

FIGURE 10-17 An illustration of how Z and XC change with frequency.

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: fc = 1/(2RC)

Normalized general response curve of a low-pass RC circuit showing the cutoff frequency and the bandwidth - 3 dB point normalized Cutoff point

Example of low-pass filtering action Example of low-pass filtering action. As frequency increases, Vout decreases

The RC lead network (Vout = VR)

Example of high-pass filtering action Example of high-pass filtering action. As frequency increases, Vout increases

High-pass filter responses.

High-pass filter responses, filters in series Each r-c combination - 20dB / decade

Observing changes in Z and XL with frequency by watching the meters and recalling Ohm’s law

RL Circuit as a Low-Pass Filter An inductor acts as a short to dc As the frequency is increased, so does the inductive reactance As inductive reactance increases, the output voltage across the resistor decreases A series RL circuit, where output is taken across the resistor, finds application as a low-pass filter

Example of low-pass filtering action Example of low-pass filtering action. Winding resistance has been neglected. As the input frequency increases, the output voltage decreases

RL Circuit as a High-Pass Filter For the case when output voltage is measured across the inductor At dc, the inductor acts a short, so the output voltage is zero As frequency increases, so does inductive reactance, resulting in more voltage being dropped across the inductor The result is a high-pass filter

FIGURE 12-39 Example of high-pass filtering action FIGURE 12-39 Example of high-pass filtering action. Winding resistance has been neglected. As the input frequency increases, the output voltage increases.