Chapter 15 – Series & Parallel ac Circuits Lecture 20 by Moeen Ghiyas 19/08/2015 1

Chapter 15 – Series & Parallel ac Circuits

(Series ac Circuits) Voltage Divider Rule Frequency Response of R-C Circuit Summary of Series ac Circuits 19/08/2015 3

The basic format for the voltage divider rule in ac circuits is exactly the same as that for dc circuits Where V x is the voltage across one or more elements in series that have total impedance Z x, E is the total voltage appearing across the series circuit, and Z T is the total impedance of the series circuit. 19/08/2015 4

Example – Using the voltage divider rule, find the unknown voltages V R, V L, V C, and V 1 for the circuit of fig Solution: 19/08/2015 5

Example – Using the voltage divider rule, find the unknown voltages V R, V L, V C, and V 1 for the circuit of fig Solution: 19/08/2015 6

Example – Using the voltage divider rule, find the unknown voltages V R, V L, V C, and V 1 for the circuit of fig Solution: 19/08/2015 7

Let us first recall the impedance-versus-frequency curve of each element

At low frequencies the reactance of the capacitor will be quite high, suggesting that the total impedance of a series circuit will be primarily capacitive in nature. At high frequencies the reactance X C will drop below the R = 5kΩ level, and the series network will start to shift toward one of a purely resistive nature (at 5 kΩ).

Frequency at which X C = R can be determined in following manner: Since X C = 1/ωC = 1/2πfC, Thus frequency at which X C = R is which for the network of interest is For frequencies Less than f1: X C > R Greater than f1: R > X C

To examine the effect of frequency on the response of an R-C series configuration, l et us first determine how the impedance of the circuit Z T will vary with frequency for the specified frequency range The magnitude of the source is fixed at 10 V in the given circuit, but the frequency range of analysis will extend from zero to 20 kHz.

We already know by now that the total impedance is determined by following equation: In rectangular form In polar form Also remember

At f = 100 Hz; At f = 1 kHz; At f = 5 kHz; At f = 10 kHz; At f = 15 kHz; At f = 20 kHz; Close to Z C = kΩ /_ 90° if circuit was purely capacitive (R = 0Ω) at 100 hz Note Z T at f = 20 kHz is approaching 5 kΩ. Also, note phase angle is approaching a pure resistive network (0°).

At f = 100 Hz; At f = 1 kHz; At f = 5 kHz; At f = 10 kHz; At f = 15 kHz; At f = 20 kHz; A plot of Z T versus frequency

At f = 100 Hz; At f = 1 kHz; At f = 5 kHz; At f = 10 kHz; At f = 15 kHz; At f = 20 kHz; The plot of θ T versus frequency suggests that Z T made transition from capacitive (θ T = 90°) to Resistive (θ T = 0°).

Applying the voltage divider rule to determine the voltage across the capacitor in phasor form Thus magnitude and phase θ C by which V C leads E is given by

To determine the frequency response, X C must be calculated for each frequency of interest Applying the open-circuit equivalent

Recall that for a purely capacitive network, current I (in phase with V R ) leads E by 90 0, and angle between E and V C is 0 0. We find that with an increase in frequency, V C begins a clockwise rotation that will in turn increase the angle θ C and decrease the phase angle between I and E eventually approaching 0°.

A plot of V C versus frequency

A plot of θ C versus frequency

An R-C circuit can be used as a filter to determine which frequencies will have the greatest impact on the stage to follow. From our current analysis, it is obvious that any network connected across the capacitor will receive the greatest potential level at low frequencies and be effectively “shorted out” at very high frequencies. Thus R-C circuit can be used as a low pass filter.

The analysis of a series R-L circuit would proceed in much the same manner as for R-C circuit, except that X L and V L would increase with frequency and the angle between I and E would approach 90° (voltage leading the current) rather than 0°. If V L were plotted versus frequency, V L would approach E, and X L would eventually attain a level at which the open circuit equivalent would be appropriate.

For series ac circuits with reactive elements: 1. The total impedance will be frequency dependent. 2. The impedance of any one element can be greater than the total impedance of the network. 3. The inductive and capacitive reactance's are always in direct opposition on an impedance diagram. 4. Depending on the frequency applied, the same circuit can be either predominantly inductive or predominantly capacitive. 5. The magnitude of the voltage across any one element can be greater than the applied voltage. 19/08/

For series ac circuits with reactive elements: 6. At lower frequencies the capacitive elements will usually have the most impact on the total impedance, while at high frequencies the inductive elements will usually have the most impact. 7. The larger the resistive element of a circuit compared to the net reactive impedance, the closer the power factor is to unity. 19/08/

(Series ac Circuits) Voltage Divider Rule Frequency Response of R-C Circuit Summary of Series ac Circuits

19/08/