Chapter 15 – Series & Parallel ac Circuits

Lecture 20

by Moeen Ghiyas

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Chapter 15 – Series & Parallel ac Circuits

(Series ac Circuits)

Voltage Divider Rule

Frequency Response of R-C Circuit

Summary of Series ac Circuits

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The basic format for the voltage divider rule in ac circuits is

exactly the same as that for dc circuits

Where

Vx is the voltage across one or more elements in series that

have total impedance Zx,

E is the total voltage appearing across the series circuit, and

ZT is the total impedance of the series circuit.19/04/23 4

Example – Using the voltage divider rule, find the unknown

voltages VR, VL, VC, and V1 for the circuit of fig

Solution:

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Example – Using the voltage divider rule, find the unknown

voltages VR, VL, VC, and V1 for the circuit of fig

Solution:

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Example – Using the voltage divider rule, find the unknown

voltages VR, VL, VC, and V1 for the circuit of fig

Solution:

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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 XC 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 XC = R can be determined in following manner:

Since XC = 1/ωC = 1/2πfC,

Thus frequency at which XC = R is

which for the network of interest is

• For frequencies

• Less than f1:

XC > R

• Greater than f1:

R > XC

To examine the effect of frequency on the response of an R-C

series configuration, let us first determine how the impedance of the

circuit ZT 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 ZC = 159.16

kΩ /_ 90° if circuit was

purely capacitive (R =

0Ω) at 100 hz

Note ZT 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 ZT 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 θT versus

frequency suggests that

ZT made transition from

capacitive (θT = 90°) to

Resistive (θT = 0°).

Applying the voltage divider rule to determine

the voltage across the capacitor in phasor form

Thus magnitude and phase θC by which VC leads E is given by

To determine the frequency response, XC 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 VR)

leads E by 900, and angle between E and VC is 00.

We find that with an increase in frequency, VC begins a clockwise

rotation that will in turn increase the angle θC and decrease the

phase angle between I and E eventually approaching 0°.

A plot of VC versus frequency

A plot of θC 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 XL and VL would increase with frequency and the angle

between I and E would approach 90° (voltage leading the current)

rather than 0°.

If VL were plotted versus frequency, VL would approach E, and XL

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.

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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.

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(Series ac Circuits)

Voltage Divider Rule

Frequency Response of R-C Circuit

Summary of Series ac Circuits

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