INDEX

UNIT 2: RC CIRCUITS

2.1           Sinusoidal Response of RC Circuits   2.2           Impedance and Phase Angle of Series RC Circuits   2.3             Analysis of Series RC Circuits 2.3.1         Phase Relationships of Current and Voltages 2.3.2         Variation of Impedance and Phase Angle with Frequency 2.3.3         The RC Lag Circuit 2.3.4         The RC Lead Circuit     2.4           Impedance and Phase Angle of Parallel RC Circuits 2.4.1         Conductance, Susceptance and Admittance   2.5         Analysis of Parallel RC Circuits   2.6         Combination of Series and Parallel RC Circuits   2.7         Power in RC Circuits   2.8         Summary   2.9          Self-Assessments

Learning Objective:

The objective of this unit is to:

1. Determine impedance and admittance in a series, parallel and combination series-parallel RC circuit and express the impedance and admittance in polar and rectangular form.
2. Analyze a series, parallel or combination series-parallel RC circuit.
3. Explain the true power, reactive power and apparent power.
4. Draw the power triangle and calculate power factor.
5. Calculate power in RC circuit.

Learning Outcome:

At the end of lesson student should be able:

1. Familiarize with the terms of impedance, conductance, susceptance and admittance of RC circuits.
2. Determine the phase relationship between current and voltage of RC circuit.
3. Analyze simple RC circuits using basic circuit law.
4. Determine power and power factor in RC circuits.

2.1 SINUSOIDAL RESPONSE OF RC CIRCUITS

When a sinusoidal signal is applied to a series RC circuit, each resulting voltage drop and the current in the circuit are also sinusoidal and have the same frequency as the applied voltages. The capacitance causes a phase shift between the voltages and current.

Figure 2.1:  RC Circuit

To explain this, consider the RC circuit with a sinusoidal voltage source shown in Figure 1. The amplitudes and the phase relationships of the voltages and current depend on the values of of the resistance and the capacitive reactance.

• If the circuit is purely resistive (the capacitive reactance is zero), the phase angle between the applied (source) voltage and the total current is zero.

Figure 2.2: V_S and I for purely resistive circuit

• If the circuit is purely capacitive (the resistance is zero), the phase angle between the applied (source) voltage and the total current is 90°, with the current leading the voltage.

    Figure 2.3:  V_S and I for purely capacitive circuit

• If the circuit is a combination of both resistance and capacitive reactance, the phase angle between the applied (source) voltage and the total current is somewhere between 0° and 90°, depending on the relative values of the resistance and the capacitive reactance.  

Figure 2.4:  V_S and I if both resistor and capacitor exist in a circuit

2.2 IMPEDANCE AND PHASE ANGLE OF SERIES RC CIRCUITS

The impedance of a series RC circuit consist of resistance and capacitive reactance. Its unit is ohm. The impedance causes a phase difference between the total current and the source voltage of the RC circuit.

• In purely resistive circuit, the impedance is simply equal to the total resistance
• In purely capacitive circuit, the impedance is equal to the total capacitive reactance.

The impedance of a series RC circuit is determined by both resistance and the capacitive reactance. The magnitude of the impedance is symbolized by Z.

Figure 2.5: Impedances

Capacitive reactance is a phasor quatity and is expressed as a complex number in rectangular form as X_C = -jX_C. X_C is the phasor quantity (representing both magnitude and angle) whereas X_C is just the magnitude.

For the series RC circuit of Figure 2.6, the total impedance is the phasor sum of R and –jX_C and is expressed as

Z = R – jX_C                  (2.1)

In a series RC circuit, the resistor voltage (voltage across resistor) is in phase with the current whereas the capacitor voltage (voltage across capacitor) lags the current by 90°. Thus, the capacitor voltage lags the resistor voltage by 90°. In ac analysis , both R and XC can be represented in a phasor diagram shown in Figure 2.7(a), in which X_C appearing at a -90° angle with respect to R. 

Figure 2.6:  Impedance, Z = R – jX_C

Since Z is the phasor sum of R and –jXC, it can be represented in a phasor diagram shown in Figure 2.7(b). By repositioning the phasors, as shown in Figure 2.7(c). Forms a right triangle known as the impedance triangle. The length of each phasor represents the magnitude in ohms, and the angle q is the phase angle of the RC circuit. The angle q represents the phase difference between the applied voltage and the current.

Figure 2.7: Development of the impedance triangle for a series RC circuit

From pythagoreon theorem (right angle trigonometry), the magnitude (length) of the impedance can be expressed in terms of the resistance and reactance as

The phase angle, q is expressed as

Combining the magnitude and angle, the phasor expression for impedance in polar form is

Example 2.1: 

For each circuit in Figure 2.8, write the phasor expression for the impedance in both rectangular form and polar form.

          Figure 2.8

Solution:       

For the circuit in Figure 2.8(a), the impedance is

The impedance is simply the resistance, and the phase angle is zero because pure resistance does not cause a phase shift between the voltage and current.

For the circuit in Figure 2.8(b), the impedance is

The impedance is simply the capacitive reactance, and the phase angle is -90° because the capacitance causes the current to lead the voltage by 90°.

For the circuit in Figure 2.8(c), the impedance is

 The impedance  in polar form is

2.3 ANALYSIS OF SERIES RC CIRCUITS

2.3.1  PHASE RELATIONSHIPS OF CURRENT AND VOLTAGES

In a series RC circuit, the current through the resistor and the capacitor is the same. Thus, the resistor voltage is in phase with the current while the capacitor voltage lags the current by 90°. Thus, the phase difference between the resistor voltage, V_R and the capacitor voltage, V_C is 90° as shown in Figure 2.9.

Figure 2.9: Phase relationship between voltages and currents in a series RC circuit

From Kirchoff’s voltage law, the sum of the voltage drops must be equal to the applied voltage. However, since V_R and V_C are not in phase with each other, they must be added as phasor quantities. Voltage phasor diagram for a series RC circuit is shown in Figure 2.10(a). From Figure 2.10(b), V_S is the phasor sum of VR and VC, which can be expressed in rectangular form as

V_S = V_R – jV_C

or in polar form as

where

is the magnitude of the souce voltage and 

is the phase angle between the source voltage and the current. As the resistor voltage and the current are in phase, also represents the phase angle between the source voltage and the current.

Figure 2.10:  Voltage phasor diagram for a series RC circuit

2.3.2     VARIATION OF IMPEDANCE AND PHASE ANGLE WITH FREQUENCY

The total impedance in series RC circuit is given by

We can observed that when XC increases, the magnitude of total impedance, Z increases and when XC decreases, the magnitude of total impedance, Z decreases. As the capacitive reactance X_C varies inversely with frequency

thus in an RC circuit, Z is inversely dependent on frequency. Figure 2.11 uses the impedance triangle to illustrate the variation of XC, Z and q as the frequency changes. R remains constant as it does not depend on frequency. As XC varies inversely with the frequency, so also do the magnitude of total impedance and the phase angle.

Figure 2.11:  As the frequency increases, X_C decreases, Z decreases and θ decreases

2.3.3  THE RC LAG CIRCUIT

An RC lag circuit is a phase shift circuit in which the output voltage lags the input voltage by a specified amount. Figure 2.12 shows a series RC lag circuit. The source voltage is the input voltage, Vin whereas the output voltage, Vout is taken accross the capacitor. Θ is the phase angle between I and Vin whereas Ф is the angle between Vout and Vin.

Figure 2.12: Basic RC lag circuit

The input voltage and current in polar form are

respectively.  The output voltage in polar form is

The resulting equation shows that the output voltage is at an angle of –90°+ө with respect to the input voltage. The angle Ф between the input and output is,

or, similarly:

The negative sign indicates that the output voltage lags the input voltage.

2.3.4 THE RC LEAD CIRCUIT

An RC lead circuit is a phase shift circuit in which the output voltage leads the input voltage by a specified amount. When the output of a series RC circuit is taken across the resistor instead of capacitor, as shown in Figure 2.13, it becomes a lead circuit.

Figure 2.13: Basic RC lead circuit

In a series RC circuit, the current leads the input voltage and the resistor voltage is in phase with the current. As the output voltage is taken across the resistor, the output leads the input.

When the input voltage is assigned a reference angle of 0°, the angle of the output voltage is is the same as ө which is the angle between the applied voltage and total current because the resistor voltage and the current are in phase with each other. Thus Ф = θ, which can be expressed by the following equation:

The positive sign indicates that the output voltage leads the input voltage.

Example 2.2:

Refer to Figure 2.14

(i)  Determine the total impedance

(ii)  Calculate the current, I

(iii)  Determine the resistor and capacitor voltages.

(iv)  Draw a phasor diagram showing all the voltages and current in the circuit.

(v)  Write the relationship between the source current and the source voltage.

Figure 2.14

Solution:

The reactance, X_C = 1/jωC =1/ j(100)(318µ) = -j31.45Ω

(i)  The total impedance can be obtained as follow:

The magnitude of total impedance;

And the phase angle of total impedance is:

            q = -tan^-1 (Xc/R) = -tan^-1(31.45/20) = -57.55^o

Therefore,

(ii)        The current, I :

            From ohm’s law,

(iii)       The resistor voltage,

V_R = IR = (1.61) 20 = 32.2V

The capacitor voltage,

            V_C = IX_c = 1.61(31.45) = 50.63V

The voltage division can also be used to determine the voltage across resistor and inductor, i.e.,

(iv)       Phasor diagram:

            Take current as a reference, as the current flowing through R and C are equal.

            For resistor : The voltage across resistor is in phase with the current flowing through  

            it.

            For capacitor : The voltage across capacitor lags current flowing through it by 90°.

            The current I leads the source voltage by 57.55°.

            Therefore, the phasor diagram is as shown below.

            Checking:  The source voltage,

equal as given in the question

(v)  The current I leads the source voltage by 57.55° or the source voltage lags the current I by 57.55°.

2.4 IMPEDANCE AND PHASE ANGLE OF PARALLEL RC CIRCUITS

A parallel RC circuit consist of resistance and capacitive reactance which are connected in parallel with each other as shown in Figure 2.15, which consist of a voltage source (Vs), a resistor (R) and a capacitor (C)

Figure 2.15: Basic Parallel RC circuit

As the capacitor and the resistor of the RC circuit in Figure 2.15 are parallel with each other, the total impedance of the circuit can be found by using ‘product over the sum formula’. If the expression for resistance and capacitive reactance in polar form are given as

then the total impedance is calculated as follows

by multiplying the magnitudes, adding the angles in the numerator, and converting the denominator to polar form, Z_tot becomes:

then, by dividing the magnitude expression in the numerator by that in the denominator, and by substracting the angle in the denominator from that in the numerator, Z_tot becomes:

similarly, the expression can be written as follows:

2.4.1   CONDUCTANCE, SUSCEPTANCE AND ADMITTANCE

            Conductance, G is :

G = 1/R                                                                                  (2.6)

            Susceptance, B :

B=1/X_C                                                                                                   (2.7)

            Admittance, Y :

Y=1/Z                                                                                    (2.8)

            The unit for G, B and Y is the Siemens (S)

In the basic parallel RC circuit shown in Figure 2.14, the admittance and phase angle of admittance are given by:

Example 2.3:

Determine the total admittance and total impedance for the circuit in Figure 2.16

Figure 2.16

Solution:

The conductance,

The susceptance;

            B_C = 1/X_C

Where X_C = 1/2pfC = 1/(2p x 1500 x 0.075 µ)  = 1.41 kW

B_C = 1/X_C  = 1/1410 = 709µS

Thus, the admittance is:

The total impedance,

2.5 ANALYSIS OF PARALLEL RC CIRCUITS

Figure 2.17 shows that the applied voltage (Vs), capacitor and resistor are parallel with each other. Thus, the applied voltage, the voltage across the capacitance and resistive branches are all of the same magnitude and in phase. Also, Figure 2.17 shows that the total current, I_total divides at the junction into the two branch currents, I_R and I_C. 

Figure 2.17: Currents in a parallel RC circuit

The current through the resistor is in phase with the voltage. The current through the capacitor leads the voltage, and thus leads the resistive current by 90°. From Kirchhoff’s current law, the total current (I_tot) is the phasor sum of the resistive branch current and the capacitive branch current.  The total current is expressed in rectangular form as follows;

The equation can be expressed in polar form as follows;

the magnitude of the total current is

and the phase angle between the resistive branch current and the total current is

Figure 2.18 shows a complete current and voltage phasor diagram. From the figure, θ also represents the phase angle between the total current and the applied voltage

Figure 2.18: Current and voltage phasor diagram for a parallel RC circuit

Example 2.4 :

For the circuit in Figure 2.19, determine the value of each current and describe the phase relationship of each current with the applied voltage. Then draw the current phase diagram.

       Figure 2.19

Solution :

The resistor current is calculated as follows:

The capacitor current is calculated as follows:

The total current (in rectangular form) is calculated as follows:

The total current in polar form

The results show that the resistor current is 54.5 mA and is in phase with the supplied voltage, V_S. The capacitor current is 80mA and leads the supplied voltage, V_S by 90°. The total current is 96.8mA and leads the voltage by 55.7°. The relationships between the currents and the suppplied voltage is illustrated in phasor diagram shown in Figure 2.20.

Figure 2.20

2.6 COMBINATION OF SERIES AND PARALLEL RC CIRCUITS

The series RC circuit and the parallel RC circuit may exist in the same circuit as shown in Figure 2.21. From section 2.3 and 2.5, the calculation of impedance of the series components is easily expressed in rectangular form whereas the calculation of impedance of the parallel components is easily expressed in polar form. Thus, to analyse a series and parallel RC circuits, first express the impedance of the series components of the circuit in rectangular form and the impedance of the parallel component in polar form. Then, convert the impedance of the parallel part to rectangular part and add it to the impedance of the series part. Then convert the resulting impedance, which is in rectangular form  to polar form in order to get the magnitude and angle of the impedance.

Figure 2.21: Combination Series-parallel RC circuit

 

Example 2.5:

For the circuit in Figure 2.22, determine:

1. total impedance
2. total current
3. phase angle

Figure 2.22

Solution:

(a)  Firstly, calculate the magnitude of capacitive reactance (X_C1 and X_C2)

To find the total impedance, calculate the series portion and the parallel portion of the impedance and add them.

The impedance of the series portion (Z₁) is

To determine the impedance of the parallel portion, first determine the admittance of the parallel portion:

Converting the admittance, Y into polar form yields:

Then, the impedance of the parallel portion (Z₂) is

Converting into rectangular form yields:

The total impedance can be calculated as follows:

Expressing the total impedance in polar form yields:

The total current can be determined by using Ohm’s law:

The total current leads the supplied voltage by 26.2°.

2.7 POWER IN RC CIRCUITS

The power in a resistor known as the true power, P_true or real power or active power is defined as the multiplication of the resistance and the square of current flow through the resistor. The expression for true or real or active power is written as follows

P_true = I²R                                                                                 (2.11)

Where I is the current through the resistor and R is the resistance of the resistor. The unit for true power is watt (W).

The power in a capacitor known as the reactive power is defined as the multiplication of the capacitive reactance and the square of current flow through the capacitor. The expression for reactive power is written as follows

Q = I²X_C                                                                                                                                (2.12)

Where I is the current through the resistor and X_C is the capacitive reactance of the capacitor. The unit for reactive power is VAR (volt-ampere reactive).

The general impedance phasor diagram for a series RC circuit is shown in Figure 2.23.

Figure 2.23

The resultant power phasor is called the apparent power, P_a, which is defined as the multiplication of the impedance and the square of current flow through the impedance. The expression for apparent power is written as follows

P_a = I²Z                                                                               (2.13)

Where I is the current through the resistor and Z is the impedance of the RC circuit. The unit for apparent power is Volt-Ampere (VA).

Alternatively, the power phasor in Figure 2.23 can be rearranged in the form of a right angle triangle, as shown in Figure 2.24 which is known as power triangle. Using the rules of trigonometry, true power can be expressed as follows:

P_true = P_acosθ                                                                         (2.14)

Or similarly

P_true = VIcosθ                                                                          (2.15)

Where V is the applied voltage and I is the total current.

For the case of a purely:

i) resistive circuit, θ=0° and cos 0°=1, thus P_true=VI

ii) capacitive circuit θ=90° and cos 90°=0, thus P_true=0

Power Factor, PF is defined as follows:

PF = cos θ.                                                                             (2.16)

The equation shows that the power factor depends on the phase angle between the applied voltage and total current. The power factor can vary from 0 for purely reactive circuit to 1 for purely resistive circuit. In an RC circuit, the power factor is referred to as a leading power factor because the current leads the voltage. 

Figure 2.24: Power Triangle

Example 2.6:

For the circuit in Figure 2.25, find the true power, the reactive power and the apparent power.

Figure 2.25

Solution :

The capacitive reactance is expressed as follows:

The current through the resistor and capacitor are calculated as follows:

The true power is:

The reactive power is:

The apparent power is:

2.8 SUMMARY

1. When sinusoidal voltage is applied to an RC circuit, the current and all voltage drops are also sine waves.

2. Total currents in an RC circuit always leads the source voltage.

3. For resistance, the voltage across it is in phase with the current through it.

4. In capacitor, the current leads the voltage by 90°.

5. The impedance of RC circuit is the ratio of source voltage to total current, i.e.

6. In an RC circuit, part of the power is resistive (real power) and part reactive (reactive power).

7. Power factor of an RC circuit is given by:

Where q is the angle between the current and voltage of the circuit and can be calculated as:

8. In an RC circuit, the power factor is said to be leading.

2.9 SELF-ASSESSMENTS

1. For each circuit in Figure Q2.1, write the phasor expression for the impedance in both rectangular form and polar form.

2. Determine the series element or elements  that must be installed in the block of figure below  to meet the following requirements. True power = 400 W and there is a leading power factor ( I_total leads V_S)

3. In a series RC circuit, the true power is 3 W, and the reactive power is 3.5 VAR. Determine the apparent power in polar form and rectangular form.

4. In a certain RC circuit, the true power is 15 W and the apparent power is 45.54Ð70.8°. Determine the reactive power and then draw the power triangle showing the true power, reactive power and apparent power.

5. For the circuit in the figure below, find the true power, the reactive power and the apparent power.