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ELECTRIC CIRCUIT ANALYSIS - I

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ELECTRIC CIRCUIT ANALYSIS - I

Chapter 15 – Series & Parallel ac Circuits

Lecture 19

by MoeenGhiyas

Chapter 15 – Series & Parallel ac Circuits

- (Series ac Circuits)
- Impedance and Phasors Diagram
- Series Configuration

- Resistive Elements- For the purely resistive circuit,
- Time domain equations:v = Vm sin ωt and i = Im sin ωt
- In phasor form:
- WhereV = 0.707Vmandwhere I = 0.707Im

- Applying Ohm’s law and using phasor algebra, we have
- Since i and v are in phase, thus, θR = 0°, if phase is to be same.
- Thus, we define a new term, ZR as impedance of a resistive element (which impedes flow of current)

- Inductive Reactance- For the inductive circuit,
- Time domain equations:v = Vm sin ωt and i = Im sin ωt
- In phasor form:
- WhereV = 0.707Vmandwhere I = 0.707Im

- Applying Ohm’s law and using phasor algebra, we have
- Since i lags v by 90°, thus, θL = 90°, for condition to be true.
- Thus, we define term, ZL as impedance of an inductive element (which impedes flow of current)

- Capacitive Reactance- For a capacitive circuit,
- Time domain equations:v = Vm sin ωt and i = Im sin ωt
- In phasor form:
- WhereV = 0.707Vmandwhere I = 0.707Im

- Applying Ohm’s law and using phasor algebra, we have
- Since i leads v by 90°, thus, θC = –90°, for condition to be true.
- Thus, we define term, ZC as impedance of a capacitive element (which impedes flow of current)

- However, it is important to realize that ZR is not a phasor, even though the format is very similar to the phasor notations for sinusoidal currents and voltages.
- The term phasor is basically reserved for quantities that vary with time, whereas R and its associated angle of 0° are fixed, i.e. non-varying quantities.
- Similarly ZL and ZC are also not phasor quantities

- Example – Find the current i for the circuit of fig. Sketch the waveforms of v and i.
- Solution:
- In phasor form
- From ohm’s law
- Converting to time domain

- Sketch of waveformandPhasor Diagram

- Example – Find the voltage v for the circuit of fig. Sketch the waveforms of v and i.
- Solution:
- In phasor form
- From ohm’s law
- Converting to time domain

- Sketch of waveformandPhasor Diagram

- Example – Find the voltage v for the circuit of fig. Sketch the waveforms of v and i.
- Solution:
- In phasor form
- From ohm’s law
- Converting to time domain

- Sketch of waveformandPhasor Diagram

- Impedance Diagram- For any network,
- Resistance is plotted on the positive real axis,
- Inductive reactance on the positive imaginary axis, and
- Capacitive reactance on the negative imaginary axis.

- Impedance diagram reflects the individual and total impedance levels of ac network.

- Impedance Diagram
- The magnitude of total impedance of a network defines the resulting current level (through Ohm’s law)
- For any configuration (series, parallel, series-parallel, etc.), the angle associated with the total impedance is the angle by which the applied voltage leads the source current.
- Thus angle of impedance reveals whether the network is primarily inductive or capacitive or simply resistive.
- For inductive networks θT will be positive, whereas for capacitive networks θT will be negative, and θT will be zero for resistive cct.

- Overall properties of series ac circuits are the same as those for dc circuits
- For instance, the total impedance of a system is the sum of the individual impedances:

- EXAMPLE - Determine the input impedance to the series network of fig. Draw the impedance diagram.
- Solution:

- EXAMPLE - Determine the input impedance to the series network of fig. Draw the impedance diagram.
- Solution:

- Current is same in ac series circuits just like it is in dc circuits.
- Ohm’s law applicability is same.
- KVL applies in similar manner.
- The power to the circuit can be determined by
- where θT is the phase angle between E and I.

- Impedance Relation with Power Factor
- We know that
- Reference to figs and equations
- θT is not only the impedance angle of ZT but also θT is the phase angle between the input voltage and current for a series ac circuit.

Impedance Diagram

Phasor Diagram

Note: θT of ZT is with reference to voltage unlike FP . Also current I is in phase with VR, lags the VL by 90°, and leads the VC by 90°.

- R-L-C Example
- Step 1 – Convert Available information to Phasor Notation

- R-L-C Example
- .Step 2 – Find ZT and make impedance diagram

- R-L-C Example
- Step 3 – Find I or E

- R-L-C Example
- Step 4 – Find phasor voltages across each element

- R-L-C Example
- I =
- VR =
- VL =
- VC =
- .Step 5 – Make phasor diagram and
- . apply KVL (for verification or if req)

Note: Current I in phase with VR, lags the VL by 90°, and leads the VC by 90°

- R-L-C Example
- Step 6 – Convert phasor values to time domain

- R-L-C Example
- Step 7 – Plot all the voltages and the current of the circuit

- R-L-C Example
- Step 8 – Calculation of total power in watts delivered to the circuit
- or
- or

- R-L-C Example
- Step 9 – The power factor of the circuit is
- or

- (Series ac Circuits)
- Impedance and Phasors Diagram
- Series Configuration