Chapter 18

AC Series-Parallel Circuits

AC Circuits

Rules and laws developed for dc circuits apply equally well for ac circuits

Analysis of ac circuits requires vector algebra and use of complex numbers

Voltages and currents in phasor form

Expressed as RMS (or effective) values

Ohm’s Law

Voltage and current of a resistor will be in phase

Impedance of a resistor is: Z_R = RÐ0°

Ohm’s Law

Voltage across an inductor leads the current by 90°(ELI the ICE man)

Ohm’s Law

Current through a capacitor leads the voltage by 90° (ELI the ICE man)

AC Series Circuits

Current everywhere in a series circuit is the same

Impedance used to collectively determine how resistance, capacitance, and inductance impede current in a circuit

AC Series Circuits

Total impedance in a circuit is found by adding all individual impedances vectorially

AC Series Circuits

Impedance vectors will appear in either the first or the fourth quadrants because the resistance vector is always positive

When impedance vector appears in first quadrant, the circuit is inductive

AC Series Circuits

If impedance vector appears in fourth quadrant

Circuit is capacitive

Voltage Divider Rule

Voltage divider rule works the same as with dc circuits

From Ohm’s law:

Kirchhoff’s Voltage Law

KVL is same as in dc circuits

Phasor sum of voltage drops and rises around a closed loop is equal to zero

Kirchhoff’s Voltage Law

Voltages

May be added in phasor form or in rectangular form

If using rectangular form

Add real parts together

Then add imaginary parts together

AC Parallel Circuits

Conductance, G

Reciprocal of the resistance

Susceptance, B

Reciprocal of the reactance

AC Parallel Circuits

Admittance, Y

Reciprocal of the impedance

Units for all of these are siemens (S)

AC Parallel Circuits

Impedances in parallel add together like resistors in parallel

These impedances must be added vectorially

AC Parallel Circuits

Whenever a capacitor and an inductor having equal reactances are placed in parallel

Equivalent circuit of the two components is an open circuit

Kirchhoff’s Current Law

KCL is same as in dc circuits

Summation of current phasors entering and leaving a node

Equal to zero

Kirchhoff’s Current Law

Currents must be added vectorially

Currents entering are positive

Currents leaving are negative

Current Divider Rule

In a parallel circuit

Voltages across all branches are equal

Series-Parallel Circuits

Label all impedances with magnitude and the associated angle

Analysis is simplified by starting with easily recognized combinations

Series-Parallel Circuits

Redraw circuit if necessary for further simplification

Fundamental rules and laws of circuit analysis must apply in all cases

Frequency Effects of RC Circuits

Impedance of a capacitor decreases as the frequency increases

For dc (f = 0 Hz)

Impedance of the capacitor is infinite

Frequency Effects of RC Circuits

For a series RC circuit

Total impedance approaches R as the frequency increases

For a parallel RC circuit

As frequency increases, impedance goes from R to a smaller value

Frequency Effects of RL Circuits

Impedance of an inductor increases as frequency increases

At dc (f = 0 Hz)

Inductor looks like a short

At high frequencies, it looks like an open

Frequency Effects of RL Circuits

In a series RL circuit

Impedance increases from R to a larger value

In a parallel RL circuit

Impedance increases from a small value to R

Corner Frequency

Corner frequency is a break point on the frequency response graph

For a capacitive circuit

w_C = 1/RC = 1/t

For an inductive circuit

w_C = R/L = 1/t

RLC Circuits

In a circuit with R, L, and C components combined in series-parallel combinations

Impedance may rise or fall across a range of frequencies

In a series branch

Impedance of inductor may equal the capacitor

RLC Circuits

Impedances would cancel

Leaving impedance of resistor as the only impedance

Condition is referred to as resonance

Applications

AC circuits may be simplified as a series circuit having resistance and a reactance

AC circuit

May be represented as an equivalent parallel circuit with a single resistor and a single reactance

Applications

Any equivalent circuit will be valid only at the given frequency of operation


	Rules and laws developed for dc circuits apply equally well for ac circuits
	Analysis of ac circuits requires vector algebra and use of complex numbers
	Voltages and currents in phasor form
		Expressed as RMS (or effective) values


	Voltage and current of a resistor will be in phase
	Impedance of a resistor is: Z_R = RÐ0°


	Voltage across an inductor leads the current by 90°(ELI the ICE man)


	Current through a capacitor leads the voltage by 90° (ELI the ICE man)


	Current everywhere in a series circuit is the same
	Impedance used to collectively determine how resistance, capacitance, and inductance impede current in a circuit


	Total impedance in a circuit is found by adding all individual impedances vectorially


	Impedance vectors will appear in either the first or the fourth quadrants because the resistance vector is always positive
	When impedance vector appears in first quadrant, the circuit is inductive


	If impedance vector appears in fourth quadrant
		Circuit is capacitive


	Voltage divider rule works the same as with dc circuits
	From Ohm’s law:


	KVL is same as in dc circuits
	Phasor sum of voltage drops and rises around a closed loop is equal to zero


	Voltages
		May be added in phasor form or in rectangular form
	If using rectangular form
		Add real parts together
		Then add imaginary parts together


	Conductance, G
		Reciprocal of the resistance
	Susceptance, B
		Reciprocal of the reactance


	Admittance, Y
		Reciprocal of the impedance
	Units for all of these are siemens (S)


	Impedances in parallel add together like resistors in parallel
	These impedances must be added vectorially


	Whenever a capacitor and an inductor having equal reactances are placed in parallel
		Equivalent circuit of the two components is an open circuit


	KCL is same as in dc circuits
	Summation of current phasors entering and leaving a node
		Equal to zero


	Currents must be added vectorially
	Currents entering are positive
	Currents leaving are negative


	Label all impedances with magnitude and the associated angle
	Analysis is simplified by starting with easily recognized combinations


	Redraw circuit if necessary for further simplification
	Fundamental rules and laws of circuit analysis must apply in all cases


	Impedance of a capacitor decreases as the frequency increases
	For dc (f = 0 Hz)
		Impedance of the capacitor is infinite


	For a series RC circuit
		Total impedance approaches R as the frequency increases
	For a parallel RC circuit
		As frequency increases, impedance goes from R to a smaller value


	Impedance of an inductor increases as frequency increases
	At dc (f = 0 Hz)
		Inductor looks like a short
		At high frequencies, it looks like an open


	In a series RL circuit
		Impedance increases from R to a larger value
	In a parallel RL circuit
		Impedance increases from a small value to R


	Corner frequency is a break point on the frequency response graph
	For a capacitive circuit
		w_C = 1/RC = 1/t
	For an inductive circuit
		w_C = R/L = 1/t


	In a circuit with R, L, and C components combined in series-parallel combinations
		Impedance may rise or fall across a range of frequencies
	In a series branch
		Impedance of inductor may equal the capacitor


	Impedances would cancel
		Leaving impedance of resistor as the only impedance
	Condition is referred to as resonance


	AC circuits may be simplified as a series circuit having resistance and a reactance
	AC circuit
		May be represented as an equivalent parallel circuit with a single resistor and a single reactance


	Any equivalent circuit will be valid only at the given frequency of operation