Lecture 04: AC SERIES/ PARALLEL CIRCUITS, VOLTAGE/ CURRENT DIVIDER

OBJECTIVES 1.Compute the total circuit impedance for series RLC circuits. 2.Explain the effects of frequency changes on RLC circuit response. 3.Determine circuit voltages and currents in series RLC circuits. 4.Compute voltage drops across components in an RLC circuit using the voltage divider formula.

Example #1: Parallel Circuits

(a) Find total impedance The circuit given is already in freq domain:The circuit given is already in freq domain:

Circuit simplification

(b) Draw Impedance Triangle ZTZTZTZT XCXCXCXC R ZTZTZTZT   j - j R

(c) Find is, i C, i RL

(d) Using Current Divider

Example #2: Find i and v c

Elements in Frequency domain =4 rad/s Time domain Freq domain

Circuit in Freq Domain

(a) The current, i The current in series circuit,The current in series circuit, The total impedance is;The total impedance is;

(b) The capacitor voltage, V C

Example #3: Find i X (time domain) Solution:

Example #4: Find Z 1 and Z 2 Solution:

Example #5: Find i(t)

Different waveform of V S  V S1 and V S2 are different:  When analyzing ac circuit, it is expedient to express both signals as either Sine or Cosine.

Change the waveform of V S1 Using identity:Using identity:

Changing Cosine to Sine:

Example #6: (a) Find i(t)

Different waveform of V S  V S1 and V S2 are different:  When analyzing ac circuit, it is expedient to express both signals as either Sine or Cosine.

Change the waveform of V S1 Using identity:Using identity:

Redraw the circuit

Elements in Frequency domain  =4 rad/s  =4 rad/s Time domain Freq domain

Find The Current

Solution:

(b) Find the v c (t), v L (t)

(c) Find the i c (t), i L (t)

(d) Sketch the waveform for v c (t), v L (t), i c (t), i L (t) (e) Explain the phase relationship between V and I for capacitor and inductor.

Example #7: Find Z T, I T, V C2, P Z2Z2

Find the total impedance:Find the total impedance: Then, the total current is:Then, the total current is: Solution

Solution P = VI cos (  T ) = (10V)(461  A) cos(-42.4 o ) = 3.40 mW

Ex: Find Z T, I L3, P Z3Z3 Z2Z2

Z 2, Z 3, Z T, and I T

I R3, I L3, P P = EI cos (  T ) = (10V)(1.01 mA) cos(31.4 o ) = 8.62 mW

TOOLS Note from the previous examples, all of our tools from DC work:Note from the previous examples, all of our tools from DC work: –Ohm’s Law –Voltage Divider –Current Divider –Kirchoff’s Current Law –Kirchoff’s Voltage Law They are the same – but we use COMPLEX NUMBERS.They are the same – but we use COMPLEX NUMBERS.