Phasors and Impedance in Circuit Analysis

EE 261, Spring 2025, Lecture 35 Best viewed in presentation mode in PowerPoint. Goals/Topics: Phasors Impedance (resistance but with complex numbers) Circuit laws in sinusoidal steady state Announcement: Homework 12 posted (due Sunday, Apr. 20th).

Real part: [ ] Phasor transform: ?[ ] Phasors (script P ) Because of Euler s formula. ??cos ?? + ? = ???? ??+? ??cos ?? + ? = ???? ??+? ??cos ?? + ? = ???? ??+? = ????????? = ????????? = ????????? (bold; non-italic) = ? ????????? = ? ????????? = ? ????????? = ? ????????? = ? ????????? = ?????= ?? ? = ? frequency domain (has ? s; no ? s) = ?????= ?? ? = ? = ?????= ?? ? = ? = ?????= ?? ? = ? = ?????= ?? ? = ? ? ??cos ?? + ? ? ??cos ?? + ? ? ??cos ?? + ? ? ??cos ?? + ? ? ??cos ?? + ? time domain (has ? s; no ? s) ? is a phasor. Phasors are simply complex numbers that provide the magnitude (??) and phase (?) of a signal. ?[ ] discards/masks ????. Given a phasor, need to have ?to reconstruct time domain signal. (???? is common to all voltages and currents. Doesn t provide much information. Okay to ignore.) ? must be specified separately (phasor does not provide that). ?[ ] converts signals from the time domain to the frequency domain.

Consider: ??(??+?) ? ? ?? ?? ?? = ???sin ?? + ? ? ? ? ?? ? ?? ?? ?? ?? ? ? ? ????????? ????????? ????????? ????????? = ??????????? = ??????????? = ??????????? = ??????????? Combine exponents. Use Euler s formula. Combine exponents. Use Euler s formula. ??cos ?? + ? ??cos ?? + ? ??cos ?? + ? ??cos ?? + ? = = = = = ????cos ?? + ? + ?sin ?? + ? = ????cos ?? + ? ???sin ?? + ? imaginary real = ???sin ?? + ? These are equal! Differentiation in the time domain (all real values) is same as multiplication by ?? in the frequency domain (complex values)! ? ?? time domain ?? frequency domain

Phasors (from previous slide) = ? ????????? = ? ????????? = ? ????????? = ? ????????? = ?????= ? = ?????= ? = ?????= ? = ?????= ? ? ??cos ?? + ? ? ??cos ?? + ? ? ??cos ?? + ? ? ??cos ?? + ? ? ?? ?? ?? ?? ? ? ? = ? ??????????? = ? ??????????? = ? ??????????? = ? ??????????? = ???????= ??? = ???????= ??? = ???????= ??? = ???????= ??? ? ? ? ? ??cos ?? + ? ??cos ?? + ? ??cos ?? + ? ??cos ?? + ? Inverse phasor transform: ? 1[ ] Reconstruct time-domain signal. Multiply phasor by ????. Take real part. ? 1?????= ?????????= ??cos ?? + ? ? 1?????= ?????????= ??cos ?? + ? ? 1?????= ?????????= ??cos ?? + ? ?[ ] transform from time domain to frequency domain. ? 1[ ] transform from frequency domain to time domain.

?-? Relationships in Sinusoidal Steady State Claim that if ? and ? varying harmonically (???? dependence), for a resistor, inductor, or capacitor, they are related by ? ?? ?? ? ? = ??? ? ? ? = ??? ? , . ? ? = ?? ? , ?? ??

?-? Relationships in Sinusoidal Steady State Claim that if ? and ? varying harmonically (???? dependence), for a resistor, inductor, or capacitor, they are related by ? ?? No more derivatives! ?? ? ? = ?? ? , ? ? = ???? ? , ? ? = ???? ? . However, we don t actually want to do things quite like this. Want to maintain a clean separation between time domain and frequency domain. We are either in the time domain (and have ? s and purely real numbers and trig functions), or we are in the frequency domain (and there are complex numbers and no ? s). Phasors Phasors and impedances. (magnitude and phase) (analogous to resistance, i.e., voltage over current )

?-? Relationships in Sinusoidal Steady State: Resistor ? ? = ?? ? ? ? ? = ? ?? ? Take phasor transform of both sides. ? = ?? ? ?= ? Time domain. Frequency domain. Ohm s law in the frequency domain. The angle (i.e., the phase ) of ? and ? must be the same. Resistance ?is real and scales magnitude but doesn t change angle.

?-? Relationships in Sinusoidal Steady State: Inductor ? ? = ??? ? = ? ??? ? ? ? ? ?? ?? Take phasor transform of both sides. ? = ???? ? ?= ??? Time domain. Frequency domain. Ohm s law in the frequency domain. Current and voltage 90 out of phase (recall ? = ?? ? 2).

?-? Relationships in Sinusoidal Steady State: Capacitor ? ? = ??? ? = ? ??? ? ? ? ? ?? ?? Take phasor transform of both sides. ? = ???? ? ?= ? ?= 1 1 1 1 ???= ? ???= ? Time domain. Frequency domain. ?? ?? Ohm s law in the frequency domain. 1 ?= 1 ?= 1 ?= 1 ?= 1 1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = = = = 1= ? 1= ? 1= ? 1= ? Current and voltage 90 out of phase (recall ? = ? ? ? 2).

Impedance For resistive circuits, given source values and resistances (? = ?/?), can use algebra to solve for voltages and currents throughout the circuit. For resistive circuits, For resistive circuits, given source values and resistances (? = ?/?), For sinusoidal steady state, define impedance as the ratio of the voltage phasor to the current phasor: ? = ?/?. For sinusoidal steady state, For an arbitrary circuit in sinusoidal steady state, given source values and impedances (? = ?/?), values and impedances (? = ?/?), can use algebra to solve for voltages and currents throughout the circuit. For an arbitrary circuit in sinusoidal steady state, given source For an arbitrary circuit in sinusoidal steady state, No more derivatives or integrals! Algebra involves complex numbers. Currents and voltages expressed in phasor form.

Ratio of Voltage to Current (phasors) Impedance, Z Note: Passive sign convention is the same as always. Change sign if current flows into the lower polarity terminal. ? ?= ? ? ?= ??? ? ?= ? ?= 1 1 1 1 ???= ? Units of ohms ( ). ???= ? ?? ?? or ? = ?? In general: ? = ? is the resistance (always positive for a passive circuit). ? is the reactance (positive [inductive] or negative [capacitive]). Aside: Impedance is a complex number, but not a phasor. ? ? = ? + ??

? =1 Inverse of resistance ? is conductance ?. ? = ?? ? = ?? ? Series resistances add. Parallel conductances add. ? =1 Inverse of impedance ? is admittance?. ? = ?? (To be shown in a bit.) ? = ?? ? (To be shown in a bit.) ? = ? + ?? Series impedances add. Parallel admittances add. ? = ? + ?? ? is conductance. ? is susceptance. Note! For complex impedances, ? = 1 ?, but ? Complex conjugate of denominator. 1 ? and ? 1 ?. 1 ?= 1 ?= 1 ?= 1 ?= 1 ?= 1 1 1 1 1 1 1 1 1 1 ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? + ??= ? + ??= ? + ??= ? + ??= ? + ??= = = = = = ?2+ ?2 ? ?2+ ?2 ? ?2+ ?2 ? ?2+ ?2 ? ?2+ ?2 ? ?2+ ?2= ? + ?? ?2+ ?2= ? + ?? ?2+ ?2= ? + ?? ?2+ ?2= ? + ?? ?2+ ?2= ? + ?? ? + ?? ? + ?? ? + ?? ? + ?? ? + ?? ? ? and ? = ? = ?2+ ?2 ?2+ ?2

Notation: ???? often written as ??. Phasor Specification of Sources 120 30 V = 60 3 + j60 V = 120?? = 120 ? 6 V ? 6 V 120cos ?? + 30 V Frequency ? not explicitly shown. Must be provided separately. 7cos ?? + ? 4 A 2 + j 7 45 A = 7 7 2 A Phasors most often (and most conveniently) expressed in polar form. = 7?? = 7 ? 4 A ? 4 A

Phasor Specification of Sources 60 0 V 60cos ?? V Note that 60 0 is the same as the purely real number 60. So why not just write 60 V ? You can, but the 0 serves as a reminder that this is a sinusoidal voltage rather than a constant one. voltage rather than a constant one. NOTE: ?? is radians. Convert any phase shift to radians before adding to ??. Ensure calculator in radian mode as appropriate. to ??. Ensure calculator in radian mode as appropriate. to ??. Ensure calculator in radian mode as appropriate. You can, but the 0 serves as a reminder that this is a sinusoidal NOTE: ?? is radians. Convert any phase shift to radians before adding NOTE: ?? is radians. Convert any phase shift to radians before adding

Circuit Analysis Bottom line: Represent sources in terms of their phasors, represent elements in terms of their impedances, represent elements in terms of their impedances, solve for unknown quantities exactly as you would with a resistive circuit. Bottom line: Represent sources in terms of their phasors, Bottom line: Represent sources in terms of their phasors, All the circuit analysis techniques previously considered for resistive circuits with constant sources for resistive circuits with constant sources pertain to ones with impedance and phasors with impedance and phasors (simply use complex numbers instead of real ones!). All the circuit analysis techniques previously considered All the circuit analysis techniques previously considered for resistive circuits with constant sources pertain to ones (Will consider power later.)

time domain ?1+ ?2+ + ??= 0 Aside: ? is a linear operator. KVL: ? ?1+ ?2+ + ?? = 0 frequency domain ? ??1+ ??2 = ? ??1+ ? ??2 ?1+ ?2+ + ??= 0 = ??1+ ??2 Sum of voltage phasors around a closed loop must be zero. ?1+ ?2+ + ??= 0 KCL: ? ?1+ ?2+ + ?? = 0 ?1+ ?2+ + ??= 0 Sum of current phasors into or out of a node must be zero. Or, sum of phasors into a node equals sum of phasors out.

Series Elements Voltage Division ??? ??? ??? ??? ??? ??? =?? ??? ??? ??? =?? =?? ??= ??? = ?? ??= ??? = ?? ??= ??? = ?? ??? ??? ??? Same form as we had with resistances. ???= ?1? + ?2? + + ??? ???= ?1+ ?2+ + ??? = ???? ???= ?1+ ?2+ + ??? = ???? ???= ?1+ ?2+ + ?? Series impedances add. Note: Doesn t matter if a ? is due to ?, ?, or ?. All combine in the same way!

Parallel Elements Current Division =???? =???? =???? =??? =??? =??? ??=? ??=? ??=? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? Same form as we had with resistances. ? =? +? + +? Parallel admittances add. ?1 ?2 ?? ???= ?1+ ?2+ + ?? 1 ?1 ?1 ?1 ?1 1 1 1 +1 +1 +1 +1 + +1 + +1 + +1 + +1 1 1 1 1 ? = ? = ? = ? = ? = ? = ? = ? = ? = ?1+ ?2+ + ??? = ???? ? = ?1+ ?2+ + ??? = ???? ? = ?1+ ?2+ + ??? = ???? ? = ?1+ ?2+ + ??? = ???? ?2 ?2 ?2 ?2 +1 ?? ?? ?? ?? ??? ??? ??? ??? 1 1 ?1 1 ?? Parallel impedance combine like parallel resistors: inverse of sum of inverses. = + + ??? ?2