BME/ISE 3511 Bioelectronics - Test Six Course Notes Fall 2016

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1 BME/ISE 35 Bioelectronics - Test Six ourse Notes Fall 06 Alternating urrent apacitive & Inductive Reactance and omplex Impedance R & R ircuit Analyses (D Transients, Time onstants, Steady State) Electrical Theory (Alternating urrent) harge Q V oulombs urrent I dq/dt Amperes Ohm s aw for A I RMS V RMS / Z where Z is the omplex Impedance Z [ (R ( - ) ] / θ tan - [ ( - ) / R ] Power Factor cos θ R / Z Joule s aw Average Power ½ V peak I peak cos θ V RMS I RMS cos θ Watts i dv/dt v / i dt for v V p sin ω t i d(v p sin ω t)/dt ω V p cos ω t ω V p sin (ω t π /) v di/dt i / v dt for i I p sin ω t v d(i p sin ω t)/dt ω I p cos ω t ω I p sin(ω t π /) EI the IE man omponent Voltage / urrent Resistor In Phase apacitor ags Inductor eads apacitive & Inductive Reactance and omplex Impedance ω πf f 0.59ω apacitive Reactance / ω / (πf ) 0.59 / f Inductive Reactance ω πf omplex Impedance R in series with series Z R j(πf - /(πf )) Impedance is a minimum at resonance R in series with parallel Z R j(πf / ( - (πf) )) Impedance is a maximum at resonance Time onstants R ircuit R ircuit Time onstant R Time onstant / R

2 Electrical Theory Quantity Symbol Unit Equation harge Q coulomb Q idt Q V urrent I ampere I dq/dt Voltage V volt V dw/dq Energy W joule W VdQ VIdt Power P watt P dw/dt IV Electrical Theory - continued Quantity Symbol Unit Equation Resistor R ohm V IR Inductor henry V di/dt I / Vdt apacitor farad V / Idt I dv/dt Electrical Theory - continued Series and Parallel omponents Ohm s aw For D: I V/R Reactance (Resistance to A) For A: I V/Z apacitive Reactance / jω -j/ω Inductive Reactance jω Series Impedance R ( - ) R j(ω /ω) R [ R ( - ) ] / θ Tan - [( - ) / R] 3 omponent Series Parallel R R Eq R R R 3 /R Eq /R /R /R 3 Z Z Eq Z Z Z 3 /R Eq /Z /Z /Z 3 Eq 3 / Eq / 4 / / 3 D ircuit omponents A Sinusoidal Analysis omponent Impedance urrent Power/Energy R R I V/R I R Zero Infinite / I Infinite Zero / V Resistor R I V/R P I R V /R Inductor jω I -jv /ω Q I V / apacitor -j/ω I jv ω Q I V / 5 urrent I I R j I Voltage V V R j V omplex Power S VI* (V R j V )(I R - j I ) omplex Power S P j Q 6

5 Equations and Relationships Inductive Reactance π f apacitive Reactance π f R ircuit R ircuit R ircuit ut-off Frequency Resonant Frequency f 0 f 0 π R π / R f 0 π Time onstant t R R / t t / R R Series Impedance Z R ( ) R& Parallel Impedance Z R R Parallel Impedance Z R R( )

6 ommon onfiguration R V in R V out Notes: When ω 0,, i.e., appears as an open circuit, so that V R out R R R V in R V out When ω >> 0, 0, i.e., appears as a short circuit, so that V 0 out R R V in R V out V in V out

7 Understanding the Behavior of omplex Impedances Understanding the Behavior of omplex Impedances at very low frequency (i.e., f 0) and at very high frequency (i.e., f >> 0 or f ). For Inductive Reactance, ω If ω 0, then ω 0 If ω >> 0, then ω >> 0 For apacitive Reactance, / ω If ω 0, then / ω >> 0 If ω >> 0, then / ω 0 R R regardless of ω. From Ohm's aw for Alternating urrent, Impedance Z v(t) / i(t) Z R j() where is the reactant component due to circuit capacitors and inductors at a given frequency. IF Z 0, then Short, i.e., like a wire conductor, very high current (I V/Z). If Z >> 0 (Z ), then Open, i.e., like a open switch, no current. A Short in parallel with any other number of elements appears as a Short overall; i.e., just a wire conductor. Short An Open in parallel with another element can be considered to be no existent, i.e., no effect. No Effect A Short in series with another element can be considered to be just a wire conductor. No Effect An Open in series with another number of elements appears as an Open Switch. Open Switch Page of.

8 Understanding the Behavior of omplex Impedances - continued If ω 0, then ω 0 and / ω >> 0. If ω >> 0, then ω >> 0 and / ω 0. R R regardless of ω. IF Z 0, then Short, i.e., like a wire conductor, very high current ( I V/R). If Z >> 0 (Z ), then Open, i.e., like a open switch, no current. For series: Z 0 Z and Z (Open) Z (0) For parallel, Z 0 0 (Short) and Z Z 0 Z( ) Z ( ) Z Z From omplex Impedance Quiz & BME/ISE 35Fall 05 Test Four A. R if ω 0, then R R 0 R (Note: 0) Overall Effect Resistive if ω >> 0, then R R 0 R (Note: 0) Overall Effect Resistive B. R R if ω 0, then R R Overall Effect Resistive R 0 R R 0 0 R R (Note: 0 and >> 0) if ω >> 0, then Overall Effect Open R 0 Open (Note: and 0) R. R R if ω 0, then R Overall Effect Open R R (0) R 0 0 Open (Note: >> 0 and 0) if ω >> 0, then R R Overall Effect Resistive R R 0 R 0 0 R R (Note: 0 and >> 0) Page of.

9 Review of omplex Numbers A complex number represents a point in a D space. The value of the complex number can be represented either by its real part (a) and imaginary part (b), or by its magnitude (M) and its phase angle (θ), as shown in the figure below. W a jb θ M e j ( M cosθ) j( M sinθ) Imaginary axis W b M θ a Real axis omputations involving two complex numbers: x a jb jθ M e and y c jd jϕ N e (a) Addition and subtraction: z x ± y (a c) j (b d) (b) Multiplication: z x y (a jb) (c jd) (ac bd) j (bc ad) z x y ( M e jθ jϕ j( θ ϕ ) )( Ne ) ( MN) e Therefore, Magnitude of z (magnitude of x) (magnitude of y) Phase angle of z (phase angle of x ) (phase angle of y) (c) Division: z x y a c jb jd ( a jb)( c ( c jd )( c jd) jd) ac bd c d bc ad j c d z x y M e N e jθ jϕ M N e j( θ ϕ ) Therefore, Magnitude of z (magnitude of x) (magnitude of y) Phase angle of z (phase angle of x ) - (phase angle of y)

BME/ISE 3511 Bioelectronics - Test Five Review Notes Fall 2015

BME/ISE 3511 Bioelectronics - Test Five Review Notes Fall 2015 BME/ISE 35 Bioelectronics - Test Five Review Notes Fall 205 Test Five Topics: RMS Resistive Power oss (I 2 R) A Reactance, Impedance, Power Factor R ircuit Analysis alculate Series R Impedance alculate