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Steady State Analysis Ch. 9 – Steady State Analysis 2 Sinusoidal Source This may be voltage or current i Im cos t Vm is the amplitude φ is the phase angle (expressed in radians) ω is the angular frequency (expressed in radians/s) T is the period (expressed in seconds) f is the frequency expressed is Hz or s-1 v Vm cos t Root Mean Square (RMS) Proof: VRMS Vm 1 2T 1 T t0 2 1 T t0 1 cos 2 t 2 V cos t dt V dt m m T t0 T t0 2 T t0 t0 dt T t0 t0 cos 2 t dt Vm VRMS Vm 2 ELEC 250 – Summer 2015 V 1 T 0 m 2T 2 Ch. 9 – Steady State Analysis 3 Angular frequency and frequency Frequency is the number of cycles per second Angular frequency corresponds to angular displacement 2π multiply by the frequency. Notes on sinusoidal All sinusoidal expressions involve the angular frequency and time and NOT the frequency f Sinusoidal are periodic with period 2π. The phase φ is the amount of shift a sinusoidal exhibits as compared to a reference and it is expressed in radians vRef Vm cos t v Vm cos t iRef Im cos t i Imcos t ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 4 Example: Response of RL circuit to a sinusoidal source Applying KVL : di L Ri Vm cos t dt The solution of this non-homogeneous differential equation comprises the solution of the homogeneous plus a particular solution of the non-homogeneous i t in i f i t Vm R L 2 2 2 cos e R / Lt tan Vm R L 2 L R ELEC 250 – Summer 2015 2 2 cos t Ch. 9 – Steady State Analysis 5 Example: Response of RC circuit to a sinusoidal source R Applying KVL: dvC RC vc Vm cos t dt v +- C The solution of this non-homogeneous differential equation comprises the solution of the homogeneous plus a particular solution of the non-homogeneous v t vn v f vC t Vm 1 R C 2 2 cos e t RC Vm 1 R C tan RC ELEC 250 – Summer 2015 2 2 cos t Ch. 9 – Steady State Analysis 6 Phasor Concept Given a sinusoidal signal v t Vm cos t e j cos j sin Euler’s identity • We can use Euler’s equality to rewrite it as v t Vm cos t Vm e jt Vme jt e j Vme j e jt The complex quantity Vm e j carries the amplitude and phase angle of a given sinusoidal signal ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 7 Phasor abbreviation This complex quantity is defined as the phasor representation or the phasor transform of v(t) V m cos t Vme j Vme Vm j We can use the phasor transform to simplify finding the steady state behavior of our circuits under sinusoidal excitation. The assumption is that under sinusoidal excitation, the steady state response is also a sinusoidal with the same angular frequency, but a different phase. ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 8 VI relationships General Assumptions We consider sinusoidal signals at steady state ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 9 Resistor IV R Note that the resistor itself doesn’t change the phase of the circuit Proof: v Vm cos t Ri V Vme j ELEC 250 – Summer 2015 i Vme j R I= V R Ch. 9 – Steady State Analysis 10 Inductance 2 V j LI Le j 90 I m e j LI m e j e j 90 LI me j 90 Note that the inductor changes the phase of the circuit Note that voltage leads the current ELEC 250 – Summer 2015 V jLI ( L ) I m 90 Ch. 9 – Steady State Analysis 11 Capacitance I jCV 2 or jC Vme j CVme j 90 CVm 90 I 1 90 I mi m i 90 C C Note that the capacitor changes the phase of the circuit Note that voltage lag the current ELEC 250 – Summer 2015 j C Ch. 9 – Steady State Analysis 12 Phasor Transform in use to solve a circuit This can be rewritten as jLI RI V ss Note that Vm R L 2 2 2 I m e j ( ) Vm e j e j t where tan L R ELEC 250 – Summer 2015 . Ch. 9 – Steady State Analysis 13 Leading and Lagging visuals ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 14 Impedance and Reactance All equations relating the phasor of the voltage to that of the current are of the form V = ZI The complex quantity Z is called the impedance while the imaginary part of it is called the reactance Circuit Element Impedance Resistor R Inductor Capacitor j L j C ELEC 250 – Summer 2015 Reactance L 1 C Ch. 9 – Steady State Analysis 15 Impedance and Reactance cont’d Impedance is a complex number, but not a phasor. That is, it does not implicitly involve time through ωt Reactance is analogous to resistance in that it is a property of circuit elements that opposes (resists) the passage of current. However, this is due to inductive (magnetic) or capacitive (electrostatic) phenomena rather than thermodynamic ones (resistance). A circuit element may exhibit both reactive and resistive behavior. This is expressed as the impedance which is a complex number, its real part corresponds to its resistivity, the imaginary part corresponds to its reactance ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 16 Kirchhoff’s laws Kirchhoff’s Voltage Law (KVL) 2 k 0 I1 2 k 0 1 Proof: Kirchhoff’s Current Law (KCL) Proof: i t 0 I cos t I e e 0 I e e 0 I e 0 I 0 k k k j k j j t mk k j k k mk mk k k j t mk k k k ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 17 Impedances in series and in parallel Wye<->Delta Same rules apply! ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 18 Impedances in series ab 1 Proof using KVL: ab 2 1 1 n 2 n 2 n 1 ab 2 ab n 1 2 n Impedances in parallel 1 1 1 Zab Z1 Z 2 1 Zn Proof using KCL: 1 2 n Zab Z1 Z2 Zn ELEC 250 – Summer 2015 1 1 1 Zab Z1 Z 2 1 Zn Ch. 9 – Steady State Analysis 19 Admittance Admittance is defined as the inverse of impedance 1 Y = = G + jB Z G is called conductance B is called susceptance Not that we can write the rule for combining impedances in parallel, in terms of the admittances as ELEC 250 – Summer 2015 Yeq Y1 Y2 Yn Ch. 9 – Steady State Analysis 20 Delta<->Wye The same rules apply as in the time domain DELTA =>WYE Zb Zc Z1 = Z a + Z b + Zc Z a Zc Z2 = Za + Zb + Zc Zb Za Z3 = Z a + Z b + Zc ELEC 250 – Summer 2015 WYE=>DELTA Za = Z1Z 2 + Z 2 Z 3 + Z 3 Z1 Z1 Z1Z 2 + Z 2 Z3 + Z 3 Z1 Z2 Z Z + Z 2 Z3 + Z3 Z1 Zc = 1 2 Z3 Zb = Ch. 9 – Steady State Analysis 21 Source Transformations Thevenin and Norton Equivalent The same techniques used for resistive circuits apply as well. Resistances are replaced by inductances and the time domain currents and voltages are replaced by their frequency domain phasors VTH = Z N I N IN = ELEC 250 – Summer 2015 VTH ZTH and ZTH Z N Ch. 9 – Steady State Analysis 22 VTH = Z N I N ZTH = Z N VTH IN = ZTH ELEC 250 – Summer 2015 Ch. 9 – Steady State Analysis 23 Node-Voltage and Mesh-Current analysis methods Exactly the same techniques are used here. We replace voltages and currents in the time domain (steady state) with their phasor representations in the frequency domain. We use impedance and admittances in place of resistance and conductance. ELEC 250 – Summer 2015