CHAPTER 8 Snusodl Stedy Stte Anlyss 8.1. Generl Approch In the prevous chpter, we hve lerned tht the stedy-stte response of crcut to snusodl nputs cn e otned y usng phsors. In ths chpter, we present mny exmples n whch nodl nlyss, mesh nlyss, Thevenn s theorem, superposton, nd source trnsformtons re ppled n nlyzng c crcuts. 8.1.1. Steps to nlyze c crcuts, usng phsor domn: Step 1. Trnsform the crcut to the phsor or frequency domn. Not necessry f the prolem s specfed n the frequency domn. Step 2. Solve the prolem usng crcut technques (e.g., nodl nlyss, mesh nlyss, Thevenn s theorem, superposton, or source trnsformtons ) The nlyss s performed n the sme mnner s dc crcut nlyss except tht complex numers re nvolved. Step 3. Trnsform the resultng phsor ck to the tme domn. 8.1.2. c crcuts re lner (they re just composed of sources nd mpednces) 8.1.3. The superposton theorem pples to c crcuts the sme wy t pples to dc crcuts. Ths s the cse when ll the sources n the crcut operte t the sme frequency. If they re opertng t dfferent frequency, see Secton 8.2. 107
108 8. SINUSOIDAL STEADY STATE ANALYSIS 8.1.4. Source trnsformton: V s = Z s I s, I s = V s Z s. Z s V s I s Z s V s = Z s I s I s = V s Z s 8.1.5. Thevenn nd Norton Equvlent crcuts: Z Th Lner crcut V Th Lner crcut I N Z N V Th = Z N I N, Z Th = Z N
8.1. GENERAL APPROACH 109 Exmple 8.1.6. Compute V 1 nd V 2 n the crcut elow usng nodl nlyss. 10 45 V V 1 4 Ω 1 2 V 2 3 0 A j3 Ω j6 Ω 12 Ω Exmple 8.1.7. Determne current I o n the crcut elow usng mesh nlyss. 4 Ω I 3 5 0 A j2 Ω j10 Ω I 2 I 0 20 90 V 8 Ω I 1 j2 Ω
The voltge-to-current converter, s shown n Fgure 8-2, produces n output current tht depends on the nput voltge nd the resstor R. In prtculr, the output current I out = V /R ndependent of the lodng resstnce R L. 110 8. SINUSOIDAL STEADY STATE ANALYSIS The current-to-voltge converter, s shown n Fgure 8-3, produces n output voltge tht depends on the nput current nd the resstor R. In prtculr, the output voltge Exmple 8.1.8. Fnd the Thevenn equvlent t termnls - of the V o = -I n R crcut elow. ndependent of the sze of the lodng resstnce R L. 6 Ω j2 Ω V V 75 20 V I out I n j4 Ω R V 10 Ω R R L V- V- R L V o - Fgure 8-2: Voltge-to-current converter. Fgure 8-3: Current-to-voltge converter. 4. An ntegrtng mplfer s shown n Fgure 8-4. Exmple 8.1.9. Op Amp AC Crcuts: Fnd the (closed-loop) gn of the crcut elow. v C - C C v n R X V V- v o - Fgure 8-4: Integrtng mplfer 2
8.2. CIRCUIT WITH MULTIPLE SOURCES OPERATING AT DIFFERENT FREQUENCIES 111 8.2. Crcut Wth Multple Sources Opertng At Dfferent Frequences A specl cre s needed f the crcut hs multple sources opertng t dfferent frequences. In whch cse, one must dd the responses due to the ndvdul frequences n the tme domn. In other words, the superposton stll works ut () We must hve dfferent frequency-domn crcut for ech frequency. () The totl response must e otned y ddng the ndvdul response n the tme domn. 8.2.1. Snce the mpednce depend on frequency, t s ncorrect to try to dd the responses n the phsor or frequency domn. To see ths note tht the exponentl fctor e jωt s mplct n snusodl nlyss, nd tht fctor would chnge for every ngulr frequency ω. In prtculr, lthough V m cos(ωt φ ) = Re { {( ) } V e jωt} = Re V e jωt, when we llow ω to e dfferent for ech snusod, generlly V m cos(ω t φ ) = Re { V e } {( ) } jω t Re V e jω t. Therefore, t does not mke sense to dd responses t dfferent frequences n the phsor domn. 8.2.2. The Thevenn or Norton equvlent crcut (f needed) must e determned t ech frequency nd we hve one equvlent crcut for ech frequency.
112 8. SINUSOIDAL STEADY STATE ANALYSIS Exmple 8.2.3. Fnd v o n the crcut elow usng the superposton theorem. 2 H 1 Ω 4 Ω v 0 10 cos 2t V 2 sn 5t A 0.1 F 5 V 1 Ω 4 Ω j4 Ω 1 Ω 4 Ω I 1 1 Ω v1 5 V 10 0 V V2 j5 Ω j10 Ω V3 2 90 A j2 Ω 4 Ω () () (c)