ELEC 2032 ELECTRONICS and SYSTEMS TUTORIAL 2 PHASOR APPROACH TO AC CIRCUIT THEORY
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1 Tutorial ELECTRONCS and SYSTEMS ELEC 3 ELEC 3 ELECTRONCS and SYSTEMS TUTORAL PHASOR APPROACH TO AC CRCUT THEORY. - Sinusoidal Steady State.. - Complex Numbers, Phasors and mpedance.. - Node and Mesh Analysis using Phasors..3 - Transformer and Mutual nductance Problems. - AC Circuit Theorems.. - Norton and Thevenin Theorems.. - Superposition/Multi-Frequency Problems Page
2 Tutorial ELECTRONCS and SYSTEMS ELEC 3. - Sinusoidal Steady State.. - Complex Numbers, Phasors and mpedance Question... Express each of the following complex numbers in three ways; rectangular form, polar form and exponential polar form. (a) 9 (b) -4 8 (c) (/4)-8 (d) 4exp(-63 ) (e) 69 (f) 4 Question... Evaluate the following expressions. Give your answer in polar form. (a) {(4 )(exp( -8 ))}/(6 - ) (b) (3)/{( ) (3 )} (c) (3-6)(3-6) * 48 (d) {( - ) (7exp( 45 ))}{ 6-5 } Question...3 For each of the sinusoidal functions below determine the corresponding RMS phasor in polar form. (a) 4cost (b) 6cos( t - 6 ) (c) -3sin( 7t 44 ) Question...4 Perform the additions indicated below by both phasor and trigonometric identity methods. (a) cos( 6t ) 3sin( 6t - 6 ) (b) cos( t ) 4sin( t - 6 ) (c) 5cos8t sin( 8t 45 ) Page
3 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...5 For the following voltage and current expressions, indicate whether the element is resistive, inductive or capacitive and find its value. (a) V(t) = 5cos( 4t 3 ), (t) = 3sin( 4t 3 ) (b) V(t) = 8sin( 9t 5 ), (t) = sin( 9t 4 ) (c) V(t) = cos( 5t 6 ), (t) = 5sin( 4t 5 ) Question...6 At an angular frequency of rad/sec a certain impedance has a value of Z = (). Given that this impedance is made up of a parallel combination of an inductance, L, and a resistance, R, find L and R. Z = Ω ωl R Question...7 Consider the circuit shown below, where L = 7mH, C = F and R = 6. Find the angular frequency for which the impedance between terminals a-b is purely resistive. Z ωc R ωl Page 3
4 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...8 n the circuit below, R = k and f = khz the admittance between terminals a-b is ms. Find L and C. a ωlω Y = ms ωc Ω RΩ b Question...9 Find V(t) in the circuit below using phasors. Ω V(t) Ω sin(t ) volts 3 cos(t ) volts Page 4
5 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question... The circuit below is operating in the sinusoidal steady state. t is known that the current in the resistor, (t), is (t) = (/)cos( t 45 ) What is the value of the capacitor, C, and the current through the capacitor, c (t)? (t) Vc(t) Ω c(t) cost.5h C L(t) Page 5
6 Tutorial ELECTRONCS and SYSTEMS ELEC Node and Mesh Analysis using Phasors Question... The circuit shown below is operating in the sinusoidal steady state. The load impedance, Z L is made up of a H inductor in series with a resistor. Find the voltage V L (t) as a function of time. H Ω.5F sin t Volts Ω Z L VL(t) Question... The circuit below is operating in the sinusoidal steady state with sources V s (t) = sint and s (t) = 5cost Find V(t). Ω H Ω F Vs(t) F Ω s(t) 5Ω V(t) Page 6
7 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...3 n the circuit below, use phasor node analysis to find V(t). t is given that V s (t) = cos( 4t - 35 ) H 8 F Vs(t) 8 F Ω C( t) 8 F V(t) C(t) Question...4 Use mesh current analysis to solve the circuit below for and 4Ω 8Ω 4 6 VoltsRMS 6 Ω 4 Ω Page 7
8 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...5 n the circuit below, g =.5S. Use phasor mesh current analysis to find V(t). V(t) F gv(t) sint Volts 5Ω H Question...6 The circuit below is operating in the sinusoidal steady state with V s (t) = cos( 3t = 5 ). The six impedances marked Z are each ( 6). Find V 3 (t) using node voltage analysis. Verify your answer by inferring the V 3 from the mesh current solution of the network. Vs(t) V V V 3 Z Z Z 3 Z Z Z Page 8
9 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...7 n the circuit below, L = 4H, C = 4F and V s (t) = cos( 5 t 44 ). Use phasor mesh current to find V(t). C 3 Ω Ω Ω Vs(t) L Ω V(t) Page 9
10 Tutorial ELECTRONCS and SYSTEMS ELEC Transformer and Mutual nductance Problems Question..3. The circuit below is operating in the sinusoidal steady state. Use mesh current analysis to find the current (t). Ω 3Ω H cos(3t 3 ) Volts H.5H 6 F Question..3. The circuit below is operating in the sinusoidal steady state. Use mesh current analysis to find the current (t). Ω H 3Ω 5Ω sin 3t Volts H H 3 sin(3t 3 ) Volts Page
11 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question..3.3 Use an equivalent circuit for the coupled inductors, followed by node voltage analysis, to find the voltage, V(t), in the circuit below. Ω Ω.H 6cost Volts.H.H 5Ω V(t) Question..3.4 Find the impedance between the terminals a-b in the circuit below. a 4Ω 3: Ω : Ω 8Ω Vout b Question..3.5 ideal ideal Find the impedance between the terminals a-b in the circuit below. a Ω Ω : :4 6Ω b Ω 8 Ω ideal ideal Page
12 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question..3.6 n the circuit below write down the phasor mesh current equations. The 4H and 6H coils are coupled with mutual inductance H and filled dots. The 6H and 5H coils are also coupled with M = 4H and open dots. The 4H and 5H coils are not coupled. Do not attempt to solve the mesh current equations. cos4tamps F Ω 4H 4H H 6H 5cos 4t 5H Volts 3 Ω F Question..3.7 Write down phasor mesh current equations for the circuit below. Do not attempt to solve by hand. mf Ω mf mh i 4Ω cos5t Volts mf 3 3Ω 8mH i 6mH 5mH Page
13 Tutorial ELECTRONCS and SYSTEMS ELEC 3. - AC Circuit Theorems.. Norton and Thevenin Theorems Question... For the phasor circuit below, find the Thevenin equivalent circuit with respect to terminals a-b. a 4Ω 8Ω 3 5 AmpsRMS Ω b Ω Question... For the phasor circuit shown below find the Norton equivalent circuit. 9cos5t Volts 6Ω V(t) mf 5 V ( t) 3Ω a b Question...3 For the phasor circuit shown below, find the Thevenin equivalent circuit. V ( t).5h cos t Amps V(t) mf a b Page 3
14 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...4 For the circuit shown below, find the Norton equivalent circuit. a a V Ω V 3 Ω 8 Ω a Ω 4 Ω Ω b Question...5 Find the Thevenin equivalent circuit for the terminals a-b in the circuit below. a Ω Ω Ω Ω b Page 4
15 Tutorial ELECTRONCS and SYSTEMS ELEC 3.. Superposition/multi-frequency problems Question... The circuit below is operating in both the DC and AC steady state. Find V(t) when V s (t) = cos( 3t 5 ) and V g = 5V DC 6Ω Vs(t).F 3H V(t) Vg 9Ω Question... The sinusoidal sources V s and s operate at = r/s and = r/s respectively. f V(t) = 4cost - sint, find V s and s. Ω Ω Vs ( t) ω = r / s s ( t) ω = r / s µf V(t) Page 5
16 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Question...3 The circuit below is operating in the sinusoidal steady state. Find the voltage V(t) if V s (t) = 5cos( t 3 ) and s (t) = sint 4cos( 3t ) Ω H H Ω Vs(t) s(t) V (t) F Ω Question...4 Determine V(t) and (t) for the circuit shown below when V s (t) = 5cost V and s = 5cost A. Ω F H (t) Vs(t) V(t) F 4 H Ω s(t) Page 6
17 Tutorial ELECTRONCS and SYSTEMS ELEC 3 ANSWERS TUTORAL... to (a) 9 = e = (b) -4 8 = = 43.86e (c) /4-8 = /4e = -/4 (d) 4e -63 = 4-63 = (e) 69 = 6e = (f) 4 = = 4.4e... (a) (b) (c) (d) (a).88 (b).35-6 (c) (a).439cos( 6t ) (b) 4.47cos( t ) (c) 5.8cos( 8t )...5 (a).5mh (b)77.8f (c)4...6 R = 5 L = 5/4H...7 = 78 r/s...8 C =.584F L =.584H cos( t )... c (t) = cos( t 45 ) C = F... 64sin( t -.94) V cos( t ) V...3 cos( 4t - 35 ) V...4 =.59.4 = cos( t ) V Vs...6 V 3 = cos( 5 t 58.4 ) 55.8 Page 7
18 Tutorial ELECTRONCS and SYSTEMS ELEC 3 Page 8 ANSWERS TUTORAL..3. to cos( 3t ) cos( 3t ) cost = / / 3 4 / = V in series with mA in parallel with V in series with cos( 3t 8. )... V s = V, s = A cos( 3t ).7cos( t ) V...4 V(t) = 4.85cos( t 8.58 ).387cos( t 46.3 ) (t) =.95cos( t -.73 ) 3.cos( t 9.75 )
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