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Unformatted text preview: 1 Electrical Engineering Technology EET 107 Introduction to Circuit Analysis # 35 Professor Robert Herrick Purdue University © EET 107 - 35 Introduction to Circuit Analysis 2 Electrical Engineering Technology ♦ Switched RC Transient ♦ Inverse Solution ♦ Multiple RC – Thévenin Model Purdue University © EET 107 - 35 Introduction to Circuit Analysis 3 RC Circuit - Sudden DC Change t=0 SW E 10 V t = 0− t=0 0 < t < 5τ t = 5τ tè∞ + vR(t) − R 1k Purdue University © iR(t) iC(t) C 1µ + vC( t) − Just before switching INITIAL - sudden dc change TRANSIENT Capacitor 99% charged STEADY STATE – final dc EET 107 - 35 Introduction to Circuit Analysis 4 RC Transient DC Circuit t=0 E +VRR C + VC - vC(t) iC(t) init = initial (t = 0) ss = steady state (t = ∞) vR(t) iR(t) vx(t) = Vss + (Vinit − Vss) e−t/τ ix(t) = Iss + (Iinit − Iss) e−t/τ Purdue University © EET 107 - 35 Introduction to Circuit Analysis 5 RC Transient DC Circuit Analyze circuit and find: •τ • Initial value • Steady state value t=0 E +VRR C + VC - Then substitute: vx(t) = Vss + (Vinit − Vss) e−t/τ Purdue University © EET 107 - 35 Introduction to Circuit Analysis 6 RC Circuit – uncharged capacitor vC(t) = Vss + (Vinit − Vss) e−t/τ Example: uncharged τ = RC = 10µs Vinit = 0V Vss = 10V vC(t) = 10V + (0V − 10V) e−t/10µs vC(t) = 10V − 10V e−t/10µs Purdue University © EET 107 - 35 Introduction to Circuit Analysis 7 RC Circuit – charged capacitor vC(t) = Vss + (Vinit − Vss) e−t/τ Example: charged τ = RC = 10µs Vinit = 3V Vss = 10V vC(t) = 10V + (3V − 10V) e−t/10µs vC(t) = 10V − 7V e−t/10µs Purdue University © EET 107 - 35 Introduction to Circuit Analysis 8 RC Transient Circuit Analysis 1. Establish capacitor voltage before switch thrown 2. Evaluate time constant after switch thrown 3. Initial model of capacitor and evaluate circuit 4. Steady state model of capacitor and evaluate circuit 5. Apply universal RC equations 6. Sketch resulting equations Purdue University © EET 107 - 35 Introduction to Circuit Analysis 9 Capacitor Models – Know These CAPACITOR - stores VOLTAGE INITIALLY - U ncharged cap 0V SHORT INITIALLY - charged cap EO inital voltage EO EO STEADY STATE - fully charged cap insulator Purdue University © EET 107 - 35 OPEN Introduction to Circuit Analysis 10 1. Example - before switch closed Example t=0 OPEN 10V +VR1k Ω 1µ F + VC OPEN - 0V t=0− • switch open for a long time is assumed • steady state capacitor è OPEN • VC = 0V Purdue University © EET 107 - 35 Introduction to Circuit Analysis 11 2. Example - time constant Example t=0 E R 1kΩ 1µF C Switch closed RC Time Constant when switch closed • τ = RC = 1kΩ × 1µF = 1ms Cap 63% charged after switch closed • 5τ = 5 × 1ms = 5ms Cap 99.3% charged after switch closed Purdue University © EET 107 - 35 Introduction to Circuit Analysis 12 3. Example - initial circuit Example t=0 10V +VR1k Ω 1µ F + VC - 0V from t = 0− t=0 • VC = 0V uncharged cap model • Capacitor acts like a SHORT • VR = 10V and Purdue University © IR = IC = 10mA EET 107 - 35 Introduction to Circuit Analysis 13 4. Example - steady state circuit Example t=0 10V +VR1k Ω 1µ F + VC OPEN - 10V t >> 5τ • capacitor acts like an open: model as OPEN • VC = 10V capacitor fully charged • VR = 0V • IC = IR = 0mA Purdue University © EET 107 - 35 Introduction to Circuit Analysis 14 5. Example - capacitor voltage equation Example t=0 10V +VR1k Ω 1µ F + VC - Capacitor Voltage τ = 1ms Vinit = 0V Vss = 10V Complete capacitor voltage expression vC (t) = Vss + (Vinit − Vss) e−t/τ vC (t) = 10V + (0V − 10V) e−t/1ms vC (t) = 10V − 10V e −t/1ms Purdue University © EET 107 - 35 Introduction to Circuit Analysis 15 5. Example - capacitor voltage equation Example vC (t) = 10V − 10V e−t/1ms Alternate form by factoring out 10V vC (t) = 10V ( 1 − e−t/1ms ) Purdue University © EET 107 - 35 Introduction to Circuit Analysis 16 5. Example - capacitor voltage equation Example Evaluating capacitor voltage - t in units of ms vC(t) = 10V − 10V e−t/1ms vC(0) = 10V − 10V e−0/1ms vC (1ms) = 10V − 10V e−1ms/1ms vC (2ms) = 10V − 10V e−2ms/1ms = 0V = 6.32V = 8.65V lll vC (5ms) = 10V − 10V e−5ms/1ms Purdue University © EET 107 - 35 = 9.93V Introduction to Circuit Analysis 17 5. Example - capacitor voltage equation Example Evaluating capacitor voltage - t in units of τ vC (t) = 10V − 10V e−t/τ vC (0) = 10V − 10V e−0/τ = 0V vC (1τ) = 10V − 10V e−1τ /τ = 6.32V vC (2τ) = 10V − 10V e−2τ /τ = 8.65V lll vC (5τ) = 10V − 10V e−5τ /τ = 9.93V Purdue University © EET 107 - 35 Introduction to Circuit Analysis 18 6. Example - capacitor voltage sketch Example vC (t) = 10V − 10V t τ = 1ms Vinit = 0V Vss = 10V e−t/τ vC (t) 0 1τ 2τ 3τ 4τ 5τ 1ms 2ms 3ms 4ms 5ms Purdue University © 0.0V 6.3V 8.7V 9.5V 9.8V 9.9V vC(t) 10V ss 0V init EET 107 - 35 0 τ 2τ 3 τ 4τ 5τ t 5ms Rising Curve Introduction to Circuit Analysis 19 5. Example - capacitor current equation Example t=0 10V +VR1k Ω 1µ F + VC - Capacitor current τ = 1ms Iinit = 10mA Iss = 0mA Complete cap current expression iC (t) = Iss + (Iinit − Iss) e−t/τ iC (t) = 0mA + (10mA − 0mA) e −t/1ms iC (t) = 10mA e −t/1ms Purdue University © EET 107 - 35 Introduction to Circuit Analysis 20 6. Example - capacitor current sketch Example τ = 1ms Iinit = 10mA Iss = 0mA iC (t) = 10mA e−t/τ t iC (t) 0 1τ 2τ 3τ 4τ 5τ 10.0mA 3.7mA 1.3mA 0.5mA 0.2mA 0.1mA 1ms 2ms 3ms 4ms 5ms Purdue University © 10mA v(t) init 0mA ss EET 107 - 35 0 τ 2τ 3τ 4τ 5τ Falling Curve t 5ms Introduction to Circuit Analysis 21 5. Example - resistor current equation Example t=0 10V +VR1k Ω 1µ F + VC - Resistor current τ = 1ms Iinit = 10mA Iss = 0mA Note: quick solution, same as the capacitor current iR(t) = iC(t) = 10mA e−t/1ms Same sketch Purdue University © EET 107 - 35 Introduction to Circuit Analysis 22 5. Example - resistor voltage equation Example t=0 10V +VR1k Ω 1µ F + VC - Resistor voltage τ = 1ms Vinit = 10V Vss = 0V Quick solution: Ohm’s Law iR(t) = 10mA e−t/1ms vR (t) = 1kΩ × 10mA e−t/1ms vR(t) = 10V e−t/1ms Purdue University © EET 107 - 35 Introduction to Circuit Analysis 23 6. Example - resistor voltage sketch Example vR(t) = 10V t τ = 1ms Vinit = 10V Vss = 0V e−t/τ VR(t) 0 1τ 2τ 3τ 4τ 5τ 1ms 2ms 3ms 4ms 5ms 10.0V 3.7V 1.3V 0.5V 0.2V 0.1V Purdue University © vR(t) 10V ss 0V init EET 107 - 35 0 τ 2τ 3τ 4τ 5τ t 5ms Falling Curve Introduction to Circuit Analysis 24 Resistor Sketches Resistor voltage and current sketch must have the same shape. same Ohm’s Law v ( t) = R i( t) Related by a constant ! Purdue University © EET 107 - 35 Introduction to Circuit Analysis 25 What if Charged Cap - initial circuit t=0 +VR1k Ω 10V 1µ F + VC - t=0 • • • • Eo Model if charged cap 4V Intial Circuit What if cap was previously charged to 4V Initial model of cap would be a 4V supply VR = 10V − 4V = 6V IR = IC = 6V/1kΩ = 6mA Purdue University © EET 107 - 35 Introduction to Circuit Analysis 26 Electrical Engineering Technology ♦ Switched RC Transient ♦ Inverse Solution ♦ Multiple RC – Thévenin Model Purdue University © EET 107 - 35 Introduction to Circuit Analysis 27 RC Circuit - universal equation vC(t) = Vss + (Vinit − Vss) e−t/τ Example τ = 10µs Vinit = −4V Vss = 10V vC(t) = 10V + (−4V − 10V) e−t/10µs vC(t) = 10V − 14V e−t/10µs Purdue University © EET 107 - 35 Introduction to Circuit Analysis 28 RC Circuit - universal equation Evaluate v(t) at t = 18µs Plug in a time and find a voltage. vC(18µs) = 10V − 14V e−18µs/10µs vC(18µs) = 7.686V Purdue University © EET 107 - 35 Introduction to Circuit Analysis 29 RC Circuit - capacitor voltage sketch At t = 18µs, VC = 7.686V vC(t) 10V -4V ss 7.7V init 0 τ = 10µs τ 2τ 3 τ 4τ 5τ 18µs t 50µ s Rising Curve Purdue University © EET 107 - 35 Introduction to Circuit Analysis 30 RC Circuit - capacitor voltage sketch Given a voltage, find the corresponding time. vC(t) 10V -4V ss 7.7V init 0 τ = 10µs τ 2τ 3 τ 4τ 5τ ? t 50µ s Rising Curve Purdue University © EET 107 - 35 Introduction to Circuit Analysis 31 RC Circuit - universal equation How much time “t” does it take to charge to a particular voltage ? For example, how much time “t” is needed to charge the capacitor to 3V ? Purdue University © EET 107 - 35 Introduction to Circuit Analysis 32 RC Circuit - universal equation vC(t) = 3V t=? 3V = 10V − 14V e−t/10µs −7V = − 14V e−t/10µs 0.5 = e−t/10µs Purdue University © EET 107 - 35 Introduction to Circuit Analysis 33 RC Circuit - universal equation Ln ( 0.5 ) = Ln ( e−t/10µs ) exponent pops out −0.693 = −t / 10µs t = 0.693 x 10µs = 6.93µs Purdue University © EET 107 - 35 Introduction to Circuit Analysis 34 RC Circuit - capacitor voltage sketch How much time is required to charge the capacitor to 3V? t = 6.93µs vC(t) 10V ss τ = 10µs 3V -4V init 0 τ 6.93µs 2τ 3 τ 4τ 5τ t 50µ s Rising Curve Purdue University © EET 107 - 35 Introduction to Circuit Analysis 35 Electrical Engineering Technology ♦ Switched RC Transient ♦ Inverse Solution ♦ Multiple RC – Thévenin Model Purdue University © EET 107 - 35 Introduction to Circuit Analysis 36 Multiple RC – simplify to Thévenin R1 3 k iC t=0 SW E 18 V C1 6 u + vC − R2 6k C2 3u C3 2 u Thévenin model Total Capacitance Charging circuit RTH 2 k ETH Ctotal 12 V 1u Purdue University © EET 107 - 35 + vC − Introduction to Circuit Analysis 37 Textbook Example 14-5 R1 3 k iC t=0 E 18 V SW R2 6k C1 6 u + vC − C2 3u C3 2 u • Uncharged capacitors before switch closed • Close switch for 4 ms then reopened • Draw vC(t) and iC(t) Purdue University © EET 107 - 35 Introduction to Circuit Analysis 38 Textbook Example 14-5 v C(t) VSS 12 V VINIT 0 V VINIT 10.4 V VSS 0 V 0 2 4 10 16 22 28 34 0 τ 2τ τ 2τ 3τ 4τ 5τ Charging curve τ = 2 ms Discharging curve t(ms) τ = 6 ms τ = 6 ms Purdue University © EET 107 - 35 Introduction to Circuit Analysis 39 Textbook Example 14-5 iC(t) IINIT 6 mA 0.8 mA ISS 0 mA 0 2 4 ISS 0 mA 10 16 22 28 34 t(ms) IINIT −1.73 mA τ = 2 ms τ = 6 ms Purdue University © EET 107 - 35 Introduction to Circuit Analysis 40 Textbook Example 14-5 t=∞ 6F R E c1 + V C1c3 3F c2 2F Larger C Smaller XC 6F drops least voltage Less voltage drop Purdue University © EET 107 - 35 Introduction to Circuit Analysis 41 Capacitance - VDR in Steady State t=∞ 6F R E 12V 1F c1 3F + V C1c3 c2 2F R è 1/C VDR steady state Purdue University © CT 1F VC1 = E= 12V = 2V C1 6F EET 107 - 35 Introduction to Circuit Analysis 42 Electrical Engineering Technology ♦ Switched RC Transient ♦ Inverse Solution ♦ Multiple RC – Thévenin Model Purdue University © EET 107 - 35 Introduction to Circuit Analysis ...
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