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Bridging Theory in Practice Transferring Technical Knowledge to Practical Applications RLC Load Characteristics and Modeling RLC Load Characteristics and Modeling Intended Audience: • Engineers with a basic knowledge of resistive circuits • Engineers desiring a more intuitive understanding of capacitive and inductive circuits Topics Covered: • Introduction to Load Modeling • Introduction to Capacitors and RC networks • Introduction to Inductors and RL networks • Example Load Models Expected Time: • Approximately 120 minutes RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay Electromechanical Power Conversion • Electrical power can be converted to mechanical power – Electrical power can turn-on a motor – Electrical power can drive a Solenoid • Electrical power can be converted to “heat” • Electrical power can a light a LED (= ) Load Modeling • “Power converters” (the loads) can be modeled by equivalent circuits composed of simple RLC passive components RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay Capacitors • Physical object with the ability to store electric charge (i.e. “electric voltage”) • Consists of two – electrically isolated – metal electrodes, typically two conductive parallel plates • Is mostly used to store energy or for filtering purposes • The isolating material – the dielectric – defines the type of capacitor: e.g. tantalum or ceramic capacitor • Circuit symbol: C Capacitors: Physical Properties • The capacitance of a parallel plate capacitor is proportional to: C~ C a d a d = Capacitance; = Area of each parallel plate; = Distance between parallel plates; d a • Larger value capacitors have larger plate areas and less spacing between plates • They can store more energy (and are more expensive) Capacitors: Physical Properties The capacitance of a parallel plate capacitor is given by: C = Capacitance Units of: F = A · s / V = Permittivity = 0· r Units of: A · s / V · m = F / m 0 = Permittivity of vacuum = 8.854x10-12 Units of: A · s / V · m = F / m r = Relative permittivity = 1 (free air) Units of: (dimensionless) ·a C= d a Permittivity1) : the ability of a dielectric to store electrical potential energy under the influence of an electric field 1) Webster’s 9th edition d Relative Size of Capacitance • Capacitance of a free air (r = 1) parallel plate capacitor with the dimensions of A=1m2 and d=1mm is: 12 2 1 8.854x10 F / m (1m ) r 0 A 9 C 8.854x10 F 3 d 1x10 m • Typically, capacitance values in the 1F range are uncommon • Capacitances typically range from microFarads to picoFarads 1 microFarad = 1mF = 10-6F 1 nanoFarad = 1nF = 10-9F 1 picoFarad = 1pF = 10-12F Capacitors Electrical Properties • The stored electrical charge Q in a capacitor is proportional to the voltage V across the capacitor: Q ~ V • The proportional factor between stored electrical charge and voltage difference is the capacitance value of the capacitor: Q=C·V Q = 8 A s = 8 Coulombs V = 16V C = Q/V = 8 A s / 16V = 0.5 Farad (F) Unit [C] = A s / V = F Parallel and Serial Capacitance Parallel capacitors C1 Serial capacitors C2 C1 C C2 C C = C1 + C2 1 1 1 = + C C1 C2 Capacitor Experiment #1 • An ideal current source is connected to a capacitor tON IC + VC - IIDEAL C The constant current causes the voltage to linearly rise across V, I the capacitor. IC IIDEAL Constant current source supplies the current regardless of the voltage drop across the load. VC tON t Capacitor Experiment #2 • An ideal current source is disconnected from a capacitor tOFF IC + VC - IIDEAL If the constant current C source is removed, the voltage across the V, I capacitor remains IIDEAL constant. VC IC tON tOFF t Capacitor Experiment #3 • An ideal current source is connected to a capacitor tON IC + VC - IIDEAL C The rate of voltage change is proportional to the current. V, I IC1 VC1 tON t Capacitor Experiment #3 • A variable ideal current source is connected to a capacitor tON IC + VC - IIDEAL C The rate of voltage change is proportional to the current. V, I IC2 VC2 IC1 VC1 tON t Capacitor Experiment #4 • A voltage source is connected to a capacitor through a resistor The peak current tON VIDEAL + - R IC in the capacitor is + VC - C limited by the resistor. The voltage across V, I VIDEAL/R the capacitor will IC VC VIDEAL tON reach VIDEAL t Ideal voltage source supplies the voltage regardless of the current load. Capacitor Experiment #5 • A voltage source is connected through a variable resistor tON VIDEAL + - R IC + VC - C Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I tON VIDEAL + - R IC VC R = R1 + C - VIDEAL R1 VC1 IC1 tON t Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I tON VIDEAL VIDEAL R1 + - R IC VC R1 > R2 + C - VIDEAL VC1 R1 IC1 tON t Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I VIDEAL R2 tON IC2 VIDEAL VIDEAL R1 + - R IC VC R1 > R2 + C - VIDEAL VC1 R1 IC1 tON t Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I VIDEAL R2 tON IC2 VIDEAL VC2 VIDEAL R1 IC VC R1 > R2 + C - VIDEAL VC1 IC1 tON + - R Capacitors are charged faster through smaller resistors t Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I tON VIDEAL VIDEAL R1 + - R IC VC R1 < R3 + C - IC1 VC1 tON t Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I tON VIDEAL VIDEAL R1 IC1 + - R IC VC R1 < R3 + C - VC1 VIDEAL R3 IC3 tON t Capacitor Experiment #5 • A voltage source is connected through a variable resistor V, I tON VIDEAL VIDEAL R1 IC1 R3 IC3 tON IC VC R1 < R3 + C - VC1 VC3 VIDEAL + - R Capacitors are charged faster through smaller resistors t Capacitor Experiment #6 • The rise time of the capacitor's voltage is monitored: tON VIDEAL VC + - R IC VC R1 tC =<RC R3 + C - VIDEAL 0.63VIDEAL 0 tC t Capacitor Experiment #6 • The rise time of the capacitor's voltage is monitored: tON VIDEAL VC + - R IC VC R1 tC =<RC R3 + C - 0.95VIDEAL 0.87VIDEAL 0.63VIDEAL 0 tC 2tC 3tC t Development of Mathematical Capacitor Model: IC vs. VC • Current is defined as the amount of charge which is transferred in a certain period of time: I = Q / t dq i or dq i dt dt (1) The relations above are derivatives for very small changes differentials can be used for quasi linear changes: i=Dq/Dt or Dq=i.Dt (1a) Development of Mathematical Capacitor Model: IC vs. VC • Current is defined as the amount of charge which is transferred in a certain period of time: I = Q / t dq i or dq i dt dt (1) • Capacitance is defined as the stored charge on a capacitor vs. the voltage across the capacitor, C = Q / V dq (2) C or dq C dv dv In differential form: C=Dq/Dt or Dq=C.Dv (2a) Development of Mathematical Capacitor Model: IC vs. VC • Current is defined as the amount of charge which is transferred in a certain period of time: I = Q / t dq i or dq i dt dt (1) • Capacitance is defined as the stored charge on a capacitor vs. the voltage across the capacitor, C = Q / V dq (2) C or dq C dv dv • Setting (2) equal to (1) results in: i dt C dv or dv iC dt Capacitors IC Voltage across VIN Capacitor VC Current through Capacitor time R VIN C VC across plates Capacitor & Resistor Networks • In general, there are two basic options for capacitor placement: C in Series with Signal Path VIN C R VOUT C from Signal Path to Ground VIN R C VOUT Capacitor & Resistor Networks C in Series with Signal Path C VOUT + VC R I V IN C from Signal Path to Ground R VIN VOUT I C + VC - • Initially a DC voltage is applied at the signal input IN. • Current passes through the capacitor and the voltage across the capacitor increases Capacitor & Resistor Networks C in Series with Signal Path C VOUT + VIN R I=0A VIN C from Signal Path to Ground R VIN VOUT I=0A C + VIN - • Initially a DC voltage is applied at the signal input IN. • Current passes through the capacitor and the voltage across the capacitor increases • When the voltage across the capacitor is equal to the input voltage the current stops Capacitor & Resistor Networks C in Series with Signal Path C 0V + VIN R VIN I=0A C from Signal Path to Ground R VIN I=0A C + VIN - VIN • Initially a DC voltage is applied at the signal input IN. • Current passes through the capacitor and the voltage across the capacitor increases • When the voltage across the capacitor is equal to the input voltage the current stops • Depending on the capacitor’s placement, the VOUT = 0V or VOUT = VIN Capacitance in Series with Signal Path VX t1 VX t2 + VC - VIN VOUT C I R I VOUT t1 t2 Capacitance in Series with Signal Path VX t1 VX t2 + VC - VIN VOUT C I R VIN I VIN/R VOUT VIN t1 t2 Capacitance in Series with Signal Path VX t1 VX t2 + VC - VIN VOUT C I R VIN I VIN/R -VIN/R VOUT VIN -VIN t1 t2 Capacitance From Signal Path to Ground t1 t2 VIN VX VX R VOUT I C + VC - I VOUT t1 t2 Capacitance From Signal Path to Ground t1 t2 VIN VX VX R VOUT I C VIN + VC - I VIN/R -VIN/R VOUT VIN t1 t2 Capacitance From Signal Path to Ground t1 t2 VIN VX VX R VOUT I C VIN + VC - I VIN/R -VIN/R VOUT VIN t1 t2 RC Networks - AC Signals • What happens when an AC input signal is applied? C in Series with Signal Path VOUT C VIN t R ? C from Signal Path to Ground VOUT R VIN t C ? Capacitors and AC signals • Capacitors act like frequency dependent resistor (capacitive reactance, XC) Xc~1/(fC) • Instead of reactance, impedance (Z) is used to characterize circuit elements: Z=1/(2pfC) Capacitors and AC signals • Act like frequency dependent resistor (capacitive 1 reactance, XC) X C fC • Instead of reactance, impedance (Z) used for circuit elements. • Impedance1): The apparent opposition in an electrical circuit to the flow of alternating current that is analogous to the actual electrical resistance to a direct current. Capacitors and AC signals • Act like frequency dependent resistor (capacitive reactance, XC) 1 XC fC • Instead of reactance, impedance (Z) used for circuit elements. • Impedance1): The apparent opposition in an electrical circuit to the flow of alternating current that is analogous to the actual electrical resistance to a direct current. • The impedance of a circuit element represents its resistive and/or reactive components Capacitors and AC signals • Act like frequency dependent resistor (capacitive reactance, XC) 1 XC fC • Instead of reactance, impedance (Z) used for circuit elements. • Impedance1): The apparent opposition in an electrical circuit to the flow of alternating current that is analogous to the actual electrical resistance to a direct current. • The impedance of a circuit element represents its resistive and/or reactive components • Besides the magnitude dependency between voltage and current the impedance, Z, gives also information about the phase shift between the two. Capacitor’s Impedance Magnitude |ZC| vs. Frequency |Z|=1/(2pfC) C=1uF |Z| (kohm) . ZC 1 fC 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 0 100 200 300 FREQUENCY (Hz) 400 500 Capacitors and AC signals iC,Max = VC,Max / |ZC| VC,Max iC,Max = +p/2 = + 90o The current leads the voltage t RC networks – AC Signals C in Series with Signal Path VOUT C VIN t R C from Signal Path to Ground VOUT R VIN t C • The capacitor acts as a frequency dependent resistor • It determines the current magnitude at a given voltage • It causes a 90 degree phase shift between the capacitor current and voltage across the capacitor RC networks – AC Signals C in Series with Signal Path VOUT C VIN t R C from Signal Path to Ground VOUT R VIN t C • For high frequency signals: – The capacitor is low impedance – Signals can pass the capacitor |ZC|=1/(2pfC) • For low frequency signals: – The capacitor is high impedance – Signals are blocked by the capacitor C in Series with Signal Path High Pass Configuration VOUT VIN VIN 6 4 C 2 R 0 -2 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 -4 |ZC|=1/(2pfC) -6 VOUT 6 4 VOUT/VINMAX Low f 0.32 Medium f 0.76 High f 0.90 2 0 -2 -4 -6 C from Signal Path to Ground Low Pass Configuration VOUT VIN VIN 6 4 R 2 C 0 -2 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 -4 |ZC|=1/(2pfC) -6 VOUT 6 4 VOUT/VINMAX Low f 0.96 Medium f 0.74 High f 0.39 2 0 -2 -4 -6 Capacitor & Resistor Networks Summary C in Series with Signal Path VOUT C VIN R C from Signal Path to Ground VOUT R VIN C Connected to DC voltages: • Capacitors will allow current to flow only until they are charged • Once charged, they block future current flow For AC signals: • Capacitors act similar to frequency dependent resistors • Low impedance at high frequencies • High impedance at low frequencies. RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay Inductors • • • • • Physical object which can store a magnetic field (electric current) Consists of a conductive wire Wire is typically a tightly wound coil around a center core (toroid) Usually used for energy conversion and for filtering purposes The inductor type is usually defined by its core material for example, air coil or ferrite coil inductors) • Circuit symbol or L Physical Properties of Inductors • The inductance of a toroid, for instance, is given by: L=m0.mrN2.a/l L N a ul0 ur A = Inductance; = Number of turns of the coil; = Coil cross section; = Average field length; = permeability of vacuum =4p10-7 V.s/(A.M) =relative permeability Core Wire • Larger value inductors have more turns and bigger cross section in less volume. They can store more energy (and may be more expensive). Inductance of a Toroid L N a m m0 mr = Inductance Units of: H = V · s/A = Number of turns of the coil = Coil cross section Units of: m2 = Average field length Units of: m = Permeability = m0· mr ; Units of: H/m = V · s/A · m = Permeability of free space = 4p10-7 Units of: H/m = V · s/A · m = Relative permeability L=mN2a/l Permeabilty1) : the property of a ferro-magnetic substance that determines the degree in which it modifies the magnetic flux in the region occupied by it in a magnetic field 1) acc. to Webster’s 9th edition a Relative Size of Inductance • Inductance of a free air toroid (mr = 1) with the cross section of a=5cm2, average field length of =10cm, and N=100 turns is L~ (1)(4p 10 H/ m) 100 5x10 m 2 7 2 10x10 m 4 2 6.28x10 5 H • Inductors in the mH range are used in switching regulators • Small relays, solenoids usually have mH values of inductance • Inductors in general typically range from a few Henries (H) to micro Henries (mH): 1 microHenry = 1mH = 10-6H 1 milliHenry = 1mH = 10-3H 1 Henry = 1H Inductors Electrical Properties • The change of magnetic field or coil flux (y in an inductor is proportional to the change of electric current (I) flowing through the inductor’s windings: y ~ I • The proportional factor between coil flux and current is given by the inductance of the coil: y = L · I y N·F I Inductors Electrical Properties • The change of magnetic field or coil flux (y in an inductor is proportional to the change of electric current (I) flowing through the inductor’s windings: y ~ I • The proportional factor between coil flux and current is given by the inductance of the coil: y = L · I I = 2A y N·F y1Vs L = y/I = 1 Vs / 2 A = 0.5 Henry (H) Unit [L] = Vs/A = H Serial and Parallel Inductance Serial inductors Parallel inductors L1 L1 L2 L L2 L L = L1 + L2 1 1 1 = + L L1 L2 Inductor Experiment #1 • An ideal voltage source is connected to an inductor tON IL VIDEAL + - + VL - The constant voltage L causes the current to increase through the inductor. V, I VL VIDEAL IL tON t Inductor Experiment #2 • An ideal voltage source is disconnected to an inductor tOFF VIDEAL + - Vsrc IL + VL - If the constant voltage L source is removed and the inductor is shorted V, I the current through VIDEAL the inductor remains IL constant. VL tON tOFF t Inductor Experiment #3 • An ideal voltage source is connected to an inductor tON IL VIDEAL + VL - + - L The rate of current change is proportional to the voltage. V, I VL1 IL1 tON t Inductor Experiment #3 • An ideal voltage source is connected to an inductor tON IL VIDEAL + VL - + - L The rate of current change is proportional to the voltage. V, I VL2 VL1 IL2 IL1 tON t Inductor Experiment #4 • A voltage source is connected to an inductor through a resistor tON VIDEAL VIDEAL + - R The peak voltage IL + VL - across the inductor L is VIDEAL. The current through VL the inductor will IL VIDEAL/R tON reach VIDEAL/R. t Inductor Experiment #5 • A voltage source is connected through a variable resistor tON VIDEAL + - R IICL + V VCL - C L Inductor Experiment #5 • A voltage source is connected through a variable resistor V, I tON VIDEAL + - R R1 > R2 IIC L + L - C L VVC VIDEAL VIDEAL/R1 IL1 VL1 tON t Inductor Experiment #5 • A voltage source is connected through a variable resistor V, I VIDEAL/R2 VIDEAL The smaller the resistor, the longer it takes the current to become steady tON VIDEAL + - R IC IL + VL - C VC R1 > R2 LL IL2 VIDEAL/R1 IL1 VL2 VL1 tON t Inductor Experiment #5 • A voltage source is connected through a variable resistor V, I The smaller the resistor, the longer it takes the current to become steady tON VIDEAL + - R R1 < R3 IC IILLV + C LL VLC - VIDEAL VIDEAL/R1 IL1 VIDEAL/R3 IL3 VL1 VL3 tON t Inductor Experiment #6 • The rise time of the capacitor's voltage is monitored: tON VIDEAL VL + - R IC IILLV + VL C tC = L/R - C LL VIDEAL 0.37VIDEAL 0 tC t Inductor Experiment #6 • The rise time of the capacitor's voltage is monitored: t ON VIDEAL VL + - R IC IILLV + VL C tC = L/R - C LL VIDEAL 0.37VIDEAL 0.14VIDEAL 0.05VIDEAL 0 tC 2tC 3tC t Development of Mathematical Inductor Model: IL vs. VL • The self induced coil voltage when exposed to an alternating magnetic field is proportional to the change of coil flux vs. time: d dy v ind N dt dt Development of Mathematical Inductor Model: IL vs. VL • The self induced coil voltage when exposed to an alternating magnetic field is proportional to the change of coil flux vs. time: v ind d dy N dt dt • The voltage v applied across an inductor is always directly opposed to the self induced voltage vind: v = -vind = N·d/dt = dy/dt (=> dy = v·dt) v v ind d dy N dt dt or dy v dt (1) Development of Mathematical Inductor Model: IL vs. VL • The self induced coil voltage when exposed to an alternating magnetic field is proportional to the change of coil flux vs. time: v ind d dy N dt dt • The voltage v applied across an inductor is always directly opposed to the self induced voltage vind: v = -vind = N·d/dt = dy/dt (=> dy = v·dt) v v ind d dy N dt dt or dy v dt (1) • The inductance is defined as coil flux vs. coil current, L=y / IL, differentially expressed as: dy L di or dy L di (2) Development of Mathematical Inductor Model: IL vs. VL v v ind d dy N dt dt dy L di or or dy v dt dy L di • Setting (1) equal to (2), the voltage - current relation for an inductor equals can be found: di v L dt (1) (2) Inductors IL,max=VIN/R VL Voltage across VIN Inductor IL Current through Inductor time VIN VL R Inductor & Resistor Networks • In general, there are two basic options for inductor placement: L in Series with Signal Path VIN L R VOUT L from Signal Path to Ground VIN R L VOUT Inductor & ResistorNetworks L in Series with Signal Path L VOUT + VL VIN R I L from Signal Path to Ground R VIN I L + VL - VOUT • Initially a DC voltage is applied at the signal input IN. • A voltage drops across the inductor and the current through the inductor increases Inductor & ResistorNetworks L in Series with Signal Path L VOUT + 0V VIN R I L from Signal Path to Ground R VIN I L + 0V - VOUT • Initially a DC voltage is applied at the signal input IN. • A voltage occurs across the inductor and the current through the inductor increases • When the current through the inductor is at its maximum and remains constant, the voltage across the inductor equals zero Inductor & ResistorNetworks L in Series with Signal Path L VIN + 0V VIN R I L from Signal Path to Ground R VIN I L + 0V - 0V • Initially a DC voltage is applied at the signal input IN. • A voltage drops across the inductor and the current through the inductor increases • When the current through the inductor is at its maximum and remains constant, the voltage across the inductor equals zero • Depending on the inductor’s placement the steady state final voltages are VOUT = VIN or VOUT = 0V Inductance in Series with Signal Path t1 VX t2 + VL - VIN VOUT L I R VX I VOUT t1 t2 Inductance in Series with Signal Path t1 VX t2 + VL - VIN VOUT L I R VX VIN I VIN/R VOUT VIN t1 t2 Inductance in Series with Signal Path t1 VX t2 + VL - VIN VOUT L I R VXV IN I VIN/R VOUT VIN t1 t2 Inductance From Signal Path to Ground t1 t2 VIN VX R VX VOUT I L + VL - I VOUT t1 t2 Capacitance From Signal Path to Ground t1 t2 VIN VX VX R VOUT I L VIN + VL - I VIN/R VOUT VIN t1 t2 Capacitance From Signal Path to Ground t1 t2 VIN VX VX R VOUT I L VIN + VL - I VIN/R VOUT VIN -VIN t1 t2 RL Networks - AC Signals • What happens when an AC input signal is applied? L in Series with Signal Path VOUT VIN t L R ? L from Signal Path to Ground VOUT R VIN t L ? Inductors and AC signals • Act like frequency dependent resistor (inductive XL=2pfL reactance, XL) • Instead of reactance, impedance (Z) used for circuit elements. Inductors and AC signals • Act like frequency dependent resistor (inductive XL=2pfL reactance, XL) • Instead of reactance, impedance (Z) used for circuit elements. • Impedance: The apparent opposition in an electrical circuit to the flow of alternating current that is analogous to the actual electrical resistance to a direct current. Inductors and AC signals • Act like frequency dependent resistor (inductive reactance, XL) X =2pfL L • Instead of reactance, impedance (Z) used for circuit elements. • Impedance: The apparent opposition in an electrical circuit to the flow of alternating current that is analogous to the actual electrical resistance to a direct current. • The impedance of a circuit element represents its resistive and/or reactive components Inductors and AC signals • Act like frequency dependent resistor (inductive reactance, XL) • • • • XL=2pfL Instead of reactance, impedance (Z) used for circuit elements. Impedance: The apparent opposition in an electrical circuit to the flow of alternating current that is analogous to the actual electrical resistance to a direct current. The impedance of a circuit element represents its resistive and/or reactive components Besides the magnitude dependency between voltage and current the impedance Z gives also information about the phase shift between the two. Inductor’s Impedance Magnitude |ZL| vs. Frequency |ZL|=2.p.f.L 35 30 25 |ZL| (ohm) 20 15 10 5 0 0 1000 2000 3000 frequency (Hz) 4000 5000 Inductors and AC signals iL,Max = VL,Max / |ZL| VL,Max iL,Max t = -p/2 = -90o The current lags the voltage RL networks – AC signals L in Series with Signal Path VOUT L VIN t R L from Signal Path to Ground VOUT R VIN t L • The inductor acts as a frequency dependent resistor • It determines the current magnitude at a given voltage • It causes a 90 degree phase shift between the inductor current and voltage across the inductor RC networks – AC signals L in Series with Signal Path VOUT L VIN t R L from Signal Path to Ground VOUT R VIN t L • For low frequency signals: – The inductor is low impedance – Signals can pass the inductor |ZL|=2pfL • For high frequency signals: – The inductor is high impedance – Signals are blocked by the inductor L in Series with Signal Path Low Pass Configuration VOUT VIN VIN 6 4 L 2 R 0 -2 0 0.01 0.02 0 0.01 0.02 0.03 0.04 0.05 -4 Z=2.p.f.L -6 VOUT 6 4 VOUT/VINMAX Low f 0.96 Medium f 0.76 High f 0.38 2 0 -2 -4 -6 0.03 0.04 0.05 L from Signal Path to Ground High Pass Configuration VOUT VIN VIN 6 4 R 2 L 0 -2 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 -4 |ZL|=2.p.f.L -6 VOUT 6 4 VOUT/VINMAX Low f 0.32 Medium f 0.74 High f 0.92 2 0 -2 -4 -6 Inductor & Resistor Networks Summary L in Series with Signal Path VIN L R VOUT L from Signal Path to Ground VIN R L Connected to DC voltages: • The voltage across an inductor changes as current increases • The voltage across inductor is 0V when current is constant For AC signals: • Inductors act similar to frequency dependent resistors • Low impedance at low frequencies • High impedance at high frequencies. VOUT Capacitor vs. Inductor Unit Comparison Capacitor Terms and Symbols Units Inductor Terms and Symbols Units Electrical Field Strength E V/m Magnetic Field Strength H A/m Charge Q As Coil Flux (=N*F) y y Vs A Voltage=-N(dF/dt)=–NDF/Dt (negative rate of change of flux times the number of turns) V V L Vs/A Current: I=dQ/dt or DQ/Dt (rate of change of charge) I Capacitance: C As/V Inductance: Permittivity of Vacuum e0=8.854.10-12 e0 As/Vm Permeability of Vacuum: m0=4p10-7 m Energy Stored in a Capacitor: E = C V2 / 2 E Constant Current ( I ) Charging a Capacitor V=It/C C=Q/V VAs J L= y /I m m00 Vs/Am Energy Stored in an Inductor: E = L I2 / 2 E Constant Voltage ( V ) Charging an Inductor I=Vt/L VAs J RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay Lamp Experiment • Turn on an incandescent light bulb and measure the current ton I 1 14V 2 Lamp Experiment • Turn on an incandescent light bulb and measure the current t • Result: I on 1 ~ 5.6A 14V 2 ~ 600mA ton Developing a RC Load Model For an Incandescent Light Bulb 14V 5.6A 600mA ton Light Bulb Developing a RC Load Model For an Incandescent Light Bulb R1 14V 5.6A V 14V R1= = I 0.6A 600mA R1= 23.3 ton Developing a RC Load Model For an Incandescent Light Bulb 23.3 14V 5.6A I =I0exp-t/RC 600mA ton Developing a RC Load Model For an Incandescent Light Bulb 23.3 14V 5.6A R2 V 23.3 R2 = I 600mA R2 = 2.80 ton Developing a RC Load Model For an Incandescent Light Bulb C 23.3 14V 5.6A 2.8 600mA C = 3.6mF ton Simulation of Lamp RC Model 6.0 1 Input Current (A) 5.0 ton 23.3 3.6mF 4.0 14V 2.80 3.0 2 2.0 1.0 0.0 0 50 ton 100 150 Time (ms) 200 250 300 350 Simulation of Lamp RC Model 6.0 1 Input Current (A) 5.0 ton 4.0 23.3 3.6mF 14V 3.0 2.80 2.0 2 1.0 0.0 0 50 ton 100 150 Time (ms) 200 250 300 350 A RC Load Model for Incandescent Light Bulbs • The model for this lamps is represented by the network below • When a lamp initially turns on, the filament is cold and has a relatively low resistance BUT as the filament warms up, the resistance increases dramatically 1 3.6mF 2.80 f(T) 2 23.3 Lamp Experiment • When a lamp initially turns on, the filament is cold and has a relatively low resistance • As the filament warms up, the resistance increases dramatically ~ 5.6A ~ 600mA RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay Switching a Relay • To the right a “high side” switching application is shown • The switch itself is modeled as a simple mechanical switch • The relay can be modeled as a low ohmic resistor and inductor connected in series VBattery S VR Relay VL IL Switching On a Relay S open closed VR time VBattery time S VL decays over time VL time + VR IL time IL = (VR-VL) / R - IL + VL - Switching Off a Relay (1) S IL closed open time VBattery time time S + VR time - IL + VL - Switching Off a Relay (2) S IL VL VR closed open time VBattery time IL cannot become zero instantaneously! VL becomes negative to force the current to 0A time (VL = -L*di/dt) For VL < 0V, VR < 0V S + VR time - IL + VL - Switching Off a Relay (3) S closed open time VBattery Arcing IL VL VR time IL cannot go to zero instantaneously! VL goes far below ground to force the current to 0A time For VL < 0V, VR < 0V (R~0) S + VR time - IL + VL - Switching Off a Relay No Arcing (1) S closed open VBattery time IL time ID S + time VR VL time ID IL VR time + VL - - Switching Off a Relay No Arcing (2) S closed open VBattery time IL time ID S Diode turns on and provides a current path time + VR VL time ID IL VR time + VL - - Switching Off a Relay No Arcing (3) S closed open VBattery time IL time S + ID time VR If R~0, VL ~ –VD VL time ID IL If R~0, VR ~ -VD VR time + VL - - Switching Off a Relay No Arcing (4) S closed open VBattery time IL time ID S + time VR VL time ID IL VR time + VL - - Switching Off a Relay No Arcing (5) S IL closed open VBattery time diL/dt = VL / L time S + ID time VR If R~0, VL ~ –3VD VL time ID VR If R~0, VR ~ -3VD time IL + VL - - RLC Load Characteristics and Modeling • Introduction to Load Modeling • Introduction to Capacitors and RC Networks • Introduction to Inductors and RL Networks • Example Load Models: – Turning on an Incandescent Lamp – Switching a Relay Thank you! www.btipnow.com