5G NR PHY Initial Power Control

4.1 RAR MAC PDU Structure

The Random Access Response (RAR) MAC PDU carries the network's initial response to the UE PRACH transmission. The RAR body is 56 bits (7 bytes) structured as follows TS 38.321 §6.1.5:

The 27-bit UL Grant field is defined in TS 38.213 §8.2 Table 8.2-1:

TPC command mapping for MSG3 TS 38.213 Table 8.2-2:

4.2 RA-RNTI Formula

The RA-RNTI identifies the PRACH occasion in time–frequency space and is used by the UE to monitor the PDCCH for the RAR during ra-ResponseWindow. Valid range: 1..65518.

Worked example — \(s_{id}=0\), \(t_{id}=8\), \(f_{id}=0\), \(\text{ul\_carrier\_id}=0\):

4.3 Timing Advance

CRITICAL: NR defines two distinct TA mechanisms. Confusing them is one of the most common errors in RACH log analysis.

4.3.1 RAR Timing Advance — Absolute Initial UL Synchronisation

TS 38.321 §5.1 TS 38.213 §4.2

The 12-bit TA_RAR field (range 0..3846) carries an absolute timing advance value. The UE converts it to a sample offset using:

Where \(T_c = \dfrac{1}{480{,}000 \times 4096} \approx 0.509\,\text{ns}\) is the NR basic time unit TS 38.211 §4.1.

Worked example — TA_RAR = 64:

The total applied timing advance includes an offset term \(N_{TA,\text{offset}}\):

where \(N_{TA,\text{offset}} = 0\) for FDD and \(N_{TA,\text{offset}} = 13792\) for TDD TS 38.213 §4.2.

4.3.2 MAC CE Timing Advance Update — Relative Ongoing Correction

TS 38.321 §6.3.2 TS 38.213 §4.2

After initial synchronisation, the network sends relative corrections via a 7-bit TA MAC CE (range 0..63, centred at 31 = no change):

Example: TA_MAC = 33 → \(N_{TA,\text{new}} = N_{TA,\text{old}} + (33-31)\times16 = N_{TA,\text{old}} + 32\) — a small positive correction advancing the UL transmission by approximately \(32 \times 0.509\,\text{ns} \approx 16.3\,\text{ns}\).

4.3.3 TA Timing Limit and Maximum Cell Radius

The corresponding theoretical maximum one-way propagation delay is \(31.3\,\mu\text{s} / 2 \approx 15.65\,\mu\text{s}\), giving a cell radius of approximately \(3\times10^8 \times 15.65\times10^{-6} \approx 4{,}695\,\text{km}\). In practice, cell radii are limited to far smaller values by link budget, PRACH preamble cyclic prefix length, and guard period design.

5.1 PUSCH Power Control Formula

The general PUSCH transmit power is defined in TS 38.213 §7.1.1 Eq. 7.1.1-1:

5.2 MSG3-Specific Power

TS 38.213 §8.3

For MSG3, the closed-loop accumulation term \(f_{b,f,c}(i,l)\) is replaced by the open-loop offset \(\Delta_{\text{msg3}}\), which combines a configured delta and the RAR TPC command:

5.3 \(\Delta_{TF}\) — MCS-Dependent Offset

TS 38.213 §7.1.1

When deltaMCS is enabled in PUSCH-Config:

Where:

• \(BPRE = \dfrac{Q_m \times R \times N_L \times M_{RB} \times N_{RE}}{M_{RE}}\) — bits per resource element
• \(K_S = 1.25\) when transform precoding is disabled (OFDM); \(K_S = 1\) when enabled (DFT-s-OFDM)
• \(\beta_{\text{offset}}^{\text{PUSCH}}\) — HARQ-ACK/CSI beta offset (linear)

5.4 Fractional Path Loss Compensation Factor \(\alpha\)

\(\alpha\) controls what fraction of the estimated path loss is compensated by the UE's transmit power. Values \(\alpha < 1\) intentionally allow cell-edge UEs to transmit below the interference target, reducing inter-cell interference at the cost of reduced UL SINR for those UEs.

5.5 Interactive Chart: MSG3 PUSCH Power vs Allocated RBs

The chart below shows the MSG3 PUSCH transmit power as a function of the number of scheduled RBs for representative \(\alpha\) and \(\mu\) values. Assumptions: \(P_{0,\text{PUSCH}} = -76\,\text{dBm}\), \(PL = 108\,\text{dB}\), \(\Delta_{TF} = 0\), \(\Delta_{\text{msg3}} = +6\,\text{dB}\) (TPC code 6), \(P_{\text{CMAX}} = 23\,\text{dBm}\).

5.6 Power Headroom Report (PHR)

TS 38.321 §6.1.3.8

MSG3 worked example: With the parameters above (\(P_0=-76\), \(PL=108\), \(\Delta_{\text{msg3}}=+6\), \(\mu=1\), \(M_{RB}=10\), \(\alpha=0.8\)): \[ P_{\text{PUSCH}} = \min\!\{23,\; -76 + 10\log_{10}(20) + 0.8\times108 + 0 + 6\} = \min\!\{23,\; -76 + 13.01 + 86.4 + 6\} = \min\!\{23,\; 29.4\} = 23\,\text{dBm} \] \[ PH = 23 - 23 = 0\,\text{dB} \quad \text{(at }P_{\text{CMAX}}\text{ boundary, no headroom)} \]

6.1 PUCCH Power Control Formula

TS 38.213 §7.2.1 Eq. 7.2.1-1

6.2 \(\Delta_{F,\text{PUCCH}}\) Format-Specific Offsets

TS 38.213 Table 7.2.1-1

The baseline value for all formats is 0 dB (Table 7.2.1-1). The network can shift these values via RRC configuration to account for format-specific power differences. For example, Format 2 (shorter duration, wider bandwidth) may need a positive offset to match performance of Format 1.

6.3 \(\Delta_{TF}\) for PUCCH — UCI Payload Compensation

TS 38.213 §7.2.1

Unlike LTE's \(h(n_{\text{CQI}})\) function, NR's \(\Delta_{TF}\) is defined as follows:

• For Formats 0 and 1 (binary signalling — SR or 1/2-bit HARQ-ACK): \(\Delta_{TF} = 0\)
• For Formats 2, 3, and 4 when \(O_{UCI} > 2\) bits:

Where \(O_{UCI}\) is the number of UCI information bits (CQI + HARQ-ACK payload) and \(\beta_{\text{offset}}^{\text{PUCCH}}\) is the PUCCH beta offset configured via PUCCH-Config.uci-OnPUSCH.betaOffsets.

6.4 Worked Example: SR on PUCCH Format 0

Parameters: \(P_{0,\text{PUCCH}} = -90\,\text{dBm}\), \(M_{RB}^{\text{PUCCH}} = 1\), \(\mu = 1\) (30 kHz), \(PL = 108\,\text{dB}\), \(\Delta_F = 0\), \(\Delta_{TF} = 0\), \(g(i) = 0\):

The UE transmits the SR at 21 dBm, which is 2 dB below \(P_{\text{CMAX}}\), leaving 2 dB headroom. If \(g(i)\) accumulates to +2 dB from TPC commands, the UE reaches \(P_{\text{CMAX}}\).

6.5 PUCCH Format Summary

6.6 SR Transmission, Counting, and Failure Recovery

TS 38.321 §5.4.4

The SR state machine at the MAC layer operates as follows:

The RACH triggered by SR failure uses the same PRACH/MSG3 procedure described in §4–5. Typical sr-TransMax deployment values: 4 for latency-sensitive configurations (e.g., low-latency slices), 8–16 for standard eMBB, 32–64 for coverage-limited scenarios.

Once the UE transitions to RRC_CONNECTED via RRCSetup or RRCReconfiguration, the uplink power control regime shifts from the common RACH parameters to the dedicated configuration provided in PUSCH-PowerControl. The key mechanism is the p0-AlphaSets structure, which allows the network to define up to 30 independent (P₀, α) pairs — each indexed by a set identifier j. TS 38.213 §7.1.1

7.1 Power Control Equation

Variable definitions:

TPC Command Mapping TS 38.213 Table 7.1.1-2

In accumulated mode (default), f grows by the δ value each subframe. In absolute mode (tpc-Accumulation = disabled), f is set to the absolute δ value directly.

7.2 p0-AlphaSets RRC Configuration

7.3 MSG3 (Common) vs Dedicated PUSCH — Parameter Comparison

7.4 Power Evolution Through the RACH Procedure

7.6 RRC IE — PUSCH-PowerControl

PUSCH-PowerControl ::= SEQUENCE {
  tpc-Accumulation            ENUMERATED {disabled} OPTIONAL, -- disable closed-loop
  msg3-Alpha                  Alpha OPTIONAL,              -- common PUSCH α for MSG3
  p0-NominalWithoutGrant      P0-PUSCH-AlphaSet OPTIONAL,  -- SRS/CG baseline
  p0-AlphaSets                SEQUENCE (SIZE(1..30))
                              OF P0-PUSCH-AlphaSet OPTIONAL, -- dedicated sets
  pathlossReferenceRSToAddModList ...                            -- PL RS config
}

P0-PUSCH-AlphaSet ::= SEQUENCE {
  p0-PUSCH-AlphaSetId         P0-PUSCH-AlphaSetId,            -- set index j (0..30)
  p0                          INTEGER (-16..15) OPTIONAL,    -- dB offset from p0-Nominal
  alpha                        Alpha OPTIONAL                  -- absent → 1.0 (full comp)
}

-- Alpha ::= ENUMERATED {0dot4, 0dot5, 0dot6, 0dot7, 0dot8, 0dot9, 1}
-- closedLoopIndex: i0 = process 0, i1 = process 1 (independent accumulators)

This section walks through a complete 12-phase RACH-to-RRC_CONNECTED procedure with all numerical calculations. Every equation maps directly to the 3GPP specification formulas covered in earlier sections. All values are traceable to RRC configuration parameters or UE measurements. TS 38.213 §6.3.3.2 TS 38.213 §7.1.1 TS 38.213 §7.2.1

9.1 System Setup Parameters

9.2 Step-by-Step Numerical Walkthrough

1. Path Loss Estimation TS 38.213 §7.1.1

   The UE estimates downlink path loss using the broadcast SSB power and its L3-filtered RSRP measurement:

   \[ PL = P_{\text{SSB}} - \text{RSRP}_{\text{L3}} = 23 - (-85) = \mathbf{108\text{ dB}} \]

   This single value governs all subsequent open-loop power calculations throughout the procedure.

2. MSG1 Attempt 1 — PRACH Preamble Transmission TS 38.213 §6.3.3.2 \[ P_{\text{PRACH}}(0) = P_{\text{preamble\_target}} + PL + \Delta_{\text{PREAMBLE}} = -104 + 108 + 0 = \mathbf{4\text{ dBm}} \]

   PRACH counter PREAMBLE_TRANSMISSION_COUNTER = 1. No RAR received within ra-ResponseWindow (power too low for reliable detection at logsA). Preamble counter increments.

3. MSG1 Attempt 2 — First Power Ramp TS 38.213 §6.3.3.2 \[ P_{\text{PRACH}}(1) = P_{\text{PRACH}}(0) + \Delta_{\text{RAMP}} \cdot 1 = 4 + 4 = \mathbf{8\text{ dBm}} \]

   RAR received successfully. logsA (base station) detects the preamble and responds with: Random Access Response (RAR) MAC CE containing TA command TA_RAR = 64, temporary C-RNTI, and UL grant for MSG3 scheduling.

4. Timing Advance Computation from RAR TS 38.213 §4.2 \[ N_{TA} = \text{TA}_{\text{RAR}} \times 16 = 64 \times 16 = 1024 \; T_c \text{ samples} \] \[ T_{TA} = N_{TA} \times T_c = 1024 \times 0.509\text{ ns} \approx 521\text{ ns} \] \[ d \approx \frac{c \cdot T_{TA}}{2} = \frac{3\times10^8 \times 521\times10^{-9}}{2} \approx \mathbf{78\text{ m}} \]

   Where \(T_c = 1/(480\times10^3 \times 4096) \approx 0.509\text{ ns}\) is the basic time unit. The UE advances its uplink transmission by \(N_{TA}\) samples to align with the base station receive window.

5. RA-RNTI Computation TS 38.213 §8.2 \[ \text{RA-RNTI} = 1 + s_{\text{id}} + 14 \cdot t_{\text{id}} + 14 \cdot 80 \cdot f_{\text{id}} + 14 \cdot 80 \cdot 8 \cdot \text{ul\_carrier\_id} \]

   For PRACH occasion: \(s_{\text{id}}=0,\; t_{\text{id}}=0,\; f_{\text{id}}=0,\; \text{ul\_carrier\_id}=0\):

   \[ \text{RA-RNTI} = 1 + 0 + 0 + 0 + 0 = \mathbf{1} \]

   The UE uses RA-RNTI = 1 to monitor PDCCH for the RAR DCI (format 1_0 with CRC scrambled by RA-RNTI).

6. MSG3 PUSCH Power Calculation TS 38.213 §7.1.1

   Bandwidth term with μ=1, M_RB=10:

   \[ 10\log_{10}(2^\mu \cdot M_{RB}) = 10\log_{10}(2^1 \times 10) = 10\log_{10}(20) = 13.0\text{ dB} \]

   Full power control equation (j=0 common, α=1.0, f=0, Δ_TF=0):

   \[ P_{\text{MSG3}} = \min\bigl\{23,\; 13.0 + (-76) + 1.0 \times 108 + 0 + 0\bigr\} = \min\{23,\; 45.0\} = \mathbf{23\text{ dBm}} \]
   P_CMAX Limited

   MSG3 hits the UE maximum transmit power ceiling. This is expected at PL = 108 dB with α = 1.0.

7. MSG4 Reception and RRCSetup

   logsA processes the MSG3 RRCSetupRequest and responds with MSG4 containing RRCSetup (via PDSCH addressed to Temporary C-RNTI). The UE decodes RRCSetup, configures SRB1, and transitions to RRC_CONNECTED. The dedicated power control parameters in PUSCH-PowerControl become active. TPC accumulator \(f_{b,f,c}(i,l)\) is initialized to 0.

8. RRCSetupComplete (MSG5) — Uplink on SRB1

   UE sends RRCSetupComplete on SRB1 via PUSCH. At this point, dedicated power control applies (p0-AlphaSet j=1) if configured; otherwise common p0-NominalWithGrant is used with no closed-loop component yet (f=0).

9. PUCCH SR Power Calculation (after RRCSetup) TS 38.213 §7.2.1

   PUCCH SR uses format 0/1 with 1 PRB. Bandwidth term with μ=1, M_RB=1:

   \[ 10\log_{10}(2^1 \times 1) = 10\log_{10}(2) \approx 3.0\text{ dB} \] \[ P_{\text{PUCCH}} = \min\bigl\{23,\; 3 + (-90) + 108 + 0 + 0\bigr\} = \min\{23,\; 21\} = \mathbf{21\text{ dBm}} \]
   3 dB headroom

   PUCCH operates 2 dB below P_CMAX — healthy headroom exists for TPC-driven power increase.

10. Dedicated PUSCH Power (j=1, α=0.8) TS 38.213 §7.1.1

    Bandwidth term same as MSG3 (10 PRBs, μ=1): 13.0 dB. Dedicated set j=1: p0=−76 dBm, α=0.8, f=0:

    \[ P_{\text{PUSCH}} = \min\bigl\{23,\; 13.0 + (-76) + 0.8 \times 108 + 0 + 0\bigr\} = \min\{23,\; 13 - 76 + 86.4\} = \min\{23,\; 23.4\} = \mathbf{23\text{ dBm}} \]
    Near P_CMAX
11. Dedicated PUSCH with TPC +3 dB

    logsA issues a DCI 0_1 TPC command (accumulated correction +3 dB). New accumulator: f=+3 dB.

    \[ P_{\text{PUSCH}} = \min\bigl\{23,\; 23.4 + 3\bigr\} = \min\{23,\; 26.4\} = \mathbf{23\text{ dBm}} \]

    UE is already at P_CMAX; TPC command has no further effect. The UE reports this saturation via Power Headroom Report (PHR = 0 dB). logsA should reduce MCS or allocated PRBs to improve link margin.

12. PHR (Power Headroom Report) TS 38.321 §6.1.3.8 \[ PH = P_{\text{CMAX}} - P_{\text{PUSCH,computed}} = 23 - 26.4 = -3.4\text{ dB} \]

    Encoded in the 6-bit PHR field (range -23..+40 dB, TS 38.321 §6.1.3.8) as level 19 (≈ −4 dB, nearest 1-dB step; encoding: 0 = -23 dB, 63 = +40 dB, so level 19 maps to -23 + 19 = -4 dB). A negative PHR indicates the UE is power-limited — the logsA scheduler should reduce M_RB or MCS to bring the UE back within its power budget.

9.3 Procedure Summary

This section provides a systematic troubleshooting guide for common power control failure modes observed during integration and field testing, followed by an RRC parameter optimization matrix for common deployment scenarios. References are to logsA (base station / network) and logsB (UE / terminal).

10.1 Common Failure Modes

10.2 RRC Parameter Optimization Matrix by Deployment Scenario

10.3 PRACH Cumulative Success Rate vs Ramping Step

10.4 Power Headroom Report (PHR) Analysis

TS 38.321 §6.1.3.8

The Power Headroom Report (PHR) MAC CE communicates the difference between the UE's maximum transmit power and the power computed by the uplink power control equation. It provides the logsA MAC scheduler with information to make grant size and MCS decisions.

This appendix provides a complete reference for all acronyms and 3GPP-defined parameters used in this notebook. Parameter ranges and references are taken directly from the 5G NR specifications.

A.1 Acronym Reference

A.2 Key Parameter Glossary

All parameters below are defined in the 3GPP 5G NR specifications and are configurable via RRC signaling unless otherwise noted. Ranges are as specified in the ASN.1 definitions.